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2,040 |
People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a technical tool. Is there a conceptual reason we want these functional equations around? Have I just not seen enough of the theory?
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There are many reasons that functional equations are important. Some background: To most varieties/schemes occurring in arithmetic geometry, you can associate a zeta function/L-function. There are two main ways of constructing these, either from l-adic cohomology, or from counting solutions in various finite fields. Usually the word L-function is used for functions associated to varieties over number fields, constructed from l-adic cohomology, and the word zeta function is used for functions associated to schemes over Z or some other ring of integers, or over a finite field. However, there is some confusion about the terminology, and there is also some overlap between the two, since there is a close relation between the L-function of a variety and the zeta function of an integral model for the variety. Over finite fields, the functional equation is part of the famous Weil conjectures, proved by Deligne. One reason that the functional equation is cool is that it reflects Poincare duality in the etale cohomology of the variety, so it is in some sense a deep geometric statement. For background on the Weil conjectures, see for example Freitag and Kiehl: Etale cohomology and the Weil conjectures, and the survey of Mazur: Eigenvalues of Frobenius acting on algebraic varieties over finite fields, in some conference proceedings. For the definition of L-functions and zeta functions and lots of other background, see Manin and Panchishkin: Introduction to modern number theory, chapter 6. There are at least two deep reasons for being interested in the functional equations for these functions. Firstly, the existence of a functional equation seems to always be directly related to the L-function coming from an automorphic representation, and the idea that "every L-function from algebraic geometry (aka motivic L-function) also comes from an automorphic representation" is in some sense the number-theoretic incarnation of the global Langlands program. See for example Bump et al: An introduction to the Langlands program. The most famous cases where this has been proved is (1) Tate's thesis, which treats Hecke L-functions, and where the corresponding automorphic representation is one-dimensional, i.e. a character on the ideles, and (2) the work by Wiles and others related to Fermat's last theorem, where they show that the L-function associated to an elliptic curve over Q also comes from a modular form, and hence satisfies the expected functional equation. See the book of Diamond and Shurman on modular forms. The other deep reason for thinking about the functional equation is that some optimistic people dream of an "arithmetic cohomology theory", which would allow us to mimick the proof of the Weil conjectures, but for zeta functions over Z or L-functions over Q. Then the functional equation should be related to Poincare duality for this cohomology. All this is related to the Riemann hypothesis, noncommutative geometry, and the field with one element. See for example Deninger's Motivic L-functions and regularized determinants ( pdf ), his more recent survey Arithmetic Geometry and Analysis on Foliated Spaces , some slides of Paugam link broken on the functional equation, and also the later chapters of Manin-Panchishkin. Some of the key names if you want to find more references: Deninger, Connes, Consani, Marcolli; most of them have lots of stuff on their webpages and on the arXiv.
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|
2,077 |
Let $SX$ be the suspension of CW complex. What are some results available to determine the homotopy groups of $SX$?
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This, in general, an incredibly difficult problem. Even we just want to compute the rational homotopy groups of the suspension of $X$ and $X$ is simply connected, where we can do everything using rational homotopy theory to reduce things to commutative DGAs, this starts to involve things like free Lie algebras and the like. Locally at the prime 2, there is actually a famous long exact sequence when $X$ is a sphere called the EHP long exact sequence. It relates the suspension homomorphism, the Hopf map, and a "whitehead product" map. This gives rise to the EHP spectral sequence that, funnily enough, starts with the 2-local homotopy groups of odd -dimensional spheres and computes the 2-local homotopy groups of spheres. Miller and Ravenel have a paper titled "Mark Mahowald's work on the homotopy groups of spheres" that covers some of this material in detail. Another approach is to say: The "stable" homotopy groups of $X$ are a first-order approximation using the Freudenthal suspension theorem that Andrea mentioned. There is then a "quadratic" correction term that you can try to use to get an approximation of the homotopy groups that is correct out to roughly three times the connectivity, and so on. These lead into the subject of Goodwillie calculus. For a classifying space $K(G,1)$, neither of these approaches work very well, because the higher homotopy groups are going to depend pretty intricately on your group itself. For instance, $\pi_3$ of the suspension of the classifying space of a free group is the set of symmetric elements in $G_{\text{ab}}\otimes G_{\text{ab}}$ where $G_{\text{ab}}$ is the abelianization of $G$, and for a general group it lives in an exact sequence between something involving such symmetric elements and the second group homology of $G$. I don't know a closed form for it but maybe someone else knows better. EDIT : Let me at least be precise, there's an exact sequence
$$\pi _4 (\Sigma BG)\rightarrow H_3 G \rightarrow (G_{\text{ab}}\otimes G_{\text{ab}})^{\mathbb Z/2} \rightarrow \pi_3 (\Sigma BG)\rightarrow H_2 G\rightarrow 0$$
Note that for $R$ a ring, an element of $(G_{\text{ab}}\otimes G_{\text{ab}})^{\mathbb Z/2}$ gives rise to an $R$-valued symmetric bilinear pairing on $\mathsf{Hom}(G_{\text{ab}},R)$. EDIT FOR THE FINAL TIME : sorry for the multiple revisions, switching back and forth between homology and cohomology gave me errors. the exact sequence above should be correct now.
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2,124 |
Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also browsed through FGA explained (Fantechi et al.). Although I find the level good, it is somewhat incomplete and I would want to see more basic examples. Unfortunately I don't read french.
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Anton live-texed notes to Martin's Olsson's course on stacks a few years ago. They are online here . Olsson's notes have been published as: Algebraic Spaces and Stacks , M. Olsson, AMS Colloquium Publications, volume 62, 2016. ISBN 978-1-4704-2798-6 My general advice is to learn algebraic spaces first. The point is that the new things you need to learn for stacks fall into two categories (which are mostly disjoint): 1) making local, functorial, and non-topological definitions (e.g. what it means for a morphism to be smooth or flat or locally finitely presented) and 2) 2-categorical stuff (e.g. what is a 2-fiber product). You don't need to do things 2-categorically for algebraic spaces, so it makes sense to learn them first. I believe it really clarifies things to learn these separately. Also, the formal notion of a stack is a generalization of functor. If you are not used to thinking of schemes functorially (e.g. as a functor from rings^op to sets) it will be hard to wrap your head around the notion of a stack. the The intermediate step of learning to think about geometry in terms of functors of points is crucial. Knutson's book Algebraic Spaces is very good for the EGA-style content, and its introduction will point you to many nice applications of algebraic spaces that are worth learning and will motivate you to learn the EGA-style stuff. Laumon and Moret-Bailly's Champs Algébriques is nice and contains more theorems that just the EGA style stuff. Its hard to point you any other particular reference without knowing what your goal in learning stacks is.
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2,144 |
Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues. Do you have such an example? Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately. And as usual let's do one per post and vote freely, vote a lot.
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Perhaps not really a paper, but i think a "must-read" is A Mathematician's Lament by Paul Lockhart.
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2,146 |
There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometric conjectures just to formulate the definition of motives . There are things that I know about motives on some level, e.g. I know what t he Grothendieck ring of varieties is or, roughy, what are the ingredients of the definition of motives. But, how would you explain the Grothendieck's yoga of motives ? What is it referring to?
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So this is a crazy question, but I will try to give at least a partial answer. This question about the Beilinson regulator is also relevant, and this is also an attempt to reply to the comments of Ilya there. I apologize for simplifying and glossing over some details, see the references for the full story. First of all, some references: A leisurely but still far from content-free exposition by Kahn on the yoga of motives is available here (in French). For Grothendieck's idea of pure motives, see Scholl: Classical motives, available on his webpage in zipped dvi format. For mixed motives, see this survey article of Levine . There is also lots of stuff in the Motives volumes, edited by Jannsen, Kleiman and Serre, here is the Google Books page . Finally, I would strongly recommend the book by André: Introduction aux motifs - this is has lots of background and "yoga", as well as precise statements about what is known and what one conjectures. Pure motives The standard way of explaining what motives are is to say that they form a "universal cohomology theory". To make this a bit more precise, let's start with pure motives. We fix a base field, and consider the category of smooth projective varieties, and various cohomology functors on this category. The precise notion of cohomology functor in this context is given by the axioms for a Weil cohomology theory, see this blog post of mine for more details. There are (at least) three key points to mention here: one is that a Weil cohomologies are "geometric" theories, as opposed to "absolute". For example, when considering etale cohomology, we are considering the functor given by base changing the variety to the absolute closure of the ground field, and then taking sheaf cohomology with respect to the constant sheaf Z/l for some prime l, in the etale topology. The "absolute" theory here would be the same, but without base changing in the beginning. In the classical literature, and in number theory, the geometric version is the most important, partly because it carries an action of the Galois group of the base field, and hence gives rise to Galois representations. On the other hand, the absolute version is important for example in the work of Rost and Voevodsky on the Bloch-Kato conjecture, and in comparison theorems with motivic cohomology. Similarly, it seems like cohomology theories in general come in geometric/absolute pairs. The second key point to mention is that the Weil cohomology groups come with "extra structure", such as Galois action or Hodge structure. For example, l-adic cohomology takes values in the category of l-adic vector spaces with Galois action, and Betti cohomology takes values in a suitable category of Hodge structures. A nice reference for some of this is Deligne: Hodge I, in the ICM 1970 volume. The third key point is that Weil cohomology theories are always "ordinary" in some sense, i.e. in some framework of oriented cohomology theories they would correspond to the additive formal group law (see Lurie: Survey on elliptic cohomology ). If we allowed more general (oriented) cohomology theories, the universal cohomology would not be pure motives, but algebraic cobordism. Now all these cohomology theories are functors on the category of smooth projective varieties, and the idea is that they should all factor through the category of pure motives, and that the category of pure motives should be universal with this property. We know how to construct the category of pure motives, but there is a choice involved, namely choosing an equivalence relation on algebraic cycles, see the article by Scholl above for more details. For many purposes, the most natural choice is rational equivalence, and the resulting notion of pure motives is usually called Chow motives. For a precise statement about the universal property of Chow motives, see André, page 36: roughly (omitting some details), any sensible monoidal contravariant functor on the category of smooth projective varieties, with values in a rigid tensor category, factors uniquely over the category of Chow motives. Now to the point of realizations raised by Ilya in the question about regulators. Given a category of pure motives with a universal property as above, there must be functors from the category of motives to the category of (pure) Hodge structures, to the category of Q_l vector spaces with Galois action, etc, simply because of the universal property. These functors are called realization functors. Mixed motives It seems like all the cohomology functors one typically considers can be defined not only for smooth projective varieties, but also for more general varieties. The right notion of cohomology here seems to be axiomatized by some version of the Bloch-Ogus axioms. One could again hope for a category which has a similar universal property as above, but now with respect to all varieties. This category would be the category of mixed motives, and in the standard conjectural framework, one hopes that it should be an abelian category. It is not clear whether this category exists or not, but see Levine's survey above for a discussion of some attempts to construct it, by Nori and others. If we had such a category, a suitable universal property would imply that there are realization functors again, now to "mixed" categories, for example mixed Hodge structures. The realization functors would induce maps on Ext groups, and a suitable such map would be the Beilinson regulator, from some Ext groups in the category of mixed motives (i.e. motivic cohomology groups) to the some Ext groups which can be identified with Deligne-Beilinson cohomology. We do not have the abelian category of mixed motives, but we have an excellent candidate for its derived category: this is Voevodsky's triangulated categories of motives. They are also presented very well in the survey of Levine. A really nice recent development is the work of Déglise and Cisinski, in which they construct these triangulated categories over very general base schemes (I think Voevodsky's original work was mainly focused on fields, at least he only proved nice properties over fields). To end by reconnecting to the Beilinson conjectures, there is extremely recent work of Jakob Scholbach (submitted PhD thesis, maybe on the arXiv soon) which seems to indicate that the Beilinson conjectures should really be formulated in the setting of the Déglise-Cisinski category of motives over Z, rather than the classical setting of motives over Q. The yoga of motives involves far more than what I have mentioned so far, for example things related to periods and special values of L-functions, the standard conjectures, and the idea of motivic (and maybe even "cosmic") Galois groups, but all this could maybe be the topic for another question, some other day :-)
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2,147 |
What are really helpful math resources out there on the web? Please don't only post a link but a short description of what it does and why it is helpful. Please only one resource per answer and let the votes decide which are the best!
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I occasionally find mathoverflow.net rather helpful. In particular, there's a good list of answers to your specific question here .
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2,173 |
I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. Now, I know that with the Ore conditions, we can localize almost commutative rings, and so we get a legitimate sheaf D to do geometry with. But how far does the analogy go? What theorems are true for commutative rings but can't be modified reasonably to be true for (nice, say, left and right noetherian or somesuch) almost commutative rings?
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Don't get too excited about the theory of algebraic geometry for almost commutative algebras. A ring can be almost commutative and still have some very weird behavior. The Weyl algebras (the differential operators on affine n-space) are a great example, since they are almost commutative, and yet: They are simple rings. Therefore, either they have no 'closed subschemes', or the notion of closed subscheme must correspond to something different than a quotient. The have global dimension n, even though their associated graded algebra has global dimension 2n. So, global dimension can jump up, even along flat deformations. There exist non-free projective modules of the nth Weyl algebra (in fact, stably-free modules!). Thus, intuitively, Spec(D) should have non-trivial line bundles, even though it is 'almost' affine 2n-space. Just having a ring of quotients isn't actually that strong a condition on a ring. For instance, Goldie's theorem says that any right Noetherian domain has a ring of quotients, and that is a pretty broad class of rings. Also, what sheaf are you thinking of D as giving you? You have all these Ore localizations, and so you can try to build something like a scheme out of this. However, you start to run into some problems, because closed subspaces will no longer correspond to quotient rings. In commutative algebraic geometry, we take advantage of the miracle that the kernel of a quotient map is the intersection of a finite number of primary ideals, each of which correspond to a prime ideal and hence a localization. In noncommutative rings, there is no such connection between two-sided ideals and Ore sets. Here's something that might work better (or maybe this is what you are talking about in the first place). If you have a positively filtered algebra A whose associated graded algebra is commutative, then A_0 is commutative, and so you can try to think of A as a sheaf of algebras on Spec(A_0). The almost commutativity requirement here assures us that any multiplicative set in A_0 is Ore in A, and so we do get a genuine sheaf of algebras on Spec(A_0). For D_X, this gives the sheaf of differential operators on X. Other algebras that work very similarly are the enveloping algebras of Lie algebroids, and also rings of twisted differential operators.
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2,185 |
I've read about model categories from an Appendix to one of Lurie's papers. What are the examples of model categories ? What should be my intuition about them? E.g. I understand the typical examples come from taking homotopy of something — but are all model categories homotopy categories?
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Here's a bit of the historical reason why model categories came up. If you have a functor on an abelian category which doesn't quite behave as you would like it to, e.g. is not left exact, there is a simple recipe to derive it: 1) Embed your category into the bigger category of complexes 2) Find a qausi-isomorphic replacement for the object you want to plug into the functor. This was so easy because the starting category was abelian. One important example of a category that is not abelian is the category of commutative rings, and there is one important functor on this category, namely the functor of Kaehler differentials, which is not quite exact. (Remember the conormal sequence from Hartshorne, it's only got a zero on one side). So what to do? Since commutative rings are not abelian, there's no way that you can work with complexes. Quillen's great move was to use simplicial objects instead of complexes. This takes care of step 1. You've just succesfully embedded the category of commutative rings into the category of simplicial commutative rings. Now you have to care of step 2. This is possible if the larger category you just moved into is a model category. The replacement is either called cofibrant or fibrant replacement, depending on your functor. The existence of such a replacement follows immediately from the axioms of a model category, since you can factor the map from the initial object to any object into a cofibration and a trivial fibration. Vice versa with final object for fibrant replacement. So we can also do step 2 now! Coming back to the example of the Kaehler differentials, we can derive them now: 1) View your commutative ring as a trivial simplicial ring. 2) Replace it cofibrantly 3)Apply \Omega levelwise. What you get is a simplicial module. If you apply the Dold-Kan correspondence you get complex, and this is the cotangent complex. Of course there's much more to Model categories, but deriving functors from non-abelian categories is already quite something.
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2,218 |
What can you say about the complexity class $\text{P}^{\text{NP}}$, i.e. decision problems solvable by a polytime TM with an oracle for SAT? This class is also known as $\Delta_2^p$. Obviously $\text{P}^{\text{NP}}$ is in $\text{PH}$ somewhere between $\text{NP} \cup \text{coNP}$, and $\Sigma_2^p \cap \Pi_2^p$. What else is known about that complexity class?
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The standard complete problem for the "function version" of P^NP is to find the lexicographically last satisfying assignment of a given boolean formula. To be more finicky, a complete language for P^NP is { n-variable boolean formulas phi : phi is satisfiable and phi's lexicographically last satisfying assignment has x_n = 1 }. This is Krentel's Theorem. Under a standard complexity assumption one can derandomize the more powerful class BPP^NP down to P^NP. With the power of BPP^NP, one can compute the number of satisfying assignments to a circuit to within a 1 +/- 1/poly(n) factor [Sipser / Stockmeyer / Valiant-Vazirani] and exactly learn unknown polynomial-size circuits [Bshouty-Cleve-Gavalda-Kannan-Tamon].
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|
2,300 |
I've heard of this many times, but I don't know anything about it. What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-dimensional, namely $\mathop{\text{Spec}}\mathbb Z$. So, what is the field with one element? And, what are typical geometric objects that descend to $\mathbb F_1$?
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As other have mentioned, F_1 does not exist of a field. Tits conjectured the existence of a "field of characteristic one" F_1 for which one would have the equality G(F_1) = W, where G is any Chevalley group scheme and W its corresponding Weyl group. Later on Manin suggested that the "absolute point" proposed in Deninger's program to prove the Riemann Hypothesis might be thought of as "Spec F_1", thus stating the problem of developing an algebraic geometry (and eventually a theory of motives) over it. There are several non-equivalent approaches to F_1 geometry, but a common punchline might be "doing F_1 geometry is finding out the least possible amount of information about an object that still allows to speak about its geometrical properties". A "folkloric" introduction can be found in the paper by Cohn Projective geometry over F_1 and the Gaussian binomial coefficients . It seems that all approaches so far contain a common intersection, consisting on toric varieties which are equivalent to schemes modeled after monoids. In the case of a toric variety, the "descent data" that gives you the F_1 geometry is the fan structure, that can be reinterpreted as a diagram of monoids (cf. some works by Kato). What else are F_1 varieties beyond toric is something that depends a lot on your approach, ranging from Kato-Deitmar (for which toric is all there is) to Durov and Haran's categorical constructions which contains very large families of examples. A somehow in-the-middle viewpoint is Soule's (and its refinement by Connes-Consani) which in the finite type case is not restricted only to toric varieties but to something slightly more general (varieties that can be chopped in pieces that are split tori). No approach is yet conclusive, so the definitions and families of examples are likely to change as the theory develops. Last month Oliver Lorscheid and myself presented an state-of-the-art overview of most of the different approaches to F_1 geometry: Mapping F_1-land: An overview of geometries over the field with one element (sorry for the self promotion).
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2,314 |
Hey. I have a few off the wall questions about topos theory and algebraic geometry. Do the following few sentences make sense? Every scheme X is pinned down by its Hom functor Hom(-,X) by the yoneda lemma, but since schemes are locally affine varieties, it is actually just enough to look at the case where "-" is an affine scheme. So you could define schemes as particular functors from CommRing^op to Sets. In this setting schemes are thought of as sheaves on the "big zariski site". If that doesn't make sense my next questions probably do not either. 2 The category of sheaves on the big zariski site forms a topos T, the category of schemes being a subcategory. It is convenient to reason about toposes in their own "internal logic". Has there been much thought done about the internal logic of T, or would the logic of T require too much commutative algebra to feel like logic? Along these lines, have there been attempts to write down an elementary list of axioms which capture the essense of this topos? I am thinking of how Anders Kock has some really nice ways to think of differential geometry with his SDG. 3 What is it about the category of commutative rings which makes it possible to put such a nice site structure on it, but not other algebraic categories? Gluing rings together lead to huge advancements in algebraic geometry. What about gluing groups? Is there a nice Grothendieck topology you could put on Groups^op, and then you could start studying sheaves on this site? If not, why not - what about rings makes them so special? 4 Why do people work with the category of schemes instead of the topos of sheaves on CommRing^op - toposes have every nice categorical property you could possibly ask for. About me: I am a 1st year grad student who is taking a first course on schemes, and I just have a lot of crazy ideas floating around. I don't feel comfortable engaging in such wild speculation with my professors. Could you offer any insight into these ideas?
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About 1: Yes! About 2: (Internal logic of Zariski topos) I don't think it has been done systematically. A glimpse of it is in Anders Kock, Universal projective geometry via topos theory, if I remember well, and certainly in some other places. But one point is that it is not at all easy to find formulas in the internal language which express what you have in mind. See my answer at "synthetic" reasoning applied to algebraic geometry About 3:You can indeed glue all sorts of things: Things fitting into the axiomatic framework of "geometric contexts":
Look at the "master course on Algebraic stacks" here: http://perso.math.univ-toulouse.fr/btoen/videos-lecture-notes-etc/ This one is great reading to understand the functorial point of view on schemes and manifolds! Commutative Monoid objects in good monoidal (model) categories: http://arxiv.org/abs/math/0509684 Commutative monads (here you can glue monoids, semirings and other algebraic structures mixing them all): http://arxiv.org/abs/0704.2030 In Shai Haran's "Non-Additive Geometry" you can even glue the monoids and semirings etc. with relations (although I wouldn't know why) You can also glue things "up to homotopy instead" of strictly - this is roughly what Lurie's infinity-topoi are about, and also the model catgeory part of the 2nd point, or any oter approaches to derived algebraic geometry (Edit in 2017) The PhD thesis by Zhen Lin Low is relevant. "The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces." One of several good points of view on what a Grothendieck topology does, is to say it determines which colimits existing in your site should be preserved under the Yoneda embedding, i.e. what glueing takes already place among the affine objects. So, if you insist on glueing groups it could be a good idea to look e.g. for a topology which takes amalgamated products (for me this means glueing groups, you may want only selected such products, e.g. along injective maps) to pushouts of sheaves...
Then feel free to develop a theory on this and send me a copy! About 4: (Why don't people work with sheaves instead of schemes) They do. One situation where they do is when taking the quotient of a scheme by a group action. The coequalizer in the category of schemes is often too degenerate. One answer is taking the coequalizer in the category of sheaves, the "sheaf quotient" (but sometimes better answers are GIT quotients and stack quotients).
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2,358 |
What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
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This isn't really a heuristic, but I hate "functions are formulas". For most students it takes a really long time to think of a function as anything other than an algebraic expression, even though natural algorithmic examples are everywhere. For example, some students won't think of \begin{gather}
f(n) = \{\text{1 if $n \bmod 2 = 0$ $\lor$ $-1$ otherwise}\}
\end{gather} as a function until you write it as $f(n) = (-1)^n$
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2,369 |
Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is the connected component of the group of invertibles $G(A)$ that contains the identity. Is it true that $e(ab)\cup\{0\} = e(ba)\cup\{0\}$ for all $a,b \in A$? Equivalently, is it true that
$1-ab$ is in $G_1(A)$ if and only if $1-ba$ is in $G_1(A)$, for all $a,b \in A$? Note: The usual spectrum has this property. Just an additional note: We have $e(ab)\cup\{0\} = e(ba)\cup\{0\}$ for all $a,b \in A$ if 1) The group of invertibles of $A$ is connected, because then the exponential spectrum of any element is just the usual spectrum of that element. 2) The set $Z(A)G(A) = \{ab: a \in Z(A), b\in G(A)\}$ is dense in $A$, where $Z(A)$ is the center of $A$. (One can prove this). In particular, we have $e(ab)\cup\{0\} = e(ba)\cup\{0\}$ for all $a,b \in A$ if the invertibles are dense in $A$. 3) $A$ is commutative, clearly. But what about other Banach algebras? Can someone provide a counterexample?
|
This isn't really a heuristic, but I hate "functions are formulas". For most students it takes a really long time to think of a function as anything other than an algebraic expression, even though natural algorithmic examples are everywhere. For example, some students won't think of \begin{gather}
f(n) = \{\text{1 if $n \bmod 2 = 0$ $\lor$ $-1$ otherwise}\}
\end{gather} as a function until you write it as $f(n) = (-1)^n$
|
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|
2,372 |
Suppose I put an ant in a tiny racecar on every face of a soccer ball. Each ant then drives around the edges of her face counterclockwise. The goal is to prove that two of the ants will eventually collide (provided they aren't allowed to stand still or go arbitrarily slow). My brother told me about this result, but I can't quite seem to prove it. Instead of a soccer ball, we should be able to use any connected graph on a sphere (provided that there are no vertices of valence 1). We may as well assume there are no vertices of valence 2 either, since you can always just fuse the two edges. I (and some people I've talked to) have come up with a number of observations and approaches: Notice that if two ants are ever on the same edge, then they will crash, so the problem is discrete. You can just keep track of which edge each ant is on, and let the ants move one at a time. Then the goal is to show that there is no way for the ants to move without crashing unless some ant only moves a finite number of times. You can assume all the faces are triangles. If there is a face with more than three edges, then you can triangulate it and make the ants on the triangles move in such a way that it looks exactly the same "from the outside". If there is a 1-gon, it's easy to show the ants will crash. If there is a 2-gon, it's easy to show that you can turn it into an edge without changing whether or not there is a crash. One approach is to induct on the number of faces. If there is a counterexample, I feel like you should either be able to fuse two adjacent faces or shrink one face to a point to get a smaller counterexample, but I can't get either of these approaches to work. If you have a counterexample on a graph, I think you get a counterexample on the dual graph. Have the dual ant be on the next edge along which a (non-dual) ant will pass through the given vertex. It feels like there might be a very slick solution using the hairy ball theorem.
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This is known as Klyachko's Car Crash Theorem. It was proved in order to prove a theorem about finitely presented groups. In fact, the result allows the ants to move at arbitrary nonzero speeds so long as they make infinitely many loops around their 2-cell. The conclusion is that there's either a collision between ants in the interior of an edge, or else there is a `complete collision', which means that there's a collision at a vertex of all ants from adjacent edges. EDIT: Oh, it's actually important that there are two complete collisions, which is somewhat harder to prove than one (a collision in the middle of an edge is also a complete collision.) You can read an expository article by Colin Rourke here .
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|
2,446 |
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc. One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes it unique or useful.
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I think Algebraic Geometry is too broad a subject to choose only one book. Maybe if one is a beginner then a clear introductory book is enough or if algebraic geometry is not ones major field of study then a self-contained reference dealing with the important topics thoroughly is enough. But Algebraic Geometry nowadays has grown into such a deep and ample field of study that a graduate student has to focus heavily on one or two topics whereas at the same time must be able to use the fundamental results of other close subfields. Therefore I find the attempt to reduce his/her study to just one book (besides Hartshorne's) too hard and unpractical. That is why I have collected what in my humble opinion are the best books for each stage and topic of study, my personal choices for the best books are then: CLASSICAL: Beltrametti-Carletti-Gallarati-Monti. "Lectures on Curves, Surfaces and Projective Varieties" which starts from the very beginning with a classical geometric style. Very complete (proves Riemann-Roch for curves in an easy language) and concrete in classic constructions needed to understand the reasons about why things are done the way they are in advanced purely algebraic books. There are very few books like this and they should be a must to start learning the subject. (Check out Dolgachev's review .) HALF-WAY/UNDERGRADUATE: Shafarevich - "Basic Algebraic Geometry" vol. 1 and 2. They may be the most complete on foundations for varieties up to introducing schemes and complex geometry, so they are very useful before more abstract studies. But the problems are hard for many beginners. They do not prove Riemann-Roch (which is done classically without cohomology in the previous recommendation) so a modern more orthodox course would be Perrin's "Algebraic Geometry, An Introduction", which in fact introduce cohomology and prove RR. ADVANCED UNDERGRADUATE: Holme - "A Royal Road to Algebraic Geometry" . This new title is wonderful: it starts by introducing algebraic affine and projective curves and varieties and builds the theory up in the first half of the book as the perfect introduction to Hartshorne's chapter I. The second half then jumps into a categorical introduction to schemes, bits of cohomology and even glimpses of intersection theory. FREE ONLINE NOTES: Gathmann - "Algebraic Geometry" (All versions are found here . The latest is of 2019.) Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch and even hinting Hirzebruch-R-R. It is the best free course in my opinion, to get enough algebraic geometry background to understand the other more advanced and abstract titles. For an abstract algebraic approach, the nice, long notes by Ravi Vakil is found here . (A link to all versions; the latest is of 2017.) GRADUATE FOR ALGEBRISTS AND NUMBER THEORISTS: Liu Qing - "Algebraic Geometry and Arithmetic Curves" . It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell's conjecture, Faltings' or even Fermat-Wiles Theorem. GRADUATE FOR GEOMETERS: Griffiths; Harris - "Principles of Algebraic Geometry" . By far the best for a complex-geometry-oriented mind. Also useful coming from studies on several complex variables or differential geometry. It develops a lot of algebraic geometry without so much advanced commutative and homological algebra as the modern books tend to emphasize. BEST ON SCHEMES: Görtz; Wedhorn - Algebraic Geometry I, Schemes with Examples and Exercises . Tons of stuff on schemes; more complete than Mumford's Red Book (For an online free alternative check Mumfords' Algebraic Geometry II unpublished notes on schemes.). It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises. UNDERGRADUATE ON ALGEBRAIC CURVES: Fulton - "Algebraic Curves, an Introduction to Algebraic Geometry" which can be found here . It is a classic and although the flavor is clearly of typed concise notes, it is by far the shortest but thorough book on curves, which serves as a very nice introduction to the whole subject. It does everything that is needed to prove Riemann-Roch for curves and introduces many concepts useful to motivate more advanced courses. GRADUATE ON ALGEBRAIC CURVES: Arbarello; Cornalba; Griffiths; Harris - "Geometry of Algebraic Curves" vol 1 and 2. This one is focused on the reader, therefore many results are stated to be worked out. So some people find it the best way to really master the subject. Besides, the vol. 2 has finally appeared making the two huge volumes a complete reference on the subject. INTRODUCTORY ON ALGEBRAIC SURFACES: Beauville - "Complex Algebraic Surfaces" . I have not found a quicker and simpler way to learn and clasify algebraic surfaces. The background needed is minimum compared to other titles. ADVANCED ON ALGEBRAIC SURFACES: Badescu - "Algebraic Surfaces" . Excellent complete and advanced reference for surfaces. Very well done and indispensable for those needing a companion, but above all an expansion, to Hartshorne's chapter. ON HODGE THEORY AND TOPOLOGY: Voisin - Hodge Theory and Complex Algebraic Geometry vols. I and II. The first volume can serve almost as an introduction to complex geometry and the second to its topology. They are becoming more and more the standard reference on these topics, fitting nicely between abstract algebraic geometry and complex differential geometry. INTRODUCTORY ON MODULI AND INVARIANTS: Mukai - An Introduction to Invariants and Moduli . Excellent but extremely expensive hardcover book. When a cheaper paperback edition is released by Cambridge Press any serious student of algebraic geometry should own a copy since, again, it is one of those titles that help motivate and give conceptual insights needed to make any sense of abstract monographs like the next ones. ON MODULI SPACES AND DEFORMATIONS: Hartshorne - "Deformation Theory" . Just the perfect complement to Hartshorne's main book, since it did not deal with these matters, and other books approach the subject from a different point of view (e.g. geared to complex geometry or to physicists) than what a student of AG from Hartshorne's book may like to learn the subject. ON GEOMETRIC INVARIANT THEORY: Mumford; Fogarty; Kirwan - "Geometric Invariant Theory" . Simply put, it is still the best and most complete. Besides, Mumford himself developed the subject. Alternatives are more introductory lectures by Dolgachev. ON INTERSECTION THEORY: Fulton - "Intersection Theory" . It is the standard reference and is also cheap compared to others. It deals with all the material needed on intersections for a serious student going beyond Hartshorne's appendix; it is a good reference for the use of the language of characteristic classes in algebraic geometry, proving Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch among many interesting results. ON SINGULARITIES: Kollár - Lectures on Resolution of Singularities . Great exposition, useful contents and examples on topics one has to deal with sooner or later. As a fundamental complement check Hauser's wonderful paper on the Hironaka theorem. ON POSITIVITY: Lazarsfeld - Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series and Positivity in Algebraic Geometry II: Positivity for Vector Bundles and Multiplier Ideals . Amazingly well written and unique on the topic, summarizing and bringing together lots of information, results, and many many examples. INTRODUCTORY ON HIGHER-DIMENSIONAL VARIETIES: Debarre - "Higher Dimensional Algebraic Geometry" . The main alternative to this title is the new book by Hacon/Kovács ' " Classifiaction of Higher-dimensional Algebraic Varieties " which includes recent results on the classification problem and is intended as a graduate topics course. ADVANCED ON HIGHER-DIMENSIONAL VARIETIES: Kollár; Mori - Birational Geometry of Algebraic Varieties . Considered as harder to learn from by some students, it has become the standard reference on birational geometry.
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|
2,520 |
Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?
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To keep things simple, let us assume we work over a perfect field.
The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying space of the multiplicative group $\mathbb G_m=\mathbb A^1-\{0\}$ has the $\mathbb A^1$-homotopy type of the infinite dimensional projective space. Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that $H^1(X,\mathbb G_m)=\text{Pic}(X)$ reads as $[X,B\mathbb G_m]=\text{Pic}(X)=CH^1(X)$, where $[?,?]$ stands for the $\text{Hom}$ in the $\mathbb A^1$-homotopy category of $k$-schemes, denoted by $H(k)$. In general, we denote by $K(\mathbb Z(n),2n)$ the $n$-th motivic Eilenberg-MacLane space, i.e. the object of $H(k)$ which represents the $n$-th Chow group in $H(k)$: for any smooth $k$-scheme $X$, one has $$[X,\Omega^i K(\mathbb Z(n),2n)]=H^{2n-i}(X,\mathbb Z(n))$$ (where $\Omega^i$ stands for the $i$-th loop space functor). For $i=0$, we just get the usual Chow groups: $$H^{2n}(X, \mathbb Z (n)))\simeq CH^n(X) .$$ Then, there are several models for $K(\mathbb Z(n),2n)$, one of the smallest being constructed as follows. What I explained above is that $K(\mathbb Z(1),2)$ is the infinite projective space.
$K(\mathbb Z(0),0)$ is simply the constant sheaf $\mathbb Z$. For higher $n$, here is a construction (this is Voevodsky's). Given a $k$-scheme $X$, denote by $L(X)$ the presheaf with transfers associated to X, that is the presheaf of abellian groups whose sections over a smooth $k$-scheme $V$ are the finite correspondences from $V$ to $X$ (i.e. the finite linear combinations of cycles
$\sum n_iZ_i$ in $V \times X$ such that $Z_i$ is finite and surjective over $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the $Z_i$; are finite and surjective over a smooth (hence normal) scheme $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U \to V$ with $U$ and $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf $L(X)$ is a sheaf for the Nisnevich topology. This construction is functorial in $X$ (I will need this functoriality only for closed immersions). Let $X$ (resp. $Y$) be the cartesian product of $n$ (resp. $n-1$) copies of the projective line.
The point at infinity gives a family of $n$ maps $u_i : Y \to X$. Then a model of $K(\mathbb Z(n),2n)$ in $H(k)$ is the sheaf of sets obtained as the quotient (in the category of Nisnech sheaves of abelian groups) of $L(X)$ by the subsheaf generated by the images of the maps $L(u_i):L(Y)\to L(X)$. If you want a more conceptual definition, there is also the direction of algebraic cobordism (but for this, you need to understand the stable homotopy of $\mathbb P^1$-spectra, but maybe it is enough at first to think of $\mathbb P^1$-spectra simply as the cohomology theories
allowed in homotopy theory of schemes): the idea is that there is an algebraic cobordism, which is represented by a $\mathbb P^1$-spectrum $MGL$ (the analog of of the spectrum $MU$ which represents complex cobordism in algebraic topology). The idea is that $MGL$ is the universal oriented cohomology theory (for short, this means that, if a cohomology theory $E$ satisfies the projective bundle formula, then the choice of an orientation, i.e. of a generating class in the second cohomology group of $\mathbb P^1$ with coefficients in $E(1)$, is the same as a map of ring spectra $MGL \to E$. The idea is that, as in algebraic topology, formal groups laws classify oriented cohomology theories ($MGL$ corresponding to the initial formal group law). The cohomology theory which corresponds to the multiplicative formal group law is $KGL$, the $\mathbb P^1$-spectrum which represents algebraic K-theory, while the cohomology theory corresponding to the additive formal group law is motivic cohomology (this latter characterization has been announced by F. Morel and M. Hopkins if $k$ is of characteristic zero, but is not published yet, and it is known for any field $k$ if we work with rational coefficients (this is a result of Spitzweck, Nauman, Ostvaer)).
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|
2,551 |
Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly. However, this heuristic breaks down in the case of many abelian groups. Abelian groups more often arise as a "receptacle for addition". What I mean is that they are more intuitively counting combinations of some generating elements. See for example: solution spaces to linear equations, the underlying group of rings and modules, (co)homology groups. They rarely act naturally on anything except themselves, which seems like a copout. It bugs me that these two intuitions are so far from each other, even though the underlying axioms differ by a single, deceptively-mild assumption. Is there a way to reconcile these two perspectives? Do abelian groups satisfy the group axioms by accident?
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Here's how I think about it: if the automorphism group of an object is abelian, this means something very strong about the object. It sort of means you can affect its structure in two different ways, in any order, and they won't affect each other. This to me hints that maybe the object consists of separate "pieces" that can only be affected independently. The first example like this which comes to mind is a direct (commutative ring) product of finite fields of different characteristics. Each field has a cyclic automorphism group, making it very "simple", and the fields can't map into each other, meaning they can only be affected "independently." So I'd posit that if you want to think of abelian groups as symmetry groups, you can imagine them as the symmetry groups of objects that "break apart" into "simple" pieces which can't interact with each other (intentionally vague, since I don't want to commit to a particular category).
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2,556 |
What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories , e.g. math.AT, math.QA, math.CO, etc. This is a community-wiki question, so please edit and improve pre-existing answers: let's keep it to a single answer for each subject area. (This is inspired by Terry Tao's recent post about a periodic table of the elements listing commercial applications. He suggested it might be fun to have such a summary for either the MSC top-level subjects or the arxiv subjects.) I'd like to propose that for areas in which the applications are either numerous, non-obvious, or generally worthy of discussion, someone volunteers to open up a new question specifically about that subject area, and takes care of providing a summary here of the best answers produced there.
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math.GR Group Theory Group theory provides methods for understanding the Rubik's cube, and for generating algorithms for solving the cube remarkably quickly from any state the cube may be in. Groups find various applications in chemistry, eg. in the study of crystal structures and spectroscopy. Cryptography - various hard algorithmic problems about groups are used to design crypto-systems. Groups of symmetries are used to reduce the dimension of parameter spaces in engineering models to make model verification more tractable. Potentially fast matrix multiplication; see this MO question . Card tricks that don't work by sleight of hand, but via the arrangements of the cards. e.g. Sim Sala Bim, see this site for a description . If you think about it, the symmetric group explains the trick and shows you how you to extend it past three piles of seven cards, but to N piles of M cards.
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2,607 |
Does anyone have an answer to the question "What does the cotangent complex measure?" Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as geometric ones (like "homology detects holes"), as are intuitions which do not exactly answer the above question. In particular: Do the degrees have a meaning? E.g. if an ideal $I$ in a ring $A$ is generated by a regular sequence, the cotangent complex of the quotient map $A\twoheadrightarrow A/I$ is $(I/I^2)[-1]$. Why does it live in degree 1?
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One thing the cotangent complex measures is what kind of deformations a scheme has. The precise statements are in Remark 5.30 and Theorem 5.31 in Illusie's article in "FGA explained". Here's the short simplified version in the absolute case: If you have a scheme $X$ over $k$, a first order deformation is a space $\mathcal{X}$ over $k[\epsilon]/(\epsilon)^2$ whose fiber over the only point of $k[\epsilon]/(\epsilon)^2$ is $X$ again. You can imagine $k[\epsilon]/(\epsilon)^2$ as a point with an infinitesimal arrow attached to it and $\mathcal{X}$ as an infinitesimal thickening of $X$. The cotangent complex gives you precise information on how many such thickenings there are: The set of such thickenings is isomorphic $\mathop{Ext}^1(L_X, \epsilon^2)$. Now let's assume that we have chosen one such infinitesimal thickening $\mathcal{X}$ over $k[\epsilon]/(\epsilon)^2$. It is not always true that you can go on and make this thickening into a thickening to the next order. Whether or not you can do this is measured precisely by the cotangent complex: There is a map that takes as input your chosen thickening $\mathcal{X}$ and spits out an element in $\mathop{Ext}^2(L_X, \epsilon^3)$. If the element in the Ext group is zero you can go on to the next level. If it is not zero, it's game over and your stuck.
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2,672 |
I made the following claim over at the Secret Blogging Seminar , and now I'm not sure it's true: Let $f: X \to Y$ and $g: X \to Y$ be two maps between finite CW complexes. If f and g induce the same map on $\pi_k$, for all k, then f and g are homotopic. Was I telling the truth? EDIT: Since I didn't say anything about basepoints, I probably should have said that f and g induce the same map $[S^k, X] \to [S^k, Y]$. This will also deal better with the situation where X and Y are disconnected. I'd be interested in knowing a result like this either with pointed maps or nonpointed maps. (Although, of course, if you work with pointed maps you have to take X and Y connected, because $[S^k, -]$ can't see anything beyond the number of components in that case.)
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This is not true. Consider, for example, a degree 1 map from a torus $S^1 \times S^1$ to $S^2$ (concretely, realize the torus as a square with identifications, and then collapse the boundary of the square to a point). This map is trivial on all homotopy groups (since for any $n>0, \pi_n$ is 0 for either the domain or the codomain), but it is not homotopically trivial because it is nonzero on $H_2$. If you want to demand that the spaces be simply connected, you can get a counterexample by considering cohomology operations: the cup square, for example, gives a map from $K(\mathbb{Z},n)$ to $K(\mathbb{Z},2n)$ which is nontrivial, but for the same reason as the previous example it must be 0 on homotopy groups. This example is not finite-dimensional, but it's probably possible to find one that is--I just don't know how because I don't know how to show a map is trivial on homotopy groups if the spaces have infinitely many nontrivial homotopy groups whose values are unknown, which is the case for most finite-dimensional examples.
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2,703 |
Continuing an amazingly interesting chain of answers about motivic cohomology , I thought I should learn about the Beilinson conjectures, referred there. I have found some references, and they seem to present the conjectures from different sides, e.g. there's the statement about vanishing but then there are also connections to motivic polylogarithms . What I miss from these articles in a general picture that would allow us to start somewhere natural. So, how would you describe an introduction into Beilinson conjectures in motivic homotopy ? Sorry for such a loaded question — I really don't know how to make it fit MathOverflow format better. One could theoreticlly post lost of specific questions on the topic, but to ask the right questions in this case you might need to know more than I do. Also, I know there are some technical developments, e.g. the language of derived stacks, and my hope would be that somebody could make a connection to these conjectures using some clear and suitable language.
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Let me talk about Beilinson's conjectures by beginning with $\zeta$-functions of number fields and $K$-theory. Space is limited, but let me see if I can tell a coherent story. The Dedekind zeta function and the Dirichlet regulator Suppose $F$ a number field, with
$$[F:\mathbf{Q}]=n=r_1+2r_2,$$
where $r_1$ is the number of real embeddings, and $r_2$ is the number of complex embeddings. Write $\mathcal{O}$ for the ring of integers of $F$. Here's the power series for the Dedekind zeta function :
$$\zeta_F(s)=\sum|(\mathcal{O}/I)|^{-s},$$
where the sum is taken over nonzero ideals $I$ of $\mathcal{O}$. Here are a few key analytical facts about this power series: This power series converges absolutely for $\Re(s)>1$. The function $\zeta_F(s)$ can be analytically continued to a meromorphic function on $\mathbf{C}$ with a simple pole at $s=1$. There is the Euler product expansion :
$$\zeta_F(s)=\prod_{0\neq p\in\mathrm{Spec}(\mathcal{O}_F)}\frac{1}{1-|(\mathcal{O}_F/p)|^{-s}}.$$ The Dedekind zeta function satisfies a functional equation relating $\zeta_F(1-s)$ and $\zeta_F(s).$ If $m$ is a positive integer, $\zeta_F(s)$ has a (possible) zero at $s=1-m$ of order
$$d_m=\begin{cases}r_1+r_2-1&\textrm{if }m=1;\\
r_1+r_2&\textrm{if }m>1\textrm{ is odd};\\
r_2&\textrm{if }m>1\textrm{ is even},
\end{cases}$$
and its special value at $s=1-m$ is
$$\zeta_F^{\star}(1-m)=\lim_{s\to 1-m}(s+m-1)^{-d_m}\zeta_F(s),$$
the first nonzero coefficient of the Taylor expansion around $1-m$. Our interest is in these special values of $\zeta_F(s)$ at $s=1-m$. At the end of the 19th century, Dirichlet discovered an arithmetic interpretation of the special value $\zeta_F^{\star}(0)$. Recall that the Dirichlet regulator map is the logarithmic embedding
$$\rho_F^D:\mathcal{O}_F^{\times}/\mu_F\to\mathbf{R}^{r_1+r_2-1},$$
where $\mu_F$ is the group of roots of unity of $F$. The covolume of the image lattice is the the Dirichlet regulator $R^D_F$. With this, we have the Dirichlet Analytic Class Number Formula . The order of vanishing of $\zeta_F(s)$ at $s=0$ is $\operatorname{rank}_\mathbf{Z}\mathcal{O}_F^\times$, and the special value of $\zeta_F(s)$ at $s=0$ is given by the formula
$$\zeta_F^{\star}(0)=-\frac{|\mathrm{Pic}(\mathcal{O}_F)|}{|\mu_F|}R^D_F.$$ Now, using what we know about the lower $K$-theory, we have:
$$K_0(\mathcal{O})\cong\mathbf{Z}\oplus\mathrm{Pic}(\mathcal{O})$$
and
$$K_1(\mathcal{O}_F)\cong\mathcal{O}_F^{\times}.$$ So the Dirichlet Analytic Class Number Formula reads:
$$\zeta_F^{\star}(0)=-\frac{|{}^{\tau}K_0(\mathcal{O})|}{|{}^{\tau}K_1(\mathcal{O})|}R^D_F,$$
where ${}^{\tau}A$ denotes the torsion subgroup of the abelian group $A$. The Borel regulator and the Lichtenbaum conjectures Let us keep the notations from the previous section. Theorem [Borel]. If $m>0$ is even, then $K_m(\mathcal{O})$ is finite. In the early 1970s, A. Borel constructed the Borel regulator maps , using the structure of the homology of $SL_n(\mathcal{O})$. These are homomorphisms
$$\rho_{F,m}^B:K_{2m-1}(\mathcal{O})\to\mathbf{R}^{d_m},$$
one for every integer $m>0$, generalizing the Dirichlet regulator (which is the Borel regulator when $m=1$). Borel showed that for any integer $m>0$ the kernel of $\rho_{F,m}^B$ is finite, and that the induced map
$$\rho_{F,m}^B\otimes\mathbf{R}:K_{2m-1}(\mathcal{O})\otimes\mathbf{R}\to\mathbf{R}^{d_m}$$
is an isomorphism. That is, the rank of $K_{2m-1}(\mathcal{O})$ is equal to the order of vanishing $d_m$ of the Dedekind zeta function $\zeta_F(s)$ at $s=1-m$. Hence the image of $\rho_{F,m}^B$ is a lattice in $\mathbf{R}^{d_m}$; its covolume is called the Borel regulator $R_{F,m}^B$. Borel showed that the special value of $\zeta_F(s)$ at $s=1-m$ is a rational multiple of the Borel regulator $R_{F,m}^B$, viz .:
$$\zeta_F^{\star}(1-m)=Q_{F,m}R_{F,m}^B.$$
Lichtenbaum was led to give the following conjecture in around 1971, which gives a conjectural description of $Q_{F,m}$. Conjecture [Lichtenbaum]. For any integer $m>0$, one has
$$|\zeta_F^{\star}(1-m)|"="\frac{|{}^{\tau}K_{2m-2}(\mathcal{O})|}{|{}^{\tau}K_{2m-1}(\mathcal{O})|}R_{F,m}^B.$$
(Here the notation $"="$ indicates that one has equality up to a power of $2$.) Beilinson's conjectures Suppose now that $X$ is a smooth proper variety of dimension $n$ over $F$; for simplicity, let's assume that $X$ has good reduction at all primes. The question we might ask is, what could be an analogue for the Lichtenbaum conjectures that might provide us with an interpretation of the special values of $L$-functions of $X$? It turns out that since number fields have motivic cohomological dimension $1$, special values of their $\zeta$-functions can be formulated using only $K$-theory, but life is not so easy if we have higher-dimensional varieties; for this, we must use the weight filtration on $K$-theory in detail; this leads us to motivic cohomology. Write $\overline{X}:=X\otimes_F\overline{F}$. Now for every nonzero prime $p\in\mathrm{Spec}(\mathcal{O})$, we may choose a prime $q\in\mathrm{Spec}(\overline{\mathcal{O}})$ lying over $p$, and we can contemplate the decomposition subgroup $D_{q}\subset G_F$ and the inertia subgroup $I_{q}\subset D_{q}$. Now if $\ell$ is a prime over which $p$ does not lie and $0\leq i\leq 2n$, then the inverse $\phi_{q}^{-1}$ of the arithmetic Frobenius $\phi_{q}\in D_{q}/I_{q}$ acts on the $I_{q}$-invariant subspace $H_{\ell}^i(\overline{X})^{I_{q}}$ of the $\ell$-adic cohomology $H_{\ell}^i(\overline{X})$. We can contemplate the characteristic polynomial of this action:
$$P_{p}(i,x):=\det(1-x\phi_{q}^{-1}).$$
One sees that $P_{p}(i,x)$ does not depend on the particular choice of $q$, and it is a consequence of Deligne's proof of the Weil conjectures that the polynomial $P_{p}(i,x)$ has integer coefficients that are independent of $\ell$. (If there are primes of bad reduction, this is expected by a conjecture of Serre.) This permits us to define the local $L$-factor at the corresponding finite place $\nu(p)$:
$$L_{\nu(p)}(X,i,s):=\frac{1}{P_{p}(i,p^{-s})}$$
We can also define local $L$-factors at infinite places as well. For the sake of brevity, let me skip over this for now. (I can fill in the details later if you like.) With these local $L$-factors, we define the $L$-function of $X$ via the Euler product expansion
$$L(X,i,s):=\prod_{0\neq p\in\mathrm{Spec}(\mathcal{O})}L_{\nu(p)}(X,i,s);$$
this product converges absolutely for $\Re(s)\gg 0$. We also define the $L$-function at the infinite prime $$L_{\infty}(X,i,s):=\prod_{\nu|\infty}L_{\nu}(X,i,s)$$
and the full $L$-function $$\Lambda(X,i,s)=L_{\infty}(X,i,s)L(X,i,s).$$ Here are the expected analytical properties of the $L$-function of $X$. The Euler product converges absolutely for $\Re(s)>\frac{i}{2}+1$. $L(X,i,s)$ admits a meromorphic continuation to the complex plane, and the only possible pole occurs at $s=\frac{i}{2}+1$ for $i$ even. $L\left(X,i,\frac{i}{2}+1\right)\neq 0$. There is a functional equation relating $\Lambda(X,i,s)$ and $\Lambda(X,i,i+1-s).$ Beilinson constructs the Beilinson regulator $\rho$ from the part $H^{i+1}_{\mu}(\mathcal{X},\mathbf{Q}(r))$ of rational motivic cohomology of $X$ coming from a smooth and proper model $\mathcal{X}$ of $X$ (conjectured to be an invariant of the choice of $\mathcal{X}$) to Deligne-Beilinson cohomology $D^{i+1}(X,\mathbf{R}(r))$. This has already been discussed here . It's nice to know that we now have a precise relationship between the Beilinson regulator and the Borel regulator. (They agree up to exactly the fudge factor power of $2$ that appears in the statement of the Lichtenbaum conjecture above.) Let's now assume $r<\frac{i}{2}$. Conjecture [Beilinson]. The Beilinson regulator $\rho$ induces an isomorphism
$$H^{i+1}_{\mu}(\mathcal{X},\mathbf{Q}(r))\otimes\mathbf{R}\cong D^{i+1}(X,\mathbf{R}(r)),$$
and if $c_X(r)\in\mathbf{R}^{\times}/\mathbf{Q}^{\times}$ is the isomorphism above calculated in rational bases, then
$$L^{\star}(X,i,r)\equiv c_X(r)\mod\mathbf{Q}^{\times}.$$
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|
2,704 |
I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links exist. What is, in basic terms, the relationship between Riemann surfaces and algebraic geometry?
|
For simplicity, I'll just talk about varieties that are sitting in projective space or affine space. In algebraic geometry, you study varieties over a base field k. For our purposes, "over" just means that the variety is cut out by polynomials (affine) or homogeneous polynomials (projective) whose coefficients are in k. Suppose that k is the complex numbers, C. Then affine spaces and projective spaces come with the complex topology, in addition to the Zariski topology that you'd normally give one. Then one can naturally give the points of a variety over C a topology inherited from the subspace topology. A little extra work (with the inverse function theorem and other analytic arguments) shows you that, if the variety is nonsingular, you have a nonsingular complex manifold. This shouldn't be too surprising. Morally, "algebraic varieties" are cut out of affine and projective spaces by polynomials, "manifolds" are cut out of other manifolds by smooth functions, and polynomials over C are smooth, and that's all that's going on. In general, the converse is false: there are many complex manifolds that don't come from nonsingular algebraic varieties in this manner. But in dimension 1, a miracle happens, and the converse is true: all compact dimension 1 complex manifolds are analytically isomorphic to the complex points of a nonsingular projective dimension-1 variety, endowed with the complex topology instead of the Zariski topology. "Riemann surfaces" are just another name for compact dimension 1 (dimension 2 over R) complex manifolds, and "curves" are just another name for projective dimension 1 varieties over any field, hence the theorem you described. As for why Riemann surfaces are algebraic, Narasimhan's book explicitly constructs the polynomial that cuts out a Riemann surface, if you are curious.
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|
2,708 |
Is there known to be an $x$ such that for all positive integers $N$ there exists some $n>N$ such that $p_{n+1}-p_n \leq x$, where $p_n$ is the $n$th prime? Or, in other words, is it known that limit as $n$ goes infinity of $p_{n+1}-p_n$ is not infinity? If such an $x$ is known to exist, what is the current best known $x$? (Showing $x=2$ would imply the Twin Prime Conjecture, of course.)
|
( Edit : things have happened since the original post, changing the short answer to yes. See for example http://arxiv.org/abs/1410.8400 for the status in 2014 where $x \leq 600$ unconditionally. GRP End Edit ) The short answer is no, though if one assumes the Elliot-Halberstam conjecture then one can take x=16. See http://arxiv.org/abs/math/0605696 for a comprehensive survey of the best known results (both conditional and unconditional). There is also the Wikipedia article at http://en.wikipedia.org/wiki/Prime_gap although this is less comprehensive.
|
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|
2,748 |
This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object of study. How would you describe rings to them in a natural way given that they like talking about groups? (Admittedly this is not really the question the title asks.)
|
Well rings are naturally the objects which act on abelian groups - indeed composition always endows the endomorphisms of an abelian group with the structure of a ring. So if one is interested in the endomorphisms of groups one is actually interested in rings. One can make this analogy more precise especially if one picks a particular ring and looks at the forgetful functor to abelian groups from its module category. This analogy can then be used again for instance to motivate the definition of plethory which are the natural objects which act on rings. To address Eric's comment about commutative rings there is an analogue in this case of something which was mentioned in the discussion of groups versus abelian groups. Indeed one can obtain commutative rings by considering the identity in an additive symmetric monoidal category. In this case the endomorphisms of the tensor unit are endowed with an abelian group structure via the augmentation over abelian groups and the Eckmann-Hilton argument applied to tensoring endomorphisms and composing endomorphims forces the composition to be abelian. So from this point of view commutative rings are the gadgets which naturally act on the hom-sets of additive symmetric monoidal categories. Since I mentioned this one can take this slightly further. If one considers such a category together with an autoequivalence (for instance if we take a tensor triangulated category) then one can consider the graded endomorphism ring of the identity. This naturally gives rise to an integer graded ring which is commutative up to some unit which squares to the identity and which has a natural action on the category.
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|
2,755 |
As Akhil had great success with his question , I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enough to care about things like G-bundles, and wants to talk about vector bundles with structure group G, and so needs to know representation theory, but wants to do it as geometrically as possible. So, in addition to the algebraic geometry, lets assume some familiarity with representations of finite groups (particularly symmetric groups) going forward. What path should be taken to learn some serious representation theory?
|
I second the suggestion of Fulton and Harris. It's a funny book, and definitely you want to keep going after you finish it, but it's a good introduction to the basic ideas. You specifically might be happier reading a book on algebraic groups. While I third the suggestion of Ginzburg and Chriss, I wouldn't call it a "second course." Maybe if what you really wanted to do was serious, Russian-style geometric representation theory, but otherwise you might want to try something a little less focused, like Knapp's "Lie Groups Beyond an Introduction." If you want Langlandsy stuff, then Ginzburg and Chriss is actually a bit of a tangent; good enrichment, but not directly what you want, since it skips over all the good stuff with D-modules. Look in the background reading for the graduate student seminar we're having in Boston this year: http://www.math.harvard.edu/~gaitsgde/grad_2009/
|
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|
2,791 |
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when they say this? The Kronecker-Weber theorem gives a good idea of what the abelianization of $G$ looks like. But in one of Richard Taylor's MSRI talks, Taylor said that he's never heard of anyone proposing a similar direct description of $G$ and that to understand $G$ one studies the representations of $G$. I know that there is a strong interest in showing the Langlands reciprocity conjecture [Edit: What I had in mind in writing this is evidently Clozel's conjecture, not the Langlands reciprocity conjecture - see Kevin Buzzard's post below] - that $L$-functions attached to $\ell$-adic Galois representations coincide with $L$-functions attached to certain automorphic representations. And I've heard people refer to the Tannakian philosophy which I understand as (roughly speaking) asserting that $G$ is determined by all of its finite dimensional representations. Here is a representation of $G$ understood not to be a representation of $G$ as an abstract group but as a group together with a labeling of some of the conjugacy classes of G by rational primes (the Frobenius elements)? When people talk about "understanding $G$" do they mean proving [Edit: Clozel's conjecture] (in view of the Tannakian philosophy)? If not, what do they mean? If so, this conceptualization seems quite abstract to me. Is this what people mean when they say "understand $G$"? Can [Edit: Clozel's conjecture] be used to give more tangible statements about $G$? Something that I have in mind as I write this is the inverse Galois problem (does every finite group occur as a Galois group of a normal extension of $\mathbb{Q}$?) and Gross' conjecture (mostly proven by now) that for each prime $p$ there exists a nonsolvable extension of $\mathbb{Q}$ ramified only at $p$. But I am open to and interested in other senses and respects in which one might "understand" $G$.
|
What would it mean to understand this Galois group? You could mean several things. You could mean trying to give the group in terms of some smallish generators and relations. This would be nice, and help to answer questions like the inverse Galois problem that Greg Muller mentioned, and having a certain family of "generating" Galois automorphisms would allow you to study questions about e.g. the representation theory in quite explicit terms. However, the Galois group is an uncountable profinite group, and so to give any short description in terms of generators and relations leads you into subtle issues about which topology you want to impose. You could also ask for a coherent system of names for all Galois automorphisms, so that you can distinguish them and talk about them on an individual basis. One system of names comes from the dessins d'enfant that Ilya mentioned: associated to a Galois automorphism we have some associated data. We have its image under the cyclotomic character, which tells us how it acts on roots of unity. By the Kronecker-Weber theorem this tells us about the abelianization of the Galois group. We also have an element in the free profinite group on two generators, which (roughly speaking) tells us something about how abysmally acting on the coefficients of a power series fails to commute with analytic continuation. These two names satisfy some relations, called the $2$-, $3$-, and $5$-cycle relation, which are conjectured to generate all relations (at least the last time I checked), but it is difficult to know whether they actually do so. If they do, then the Galois group is the so-called Grothendieck-Teichmüller group. The problem with this perspective is that the names aren't very explicit (and we don't expect them to be: we may need the axiom of choice to show they exist, and there are only two Galois automorphisms of $\mathbb{C}$ that are measurable functions!) and it seems to be a difficult problem to determine whether the Grothendieck-Teichmuller group really is the whole thing. (Or it was the last time I checked.) However, the cyclotomic character is a nice, and fairly canonical, name associated for Galois automorphisms. We could try to generalize this: there are Kummer characters telling us what a Galois automorphism does to the system of real positive roots of a positive rational number number (these determine a compatible system of roots of unity, or equivalent an element of the Tate module of the roots of unity). This points out one of the main difficulties, though: we had to make choices of roots of unity to act on, and if Galois theory taught us nothing else it is that different choices of roots of an irreducible polynomial should be viewed as indistinguishable. Different choices differ by conjugation in the Galois group. This brings us to the point JSE was making: if we take the "symmetry" point of view seriously, we should only be interested in conjugacy-invariant information about the Galois group. Assigning names to elements or giving a presentation doesn't really mesh with the core philosophy. So this brings us to how many people here have mentioned understanding the Galois group: you understand it by how it manifests, in terms of its representations (as permutations, or on dessins, or by representations, or by its cohomology), because this is how it's most useful. Then you can study arithmetic problems by applying knowledge about this. If I have two genus $0$ curves over $\mathbb{Q}$, what information distinguishes them? If I have two lifts of the same complex elliptic curve to $\mathbb{Q}$, are they the same? How can I get information about a reduction of an abelian variety mod $p$ in terms of the Galois action on its torsion points? Et cetera.
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|
2,795 |
Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a tool for studying representations of a group; they classify modules up to isomorphism, the characters of irreducible modules form an orthonormal basis for the space of class functions, and other facts of that sort. To me, characters seem an arbitrary object of study (why take the trace and not any other coefficient of the characteristic polynomial?), and I've never seen any intuition given for their development other than "we try this, and it works great". The proof of the orthogonality relations is elementary but, to me, casts no light; I do get the nice behaviour of characters with respect to direct sums and tensor products, and I understand its desirability, but that's clearly not all that's going on here. So, my question: is there a high-level reason why group characters are as magic as they are, or is it all just coincidence? Or am I simply unable to see how well-motivated the proof of orthogonality is?
|
Orthogonality makes sense without character theory. There's an inner product on the space of representations given by $\dim \operatorname {Hom}(V, W)$. By Schur's lemma the irreps are an orthonormal basis. This is "character orthogonality" but without the characters. How to recover the usual version from this conceptual version? Notice $$\dim \operatorname{Hom}(V,W) = \dim \operatorname{Hom}(V \otimes W^*, 1)$$ where $1$ is the trivial representation. So in order to make the theory more concrete you want to know how to pick off the trivial part of a representation. This is just given by the image of the projection $\frac1{|G|} \sum_{g\in G} g$. The dimension of a space is the same as the trace of the projection onto that space, so
$$
\def\H{\rule{0pt}{1.5ex}H}
\dim \operatorname{Hom}(V \otimes W^*, 1) = \operatorname{tr}\left(\frac1{|G|} \sum_{g\in G} {\large \rho}_{\small V \otimes W^*}(g)\right)
= \frac1{|G|} \sum_{g\in G} {\large\chi}_{V}(g)\ {\large\chi}_{W}\left(g^{-1}\right)
\\
$$ using the properties of trace under tensor product and duals.
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|
2,809 |
It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace Transforms). My question: Is there any intuition why this is so? It can be proved, ok - but can somebody please explain the big picture (please not too technical - I might need another intuition to understand that one then, too ;-)
|
The Fourier and Laplace transforms are defined by testing the given function f by special functions (characters in the case of Fourier, exponentials in the case of Laplace). These special functions happen to be eigenfunctions of translation: if one translates a character or an exponential, one gets a scalar multiple of that character or an exponential. As a consequence, the Fourier or Laplace transforms diagonalise the translation operation (formally, at least). Whenever two linear operations commute, they are simultaneously diagonalisable (in principle, at least). As such, one expects the Fourier or Laplace transforms to also diagonalise other linear, translation-invariant operations. Differentiation and integration are linear, translation-invariant operations. This is why they are diagonalised by the Fourier and Laplace transforms. Diagonalisation is an extremely useful tool; it reduces the non-abelian world of operators and matrices to the abelian world of scalars.
|
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|
2,861 |
Given a high precision real number, how should I go about guessing an algebraic integer that it's close to? Of course, this is extremely poorly defined -- every real number is close to a rational number, of course! But I'd like to keep both the coefficients and the degree relatively small. Obviously we can make tradeoffs between how much we dislike large coefficents and how much we dislike large degrees. But that aside, I don't even know how you'd start. Any ideas? Background : this comes out of the project that prompted Noah's recent question on solving large systems of quadratics. We're trying to find subfactors inside their graph planar algebras. We've found that solving quadratics numerically, approximating the solutions by algebraic integers, and then checking that these are actually exact solutions is very effective. So far, we've been making use of Mathematica's " RootApproximant " function which does exactly what I ask here, but it's an impenetrable black box, like everything out of Wolfram.
|
The Lenstra-Lenstra-Lovasz lattice basis reduction algorithm is what you need. Suppose that your real number is a and you want a quadratic equation with as small coefficients as possible, of which a is nearly a root. Then calculate 1,a,a^2 (to some precision), find a nontrivial integer relation between them, and use the LLL algorithm to find a much better one from the first one. Exactly this example is discussed in the Wikipedia entry on the LLL algorithm, applied to the Golden Section number. And there is a big literature on the algorithm and its many applications. (For higher degree, calculate 1,a,a^2,...,a^n).
|
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|
2,904 |
Inspired by this question , I was curious about a comment in this article : In many situations, it can be easy to
apply Kolmogorov's zero-one law to
show that some event has probability 0
or 1, but surprisingly hard to
determine which of these two extreme
values is the correct one. Could someone provide an example?
|
There's a set of good examples from percolation theory: http://en.wikipedia.org/wiki/Percolation_theory If you create a "random network" with a certain probability p of edges between nodes (see article above for precise definitions) then there is an infinite cluster with probability either zero or one. But for a given value of p it can be nontrivial to determine which.
|
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|
2,917 |
My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn quantum for some time now, and after seeing it come up repeatedly in mathematical contexts like Scott Aaronson's blog or John Baez's TWF, I figure I might as well do it now. Unfortunately, my physics background is a little lacking. I know some mechanics and some E&M, but I can't say I've mastered either (for example, I don't know either the Hamiltonian or the Lagrangian formulations of mechanics). I also have a relatively poor background in differential equations and multivariate calculus. However, I do know a little representation theory and a little functional analysis, and I like q-analogues! (This last comment is somewhat tongue-in-cheek.) Given this state of affairs, what's my best option for learning quantum? Can you recommend me a good reference that downplays the historical progression and emphasizes the mathematics? Is it necessary that I understand what a Hamiltonian is first? (I hope this is "of interest to mathematicians." Certainly the word "quantum" gets thrown around enough in mathematics papers that I would think it is.)
|
It could be just my own personal bias, but I think it is difficult to learn quantum mechanics without first learning classical mechanics. I recommend taking a 1 semester course, either graduate or advanced undergraduate, in classical mechanics and then taking a quantum mechanics course. I also think it would be a mistake to start with an overly mathematically-oriented QM course. You want to learn how physicists think and how they use this stuff to come up with real physical predictions. Otherwise, you're just learning math packaged as "physics". You shouldn't have much trouble later figuring out how to translate the physics back into math. But if you focus too much on the math at the beginning, you make it less likely you'll ever understand the physics.
|
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|
2,944 |
Let $\{a_n\}$ be a sequence of complex numbers indexed by the positive integers. Does there always exist an analytic function $f$ such that $f(n) = \{a_n\}$ for $n=1,2,...$? If not, are there any simple necessary or sufficient conditions for the existence of such $f$? This analytic function should be defined on some connected domain in the complex plane containing the positive integers. To make this concrete, consider Ackermann's function, which is defined recursively: first define the sequence of functions $A_k$, $k=1,2,...$, as $A_1(n) = 2n$, $A_k(1) = 2$,
$A_k(n) = A_{k-1}(A_k(n-1))$, and then define Ackermann's function as the diagonal $A(n) = A_n(n)$ for $n \geq 1$. Does there exist an analytic function $f$ such that $f(n) = A(n)$ for $n = 1,2,...$? Actually, the individual functions $A_k$ are interesting as well. $A_1(n) = 2n$, as given above; $A_2(n) = 2^n$, and $A_3(n) = 2^{2^{\ldots^{2^2}}}$ (with $n$ twos in the expression). Obviously, $A_1$ and $A_2$ have analytic extensions. According to Wikipedia (which uses a slightly different definition and notation), analytic extensions of $A_3$ or $A_k$ for any other $k$ aren't known, but from the language, it isn't clear whether the existence of an extension is itself in question, or whether one simply hasn't been found yet. Also, it doesn't say anything about the diagonal $A(n)$ (unless I missed it). There are many other obvious sequences that don't seem to have obvious analytic extensions, like the prime-counting function (just to name one!). As far as my knowledge is concerned, this seems more like the rule than the exception. My knowledge here is admittedly very limited, though, so anything at all that you can share will probably teach me something.
|
It's a standard theorem in complex analysis that if $z_n$ is a sequence that goes to infinity, there is an entire function taking any prescribed values at the $z_n$ . There is a function $f$ vanishing to order 1 at each $z_n$ (for $z_n=n$ , you could take $f(z)=\sin \pi z$ ), and then consider $\sum_n a_nf(z)/(f'(z_n)(z-z_n))$ . This may not converge, but you can tweak it by multiplying each term by something that is 1 at $z_n$ (eg, $\exp(c_n(z-z_n))$ for $c_n$ chosen appropriately) to make it converge. (I don't know off the top of my head how to choose the $c_n$ ; this is copied from Exercise 1 on page 197 of Ahlfors's Complex Analysis.) EDIT: It's easy to show that such $c_n$ exist. If you write $b_n=a_n/(z_n f'(z_n))$ , then for any fixed $z$ , the terms of the sum will be approximately $b_n \exp(c_n z_n)$ for $n$ large. You can obviously pick $c_n$ so that this converges.
|
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|
3,041 |
That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French . This raises the question if a student, like me, should even bother reading EGA. There must nowadays be less time-consuming ways to absorb the "required knowledge" needed to do "serious" (non-arithmetic) algebraic geometry. I would like to hear what professional algebraic geometers would recommend their students in this matter.
|
I'm surprised, reading the various answers and comments to this question, how much support
there is for the idea of reading EGA. It is evidently a mathematical masterpiece of a certain kind, but I would never recommend it to a student to study. In response to a similar question asked on Terry Tao's blog, I posted the following advice : As to how much time to spend on EGA and SGA, this is something that you ultimately have to decide for yourself, hopefully with the guidance of your thesis adviser. But one thing to remember is that many very clever people have pored over the details of EGA and SGA for many years now, and it so it is going to be hard for anyone to find interesting new results that can be obtained just by applying the ideas from these sources alone (as important as those ideas are). Even if you want to make progress in a very general, abstract setting, you will need ideas to come from somewhere, motivated perhaps by some new phenomenon you observe in geometry, or number theory, or arithmetic, or … . This is part of the reason why I advise against spending too much time just holed up with EGA and SGA. By themselves, they are not likely (for most people) to provide the inspiration for new results. On the other hand, when you are trying to prove your theorems, you might well find techincal tools in them which are very helpful, so it is useful to have some sense of what is in them and what sort of tools they provide. But you will likely have to find your inspiration elsewhere. I think it is worth thinking about two of the most significant recent theorems in algebraic geometry: the deformation invariance of plurigenera for varieties of general type, proved by Siu, and the finite generation of the canoncial ring for varieties of general type, proved by Birkar, Cascini, Hacon, and McKernan, and independently by Siu. Siu's methods are analytic; one might summarize them as $L^2$ -methods. I'm not sure what text one would begin with to learn these methods. Certainly Griffiths and Harris for the very basics, but then ... ? The methods of BCHM are techniques of projective and birational geometry. A careful reading of Hartshorne, especially the last two chapters, would be a good preparation for entering the research literature in this subject, I think. It's not clear to me that one gains more essential background by reading EGA. An aside: there is much more to EGA than just handling non-Noetherian schemes, but the spectre of non-Noetherian situations seems to loom a little large over this discussion. Thus it seems worthwhile to mention that,
while non-Noetherian schemes arise naturally in certain contexts (as Kevin Buzzard noted in a comment on David Levahi's answer), I think these contexts are pretty uncommon unless one is doing a certain style of arithmetic geometry. Somewhat more generally, I don't think that flat descent should be the focus for most students when learning algebraic geometry.
It is a basic foundational technique, but I don't expect most interesting new work in algebraic geometry to occur in the foundations. (More generally still, this is probably a good summary for my case against spending time reading EGA.)
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|
3,103 |
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π 1 is trivial because Spec Z has no unramified étale covers, but what is known about the higher homotopy groups?
|
$Spec(\mathbb{Z})$ should only be considered as $S^3$ , if you "compactify" that is add the point at the real place. This is demonstrated by taking cohomology with compact support.
The étale homotopy type of $Spec(\mathbb{Z})$ is however contractible (indeed what do you get by removing a point form a sphere?) to see this (all results apper in Milne's Arithmetic Dualities Book) (let $X=Spec(\mathbb{Z})$ ) $H^r_c(X_{fl},\mathbb{G}_m)=H^r_{c}(X_{et},\mathbb{G}_m) = 0$ for $r \neq 3$ . $H^3_c(X_{fl},\mathbb{G}_m)=H^3_{c}(X_{et},\mathbb{G}_m) = \mathbb{Q}/\mathbb{Z}$ by 2+1, we have: $H^3_c(X_{fl},\mu_n)= \mathbb{Z}/n$ $H^r_c(X_{fl},\mu_n)= 0$ for $r \neq 3$ . since we have a duality $$H^r(X_{fl},\mathbb{Z}/n)\times H^{3-r}_c(X_{fl},\mu_n) \to \mathbb{Q}/\mathbb{Z} $$ we have $H^0(X_{fl},\mathbb{Z}/n) = H^0(X_{et},\mathbb{Z}/n) = \mathbb{Z}/n,$ $H^r(X_{fl},\mathbb{Z}/n) = H^r(X_{et},\mathbb{Z}/n) = 0$ , $r >0$ Now since $\pi_1$ is trivial we have by the Universal Coefficient Theorem, the Hurewicz Theorem and the profiniteness theorem for 'etale homotopy that all homotopy groups are zero.
|
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|
3,134 |
Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule . What's your favorite equation, formula, identity or inequality?
|
$e^{\pi i} + 1 = 0$
|
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|
3,165 |
If I remember correctly, I read that given a presheaf $P:\mathcal{C}^{op} \to Set$ , it is possible to describe it as a limit of representable presheaves. Could someone give a description of the construction together with a proof?
|
You mean "colimit of representable presheaves", not limit. Any limits that C has are preserved by the Yoneda embedding. So if C is, say, a complete poset like • → •, so that it is small and has all limits, you won't be able to produce any non-representable presheaves by taking limits of representable ones. The way to write any presheaf as a colimit of representables is, like all things Yoneda-related, somewhat tautological, and should be worked out for oneself; but anyways it's explained to some extent at this nlab page . Rather than write out formulas, I usually think of the example of simplicial sets: every simplicial set X can be formed as a colimit of its simplices, i.e., a diagram of representables which is indexed on the "category of simplices of X", whose objects are pairs (n, x) where n is in the indexing category and x is an object of X n . The same works in any presheaf category.
|
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"https://mathoverflow.net/questions/3165",
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|
3,184 |
The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning how it has deeper implications into how to think about representable functors. What are some examples of this? How should one think of the Yoneda Lemma?
|
One way to look at it is this: for $C$ a category, one wants to look at presheaves on $C$ as being "generalized objects modeled on $C$ " in the sense that these are objects that have a sensible rule for how to map objects of $C$ into them. You can "probe" them by test objects in $C$ . For that interpretation to be consistent, it must be true that some $X$ in $C$ regarded as just an object of $C$ or regarded as a generalized object is the same thing. Otherwise it is inconsistent to say that presheaves on $C$ are generalized objects on $C$ . The Yoneda lemma ensures precisely that this is the case. I wrote up a more detailed expository version of this story at motivation for sheaves, cohomology and higher stacks .
|
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"source": [
"https://mathoverflow.net/questions/3184",
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|
3,188 |
Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-meager. So Y must be dense. If Y has the Baire property, then it follows from Pettis Lemma that Y is open and hence closed (since the complement of Y is the union of translates of Y), contradicting Y being proper. Thus, Y must be dense and not have the Baire property. My question is: is there a Banach space X with a proper non-meager linear subspace Y? Such a Y must be dense and not have the Baire property. Any such Y must be difficult to construct since all Borel sets and even all continuous images of separable complete metric spaces have the Baire property. More info: 1. Meager is just another word for first category, i.e. the countable union of nowhere dense sets. 2. A set A in a topological space has the Baire property if for some open set V (possibly empty) the set (A-V)U(V-A) is meager. 3. The collection of sets with the Baire property form a sigma-algebra. All open sets trivially have the Baire property, thus all Borel sets have the Baire property. All analytic sets also have the Baire property. 4. Pettis Lemma: Let G be a topological group and let A be a non-meager subset of G with the Baire property. Then the set A*A^{-1} (element-wise multiplication) contains an open neighborhood of the identity. This is an analog to a similar theorem about Lebesgue measure: If A is a Lebesgue measurable subset of the reals with positive Lebesgue measure, then A - A (element-wise subtraction) contains an open set around 0.
|
I am afraid that Konstantin's accepted answer is seriously flawed. In fact, what seems to be proved in his answer is that $\ker f$ is of second category, whenever $f$ is a discontinuous linear functional on a Banach space $X$. This assertion has been known as Wilansky-Klee conjecture and has been disproved by Arias de Reyna under Martin's axiom (MA).
He has proved that, under (MA), in any separable Banach space there exists a discontinuous linear functional $f$ such that $\ker f$ is of first category.
There have been some subsequent generalizations, see Kakol et al. So, where is the gap in the above proof? It is implicitly assumed that $\ker f = \bigcup A_i$. Then $f$ is bounded on $B_i=A_i+[-i,i]z$. But in reality, we have only $\ker f \subset \bigcup A_i$ and we cannot conclude that $f$ is bounded on $B_i$. And finally, what is the answer to the OP's question? It should not be surprising (remember the conjecture of Klee and Wilansky) that the answer is:
in every infinite dimensional Banach space $X$ there exists a discontinuous linear form $f$ such that $\ker f$ is of second category. Indeed, let $(e_\gamma)_{\gamma \in \Gamma}$ be a normalized Hamel basis of $X$.
Let us split $\Gamma$ into countably many pairwise disjoint sets $\Gamma =\bigcup_{n=1}^\infty \Gamma_n$, each of them infinite. We put $X_n=span\{e_\gamma: \gamma \in \bigcup_{i=1}^n \Gamma_i\}$. It is clear (from the definition of Hamel basis) that $X=\bigcup X_n$. Therefore there exists $n$ such that $X_n$ is of second category. Finally, we define $f(e_\gamma)=0$ for every $\gamma \in \bigcup_{i=1}^n\Gamma_i$ and $f(e_{\gamma_k})=k$ for some sequence $(\gamma_k) \subset \Gamma_{n+1}$. We extend $f$ to be a linear functional on $X$. It is clearly unbounded, $f\neq 0$, and $X_n \subset \ker f$. Hence $\ker f\neq X$ is dense in $X$ and of second category in $X$.
|
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|
3,190 |
Suppose G is an algebraic group with an action G×X→X on a scheme. Does the fixed locus (the set of points x∈X fixed by all of G) have a scheme structure? You can obviously define the functor Fix(T)={t∈X(T)|t is fixed by every element of G(T)}. Is this functor always representable? (This question was "broken off" of a compound question of mine after Scott Carnahan answered the other part so wonderfully that I had to accept his answer.)
|
The question gives the "wrong" definition of Fix(T), hence the resulting confusion. A more natural definition of the subfunctor X^G of "G-fixed points in X" is (X^G)(T) = {x in X(T) | G_T-action on X_T fixes x} = {x in X(T) | G(T')-action on X(T') fixes x for all T-schemes T'}. (Of course, can just as well restriction to affine T and T' for "practical" purposes.) By way of analogy with more classical situations, if the base is a field k then a moment's reflection with the case of finite k shows that {x in X(k) | G(k) fixes x} is the "wrong" notion of (X^G)(k), whereas {x in X(k) | G-action on X fixes x} is a "better" notion, and is what the above definition of (X^G)(k) says. From this point of view, if (for simplicity of notation) the base scheme is an affine Spec(k) for a commutative ring k then the "scheme of G-fixed points" exists whenever G is affine and X is separated provided that k[G] is k-free (or becomes so after faithfully flat extension on k). So this works when k is a field, or any k if G is a k-torus (or "of multiplicative type"). See Proposition A.8.10(1) in the book Pseudo-reductive groups .
|
{
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|
3,204 |
It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I can't remember where I heard this) that the harmonic series is more or less the only important series with this property. What other methods besides analytic continuation and zeta regularization exist for summing divergent series? Do they work on the harmonic series? And are there other well-known series which also don't have obvious regularizations?
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One common regularization method that wasn't mentioned in the Everything Seminar post is to take the constant term of a meromorphic continuation. While the Riemann zeta function has a simple pole at 1, the constant term of the Laurent series expansion is the Euler-Mascheroni constant gamma = 0.5772156649... It is reasonable to claim that most divergent series don't have interesting or natural regularizations, but you could also reasonably claim that most divergent series aren't interesting. Any function with extremely rapid growth (e.g., the Busy Beaver function) is unlikely to have a sum that is regularizable in a natural way.
|
{
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|
3,237 |
I'd like to learn to read math articles in Japanese or Chinese, but I am not interested in learning these languages from usual textbooks. Exist suitable texts, specialized for the needs for reading mathematics? What do you suggest? I look for something similar to "Russian for the mathematician" , which was very usefull when I was interested in some russian articles. In the language books I know, most of the vocabulary is irrelevant for reading mathematics, but needed terminology is missing. A collection of mathematical vocabulary and training texts with translation would be usefull. I know good books, e.g. Bowring "An introduction to modern Japanese" or Lewin "Textlehrbuch der japanischen Sprache" and could read articles about history or humanities after having read them, but not mathematics (resulting in forgetting the language by lack of training). Edit: F. Orgogozo's dictionary . (BTW, giving the direct link did not work, app. jap./chin. characters not accepted within a url by the MO-software) Edit: Zagier's dictionary .
|
Here are a few for chinese: Commercial Press Staff. English-Chinese Dictionary of Mathematical Terms. New York: French & European Publications, Incorporated, 1980. De Francis, John F. Chinese-English Glossary of the Mathematical Sciences. Reprint. Ann Arbor, MI: Books on Demand. Dictionary of Mathematics. New York: French & European Publications, Incorporated, 1974. He Xiuhuang. A Glossary of Logical Terms. Hong Kong: Chinese University Press, 1982. Science Press Staff. English - Chinese Mathematical Dictionary. Second Edition. New York: French & European Publications, Incorporated, 1989. Science Press Staff. Chinese-English Mathematical Dictionary. New York: French & European Publications, Incorporated, 1990. Science Press Staff. New Russian - Chinese Dictionary of Mathematical Terms. New York: French & European Publications, Incorporated, 1988. Silverman, Alan S. Handbook of Chinese for Mathematicians. Berkeley, CA: University of California, Institute of East Asian Studies, 1970. Source: here I have never read any of these books, and I honestly doubt it that they have all the mathematical terms (especially in higher more sophisticated fields). Don't expect to be able to write "diffeomorphism between manifolds" in chinese or japanese immediately. I suggest you take a look at these references in your public library and get one that helps you the most. To be honest, I am also interested I have several chinese papers I really want to read. I would first try anything with the latest jedict/edict/cedict, and then try something else like the above references.
|
{
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|
3,239 |
There are many proofs based on a "tertium non datur"-approach (e.g. prove that there exist two irrational numbers a and b such that a^b is rational). But according to Gödel's First Incompleteness Theorem, where he provides a constructive example of a contingent proposition, which is neither deductively (syntactically) true nor false, we know that there can be a tertium. My question: Are all proofs that are based on that principle useless since now we know that a tertium can exist?
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You are confused. The best way out of your confusion is to maintain a very careful distinction between strings of formal symbols and their mathematical meanings. Godel's theorem is, on its most primitive level, a theorem about which strings of formal symbols can be obtained from other strings by certain formal manipulations. These formal manipulations are called proofs, and the strings which are obtainable in this way are called theorems. For clarity, I'll call them formal proofs and formal theorems. In particular, let G be a string such that G is not a formal theorem and neither is NOT(G). It is still true that G OR NOT(G) is a formal theorem. Moreover, if G IMPLIES H and NOT(G) IMPLIES H are both formal theorems, then H will be a formal theorem; because there are rules of formal manipulation that allow you to take the first two strings and produce the third. I believe that Douglass Hofstader discusses this in a fair bit of detail when he goes over Godel's theorem. The above is mathematics. Next, some philosophy. I don't find it helpful to say that G is neither true nor false. It find it more helpful to say that our systems of formal symbols and formal manipulation rules can describe more than one system. For example, Euclid's first four axioms can describe both Euclidean and non-Euclidean geometries. This doesn't mean that Euclid's fifth postulate has some bizarre third state between truth and falsehood. It means that there are many different universes (the technical term is models) described by the first four axioms, and the fifth postulate is true in some and false in others. However, in any particular one of those universes, either the fifth postulate is true or it is false. Thus, if we prove some theorem on the hypothesis that the fifth postulate holds, and also that the fifth postulate does not hold, then we have shown that this theorem holds in every one of those universes. There are fields of mathematical logic, called constructivist, where the law of the excluded middle does not hold. As far as I understand, that issue is not related to Godel's theorem.
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|
3,269 |
Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (8n + 7)$ : first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that $a$ is the sum of three rational squares. Then something magical happens: Let $C$ denote the circle $x^2 + y^2 + z^2 = a$. We are given a rational point $p$ on this circle. Round the coordinates of $p$ to the closest integer point $q$, then draw the line through $p$ and $q$, which intersects $C$ at a rational point $p'$. Round the coordinates of $p'$ to the closest integer point $q'$, and repeat this process. A straightforward calculation shows that the least common multiple of the denominators of the points $p'$, $p''$, ... are strictly decreasing, so this process terminates at an integer point on $C$. Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply? Edit: Let me suggest a possible reformulation of the question as follows. Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________.
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A few days ago Serre told me about some modest improvements to the proof, based on Weil's book Number theory: an approach through history from Hammurapi to Legendre and on a 1998 letter from Deligne to Serre; I will paraphrase these below. According to Weil (p. 292), the ``magical'' argument is due to an amateur mathematican: L. Aubry, Sphinxe-Oedipe 7 (1912), 81--84. Here is a generalization that allows for a clearer proof. Lemma: Let $f = f_2+f_1+f_0 \in \mathbf{Z}[x_1,\ldots,x_n]$, where $f_i$ is homogeneous of degree $i$. Suppose that for every $x \in \mathbf{Q}^n-\mathbf{Z}^n$, there exists $y \in \mathbf{Z}^n$ such that $0<|f_2(x-y)|<1$. If $f$ has a zero in $\mathbf{Q}^n$, then it has a zero in $\mathbf{Z}^n$. Proof: If $x=(x_1,\ldots,x_n) \in \mathbf{Q}^n$, let $\operatorname{den}(x)$ denote the lcm of the denominators of the $x_i$. By iteration, the following claim suffices: If $x \in \mathbf{Q}^n - \mathbf{Z}^n$ and $y \in \mathbf{Z}^n$ satisfy $0<|f_2(x-y)|<1$, and the line $L$ through $x$ and $y$ intersects $f=0$ in $x,x'$, then $\operatorname{den}(x')<\operatorname{den}(x)$. By an affine change of variable over $\mathbf{Z}$, we may assume that $y$ is $0$ and that $L$ is the $x_1$-axis. By restricting to $L$, we reduce to proving the following: given $f(t)=At^2+Bt+C \in \mathbf{Z}[t]$ with zeros $x,x' \in \mathbf{Q}$ such that $0<|Ax^2|<1$, we have $\operatorname{den}(x')<\operatorname{den}(x)$. Proof: Factor $f$ over $\mathbf{Z}$ as $E(Dt-N)(D't-N')$ with $x=N/D$ and $x'=N'/D'$ in lowest terms. Then $0<|Ax^2|<1$ implies $0<|A|<D^2$. On the other hand, $DD'$ divides $EDD'=A$, so $DD' \le |A| < D^2$. Hence $D'<D$.
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|
3,278 |
Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large . One of many examples of a large category is Set , for Russell's paradox reasons. A category C is locally small if the class of morphisms between any two of its objects is a set. Of course, a small category is necessarily locally small. The converse is not true, as Set is a counterexample. Now, I can construct categories that are not locally small. However, what's the most common or most reasonable such category?
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The category of multi-spans spans (thanks to everyone below for correcting my terminology). The objects are sets, and a map from $A$ to $B$ is a set $X$ equipped with a map $X → A × B$. The composition of $X → A × B$ and $Y → B × C$ is $X ×_B Y → A × C$. I am stealing notation from algebraic geometry here: $X ×_B Y$ is the limit of the diagram $X → B ← Y$. Admittedly, I've never wanted to allow $X$ to be an arbitrary set. I usually want it to be something like a finite set, a finite simplicial complex or a scheme of finite type. But it is certainly natural to define the category without any restrictions.
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|
3,283 |
I've recently started my personal wiki to organize my notes and thoughts. I use the wiki program instiki which I believe is the same as the n-lab uses. Instiki can upload svg 's. I want to be able to create nice looking pictures of some (not to complicated) geometric objects, e.g. knots, pair of pants, etc. My question is twofold. Do people draw these things using svg format? For example people who do diagram algebra, do you use svg format? If so, what are some good free/open source programs for creating these pictures?
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I am a huge fan of the open source program Inkscape . I mostly use it to produce pictures for my papers in the eps format, but its native format is svg.
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|
3,446 |
The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, why should this be enough information to recover the group? And does this work for other base fields (or rings?)?
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A very nice and very general version of Tannakian formalism is in Jacob Lurie's paper,
Tannaka duality for geometric stacks, arXiv:math/0412266 . I like to think of Tannaka duality as recovering a scheme or stack from its category of coherent (or quasicoherent) sheaves, considered as a tensor category.
From this POV the intuition is quite clear: having a faithful fiber functor to Vect (or more generally to R-modules) means your stack is covered (in the flat sense) by a point (or by Spec R). This is why you get (if you're over an alg closed field) that having a faithful fiber functor to Vect_k means you're sheaves on the quotient BG of a point by some group G, i.e. Rep G.. Over a more general base, you only locally look like a quotient of Spec R (or Spec k for k non-alg closed) by a group ---- ie you're a BG-bundle over Spec R, aka a G-gerbe. Even more generally, the kind of Tannakian theorem Jacob explains basically says that any stack with affine diagonal can be recovered from its tensor category of quasicoherent sheaves.. Actually the construction of the stack from the tensor category is just a version of the usual functor Spec from rings to schemes. Recall that as a functor, Spec R (k) = homomorphisms from R to k. So given a tensor category C let's define Spec C as the
stack with functor of points Spec C(k) = tensor functors from C to k-modules (for any ring k, or algebra over the ground field etc). The Tannakian theorems then say for X reasonable
(ie a quasicompact stack with affine diagonal), we have X= Spec Quasicoh(X) ---
so X is "affine in a quasicoherent-sheaf sense". Again, the usual Tannakian story is the case X=BG or more generally a G-gerbe.
|
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|
3,448 |
If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?
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John Jones has computed tables of number fields of low degree with prescribed ramification . Though the tables just list the defining polynomials and the set of ramified primes, and not any other invariants, it's not hard to search them to find, e.g., that the three quartic fields obtained by adjoining a root of $x^4 - 6$, $x^4 - 24$, and $x^4 - 12x^2 - 16x + 12$ respectively all have degree $4$, class number $1$, and discriminant $-2^{11} \cdot 3^3$. On the other hand these three fields are non-isomorphic (e.g. the regulators distinguish them, the splitting fields distinguish them...).
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|
3,455 |
For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication -
$$(f\cdot g)(x) = f(x)g(x),$$ and convolution -
$$(f*g)(x) = \int_{y+z=x}f(y)g(z)$$ Both operations define commutative ring structures (possibly without identity) with the usual addition. (For that to make sense, we have to find a subset of functions that is closed under addition, multiplication, and convolution. If $G$ is finite, this is not an issue, and if G is compact, we can consider infinitely differentiable functions, and if $G$ is $\mathbb R^d$, we can consider the Schwarz class of infinitely differentiable functions that decay at infinity faster than all polynomials, etc. As long as our class of functions doesn't satisfy any additional nontrivial algebraic identities, it doesn't matter what it is precisely.) My question is simply: do these two commutative ring structures satisfy any additional nontrivial identities? A "trivial" identity is just one that's a consequence of properties mentioned above: e. g., we have the identity
$$f*(g\cdot h) = (h\cdot g)*f,$$ but that follows from the fact that multiplication and convolution are separately commutative semigroup operations. Edit: to clarify, an "algebraic identity" here must be of the form $A(f_1, ..., f_n) = B(f_1, ..., f_n)$," where $A$ and $B$ are composed of the following operations: addition negation additive identity (0) multiplication convolution (Technically, a more correct phrasing would be "for all $f_1, ..., f_n$: $A(f_1, ..., f_n) = B(f_1, ..., f_n)$," but the universal quantifier is always implied.) While it's true that the Fourier transform exchanges convolution and multiplication, that doesn't give valid identities unless you could somehow write the Fourier transform as a composition of the above operations, since I'm not giving you the Fourier transform as a primitive operation. Edit 2 : Apparently the above is still pretty confusing. This question is about identities in the sense of universal algebra. I think what I'm really asking for is the variety generated by the set of abelian groups endowed with the above five operations. Is it different from the variety of algebras with 5 operations (binary operations +, *, .; unary operation -; nullary operation 0) determined by identities saying that $(+, -, 0, *)$ and $(+, -, 0, \cdot)$ are commutative ring structures?
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I think the answer to the original question (i.e. are there any universal algebraic identities relating convolution and multiplication over arbitrary groups, beyond the "obvious" ones?) is negative, though establishing it rigorously is going to be tremendously tedious. There are a couple steps involved. To avoid technicalities let's restrict attention to discrete finite fields G (so we can use linear algebra), and assume the characteristic of G is very large. Firstly, given any purported convolution/multiplication identity relating a bunch of functions, one can use homogeneity and decompose that identity into homogeneous identities, in which each function appears the same number of times in each term. (For instance, if one has an identity involving both cubic expressions of a function f and quadratic expressions of f, one can separate into a cubic identity and a quadratic identity.) So without loss of generality one can restrict attention to homogeneous identities. Next, by depolarisation , one should be able to reduce further to the case of multilinear identities: identities involving a bunch of functions $f_1, f_2, ..., f_n$, with each term being linear in each of the functions. (I haven't checked this carefully but it should be true, especially since we can permit the functions to be complex valued.) It is convenient just to consider evaluation of these identities at the single point 0 (i.e. scalar identities rather than functional identities). One can actually view functional identities as scalar identities after convolving (or taking inner products of) the functional identity with one additional test function. Now (after using the distributive law as many times as necessary), each term in the multilinear identity consists of some sequence of applications of the pointwise product and convolution operations (no addition or subtraction), evaluated at zero, and then multiplied by a scalar constant. When one expands all of that, what one gets is a sum (in the discrete case) of the tensor product $f_1 \otimes ... \otimes f_n$ of all the functions over some subspace of $G^n$. The exact subspace is given by the precise manner in which the pointwise product and convolution operators are applied. The only way a universal identity can hold, then, is if the weighted sum of the indicator functions of these subspaces (counting multiplicity) vanishes. (Note that finite linear combinations of tensor products span the space of all functions on $G^n$ when $G$ is finite.)
But when the characteristic of $G$ is large enough, the only way that can happen is if each subspace appears in the identity with a net weight of zero. (Indeed, look at a subspace of maximal dimension in the identity; for $G$ large enough characteristic, it contains points that will not be covered by any other subspace in the identity, and so the only way the identity can hold is if the net weight of that subspace is zero. Now remove all terms involving this subspace and iterate.) So the final thing to do is to show that a given subspace can arise in two different ways from multiplication and convolution only in the "obvious" manner, i.e. by exploiting associativity of pointwise multiplication and of convolution. This looks doable by an induction argument but I haven't tried to push it through.
|
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|
3,477 |
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion. Background I think I understand the basic idea on $\mathbb{R}^n$ , so for readers who know as little as I do, I will provide some ideas. Any differential operator on $\mathbb{R}^n$ is (uniquely) of the form $\sum p_{i_1,\dotsc,i_k}(x)\frac{\partial^k}{\partial x_{i_1}\dots\partial x_{i_k}}$ , where $x_1,\dotsc,x_n$ are the canonical coordinate functions on $\mathbb{R}^n$ , the $p_{i_1,\dotsc,i_k}(x)$ are smooth functions, and the sum ranges over (finitely many) possible indexes (of varying length). Then the symbol of such an operator is $\sum p_{i_1,\dotsc,i_k}(x)\xi^{i_1}\dotso\xi^{i_k}$ , where $\xi^1,\dotsc,\xi^n$ are new variables; the symbol is a polynomial in the variables $\{\xi^1,\dotsc,\xi^n\}$ with coefficients in the algebra of smooth functions on $\mathbb{R}^n$ . Ok, great. So symbols are well-defined for $\mathbb{R}^n$ . But most spaces are not $\mathbb{R}^n$ — most spaces are formed by gluing together copies of (open sets in) $\mathbb{R}^n$ along smooth maps. So what happens to symbols under changes of coordinates? An affine change of coordinates is a map $y_j(x)=a_j+\sum_jY_j^ix_i$ for some vector $(a_1,\dotsc,a_n)$ and some invertible matrix $Y$ . It's straightforward to describe how the differential operators change under such a transformation, and thus how their symbols transform. In fact, you can forget about the fact that indices range $1,\dotsc,n$ , and think of them as keeping track of tensor contraction; then everything transforms as tensors under affine coordinate changes, e.g. the variables $\xi^i$ transform as coordinates on the cotangent bundle. On the other hand, consider the operator $D = \frac{\partial^2}{\partial x^2}$ on $\mathbb{R}$ , with symbol $\xi^2$ ; and consider the change of coordinates $y = f(x)$ . By the chain rule, the operator $D$ transforms to $(f'(y))^2\frac{\partial^2}{\partial y^2} + f''(y) \frac{\partial}{\partial y}$ , with symbol $(f'(y))^2\psi^2 + f''(y)\psi$ . In particular, the symbol did not transform as a function on the cotangent space. Which is to say that I don't actually understand where the symbol of a differential operator lives in a coordinate-free way. Why I care One reason I care is because I'm interested in quantum mechanics. If the symbol of a differential operator on a space $X$ were canonically a function on the cotangent space $T^\ast X$ , then the inverse of this Symbol map would determine a "quantization" of the functions on $T^\ast X$ , corresponding to the QP quantization of $\mathbb{R}^n$ . But the main reason I was thinking about this is from Lie algebras. I'd like to understand the following proof of the PBW theorem: Let $L$ be a Lie algebra over $\mathbb{R}$ or $\mathbb{C}$ , $G$ a group integrating the Lie algebra, $\mathrm{U}L$ the universal enveloping algebra of $L$ and $\mathrm{S}L$ the symmetric algebra of the vector space $L$ . Then $\mathrm{U}L$ is naturally the space of left-invariant differential operators on $G$ , and $\mathrm{S}L$ is naturally the space of symbols of left-invariant differential operators on $G$ . Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism $\mathrm{U}L\to\mathrm{S}L$ .
|
The principal symbol of a differential operator
$\sum_{|\alpha| \leq m} a_\alpha(x) \partial_x^\alpha$
is by definition the function $\sum_{|\alpha| = m} a_\alpha(x) (i\xi)^\alpha$
Here $\alpha$ is a multi-index (so $\partial_x^\alpha$ denotes $\alpha_1$ derivatives
with respect to $x_1$, etc.)
At this point, the vector $\xi = (\xi_1, \ldots, \xi_n)$ is merely a formal variable.
The power of this definition is that if one interprets $(x,\xi)$ as variables in the
cotangent bundle in the usual way -- i.e. $x$ is any local coordinate chart, then
$\xi$ is the linear coordinate in each tangent space using the basis $dx^1, \ldots, dx^n$,
then the principal symbol is an invariantly defined function on $T^*X$, where $X$ is
the manifold on which the operator is initially defined, which is homogeneous of degree
$m$ in the cotangent variables. Here is a more invariant way of defining it: fix $(x_0,\xi_0)$ to be any point in $T^*X$
and choose a function $\phi(x)$ so that $d\phi(x_0) = \xi_0$. If $L$ is the differential operator,
then $L( e^{i\lambda \phi})$ is some complicated sum of derivatives of $\phi$, multiplied together, but always with a common factor of $e^{i\lambda \phi}$. The `top order part' is the one which has a $\lambda^m$, and if we take only this, then its coefficient has only first derivatives of $\phi$ (lower order powers of $\lambda$ can be multiplied by higher derivatives of $\phi$).
Hence if we take the limit as $\lambda \to \infty$ of $\lambda^{-m} L( e^{i\lambda \phi})$
and evaluate at $x = x_0$, we get something which turns out to be exactly the principal symbol of $L$ at the point $(x_0, \xi_0)$. There are many reasons the principal symbol is useful. There is indeed a `quantization map'
which takes a principal symbol to any operator of the correct order which has this as its principal symbol. This is not well defined, but is if we mod out by operators of one order lower. Hence the comment in a previous reply about this being an isomorphism between filtered algebras. In special situations, e.g. on a Riemannian manifold where one has preferred coordinate
charts (Riemann normal coordinates), one can define a total symbol in an invariant fashion
(albeit depending on the metric). There are also other ways to take the symbol, e.g. corresponding to the Weyl quantization, but that's another story. In microlocal analysis, the symbol captures some very strong properties of the operator $L$.
For example, $L$ is called elliptic if and only if the symbol is invertible (whenever $\xi \neq 0$). We can even talk about the operator being elliptic in certain directions if the principal symbol is nonvanishing in an open cone (in the $\xi$ variables) about those directions. Another interesting story is wave propagation: the characteristic set of the operator is the set of $(x,\xi)$ where the principal symbol $p(L)$ vanishes. If its differential (as a function on the cotangent bundle) is nonvanishing there, then the integral curves of the Hamiltonian flow associated to $p(L)$, i.e. for the Hamiltonian vector field determined by $p(L)$ using the standard symplectic structure on $T^*X$, ``carries'' the singularities of solutions of $Lu = 0$. This is the generalization of the classical fact that singularities of solutions of the wave equation propagate along light rays.
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|
3,512 |
In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA. What journals play a similar role in other subjects? Let me be more specific. Suppose that I'm an analyst (or a representation theorist, or a number theorist, etc.) and I've written a paper that I judge as being not quite good enough for a top journal like the Annals or Inventiones or Duke, but still very good. If I want to be ambitious, where would I submit it? Since the answer will depend on the subject, I marked this "community wiki".
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The following is my personal (i.e., includes all of my mathematical prejudices) ranked list of subject area journals in number theory. From best to worst: 1) Algebra and Number Theory 2) International Journal of Number Theory 3) Journal de Theorie des Nombres de Bordeaux 4) Journal of Number Theory 5) Acta Arithmetica 6) Integers: The Journal of Combinatorial Number Theory 7) Journal of Integer Sequences 8) JP Journal of Algebra and Number Theory For a slightly longer list, see http://www.numbertheory.org/ntw/N6.html but I don't have any personal experience with the journals listed there but not above. Moreover, I think 1) is clearly the best (a very good journal), then 2)-5) are of roughly similar quality (all quite solid), then 6) and 7) have some nice papers and also some papers which I find not so interesting, novel and/or correct; I have not seen an interesting paper published in 8). But I don't think that even 1) is as prestigious as the top subject journals in certain other areas, e.g. JDG or GAFA. There are some other excellent journals which, although not subject area journals, seem to be rather partial to number theory, e.g. Crelle, Math. Annalen, Compositio Math. Finally, as far as analytic and combinatorial number theory goes, I think 4) and 5) should be reversed. (Were I an analytic number theorist, this would have caused me to rank 5) higher than 4) overall.)
|
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|
3,524 |
Start with A an abelian category and form the derived category D(A).
Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the triangle is distinguished it is exact. How about the converse: if the long sequence in cohomology is exact does it follow that the triangle is distinguished? (My guess is no, but I can't find a counter-example).
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An important property of the derived category is that distinguished triangles don't just produce long exact sequences in cohomology. If A -> B -> C -> A[1] is an exact triangle and E is another object in the derived category, then you get a long exact sequence ... Hom(A[1],E) -> Hom(C,E) -> Hom(B,E) -> Hom(A,E) -> Hom(C[-1],E) -> ... where these Hom-sets are sets of maps in the derived category. A particular counterexample is as follows. We can view any abelian group as a chain complex concentrated in degree zero. There is a distinguished triangle as follows: ℤ -> ℤ -> ℤ/2 -> ℤ[1] However, we can take the last map ℤ/2 -> ℤ[1] in the sequence (which is not zero in the derived category) and replace it with the zero map. This still gives us a long exact sequence on (co)homology groups. However, if we let E = ℤ/2, then applying maps in the derived category from our new non-distinguished triangle gives us the sequence ... 0 -> Hom(ℤ/2,ℤ) -> Hom(ℤ,ℤ) -> Hom(ℤ,ℤ) -> Ext(ℤ/2,ℤ) -> 0 ... where the last map is induced by the zero map from ℤ/2[-1] to ℤ, and so it must be zero. This sequence can't possibly be exact, and so the new triangle is not distinguished. EDIT : A more subtle question this suggests is: "Suppose I have a triangle and, for any E, applying Map(-,E) or Map(E,-) gives a long exact sequence. Is this a distinguished triangle?" The answer to this is actually still no. Still considering chain complexes of abelian groups, take the distinguished triangle ℤ -> ℤ -> ℤ/3 -> ℤ[1] where I'll call the last map β (for Bockstein). You can take this distinguished triangle and replace β with its negative -β. The new triangle still induces long exact sequences on maps in or maps out (because the maps in the triangle have the same kernel and image). However, as an exercise show that this can't be a disntinguished triangle because it's not isomorphic to the original distinguished triangle.
|
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|
3,525 |
If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. Does anyone know an example of two different probability distributions with identical moments? The less pathological the better. Edit: Is it unconditionally true if I specialize to discrete distributions? And a related question: Suppose I ask the same question about Renyi entropies. Recall that the Renyi entropy is defined for all $a \geq 0$ by $$ H_a(p) = \frac{\log \left( \sum_j p_j^a \right)}{1-a} $$ You can define $a = 0, 1, \infty$ by taking suitable limits of this formula. Are two distributions with identical Renyi entropies (for all values of the parameter $a$) actually equal? How "rigid" is this result? If I allow two Renyi entropies of distributions $p$ and $q$ to differ by at most some small $\epsilon$ independent of $a$, then can I put an upper bound on, say, $||p-q||_1$ in terms of $\epsilon$? What can be said in the case of discrete distributions?
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Roughly speaking, if the sequence of moments doesn't grow too quickly, then the distribution is determined by its moments. One sufficient condition is that if the moment generating function of a random variable has positive radius of convergence, then that random variable is determined by its moments. See Billingsley, Probability and Measure , chapter 30. A standard example of two distinct distributions with the same moment is based on the lognormal distribution: $$f_0(x) = \sqrt{2 \pi} x^{-1} e^{-(\log x)^2 /2}$$ which is the density of the lognormal, and the perturbed version $$f_a(x) = f_0(x)(1 + a \sin(2 \pi \log x))$$ These have the same moments; namely the nth moment of each of these is exp(n 2 /2). A condition for a distribution over the reals to be determined by its moments is that lim sup k → ∞ (μ 2k ) 1/2k /2k is finite, where μ 2k is the (2k)th moment of the distribution. For a distribution supported on the positive reals, lim sup k → ∞ (μ k ) 1/2k /2k being finite suffices. This example is from Rick Durrett, Probability: Theory and Examples , 3rd edition, pp. 106-107; as the original source for the lognormal Durrett cites C. C. Heyde (1963) On a property of the lognormal distribution, J. Royal. Stat. Soc. B. 29, 392-393.
|
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|
3,540 |
Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth category?
|
There are such spaces, for example $X = S^2 \times \mathbb{R}P^3$, $Y = S^3 \times \mathbb{R}P^2.$
(These are both smooth and CW-complexes.) Whitehead's Theorem says that for CW-complexes, if a map $f : X \to Y$ induces an isomorphism on all homotopy groups then it is a homotopy equivalence. But, as the example above shows,
you need the map. Such a map is called a weak homotopy equivalence. (Whitehead's Theorem is not true for spaces wilder than CW-complexes. The Warsaw circle
has all of its homotopy groups trivial but the unique map to a point is not a homotopy equivalence.)
|
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|
3,551 |
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far as intuition goes, I might as well add "...of characteristic 0" at the end of that. I know the complex numbers from kindergarten algebra, so I have a fairly good idea of how at least one algebraically closed field of characteristic 0 looks and feels. And while I don't have nearly the same handle on the field of algebraic numbers, I can pretty much do arithmetic in it, so that's two examples. But I've never ( really ) seen an algebraically closed field of characteristic p > 0! I can build one just fine, and if you put a gun to my head I could probably even do some arithmetic in it, but there's no intuition of the sort that you get with the complex numbers. So: does anyone know of an intuitive description of such a field, that it's possible to get a real sense of in the same way as C?
|
It's certainly not too hard to understand everything there is to understand about the algebraic closure of Fp. Perhaps the reason this is unsatisfying as an example for founding intuition is because it doesn't really have a nice topological structure; it lacks anything like a natural metric. So here's an attempt to explain why what is in some sense the next simplest example puts you in a better situation, intuition-wise. If you have some intuition about the p-adic numbers look and feel (for example, topologically), then you secretly have intuition for the t-adic topology on the complete local field K=Fp((1/t)). Now, as far as characteristic p fields go, this sort of puts you in the position of (in your parlance) a "preschooler" who knows about R but hasn't yet gotten to kindergarten to learn about C. Why is K like R? First, it is locally compact. Second, it is at least analogous to completing Fp(t), which is very much like Q with Fp[t] as the analogue of Z, at an "infinite" valuation, namely the degree or (1/t)-adic valuation, rather than a "finite" place like a prime polynomial in Fp[t]. (The (1/t)-adic valuation corresponds to the point at infinity on the projective line over F_p. Likewise, number theorists love to say, perhaps partly to annoy John Conway, that the real and complex absolute values on Q correspond to "archimedean primes" or "primes dividing infinity". This is actually a pretty lame analogy, though, since K=Fp((1/t)) looks a lot more like Fp((t)), say, than R or C looks like Q2.) Unfortunately there are two extra difficulties in the characteristic p case. First, upon passing to the algebraic closure L of K we lose completeness. Second, we make an infinite field extension, unlike the degree 2 extension C/R. Thus, while L is an algebraically closed field of characteristic p, it bears little resemblance to R. In fact, it's a lot more like an algebraically closed field of characteristic 0 that is a bit scarier (at least to me) than C, namely Cp, or what you get when you complete the algebraic closure of Qp with respect to the topology coming from the unique extension of the p-adic valuation.
While this may seem bad, I think it's actually good, because one can really get a handle on some of the properties of Cp. [Note that as another answerer pointed out, Cp = C as a field, but not as a topological or valued field, which is really a more interesting structure to consider from the viewpoint of intuition anyway.] For example of some similarities, miraculously Cp turns out to still be algebraically closed, and I believe the same proof goes through for L above. Another property L and Cp share is that in addition to "geometric" field extensions K'/K obtained by considering function fields of plane curves over Fp, there are also "stupider" extensions coming from extending the coefficient field. This is like passing to unramified extensions of p-adic fields, where one ramps up the residue field. (In fact, it's exactly the same thing.) Both L and Cp are complete valued fields with residue field the algebraic closure of Fp. (But the valuation is NOT discrete; it takes values in Q.) There are some dangerous bends to watch out for topologically, however. Some cursory googling tells me that Cp is not locally compact, although it is topologically separable. In addition, positive characteristic inevitably brings along the problem of inseparable field extensions sitting side L. This is, of course, an aspect where L/K is unlike Cp/Qp. Notwithstanding such annoyances, I would argue that the picture sketched above actually does give an example of an algebraically closed field of characteristic p for which it is possible to have some real intuition.
|
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|
3,557 |
The two that come to mind are splitting epics in Set and taking the Skel of a category. Surely there are lots of other interesting (and maybe upsetting) places where this comes up.
|
Using the usual definition of "functor," almost any functor
constructed using only universal properties requires the axiom of
choice. For instance, if a category C has products, then one wants a
"product assigning" functor C×C → C, but in order to define this you
have to choose a product for each pair of objects. If C is a large
category, then you need an axiom of choice for proper classes. However, this sort of thing is arguably not a "real" use of the axiom
of choice. It's more accurate to say that in the absence of the axiom
of choice the usual definition of "functor" is not sufficient, and one
must use anafunctors instead. Proving that fully faithful + essentially surjective =
equivalence is the same. Most often in category theory when we want
to "choose" something, that thing is in fact determined up to unique
isomorphism (though not uniquely on the nose) and in that case using
anafunctors is sufficient to avoid choice. The axiom of choice does, however, come up in the study of particular
properties of the category Set. One interesting consequence of the
fact that epics split in Set is that all functors defined on Set
preserve epics . I think this is an important part of Blass' proof
that the existence of nontrivial (left and right) exact endofunctors
of set is equivalent to the existence of measurable cardinals.
Another interesting consequence is that Set is its own "ex/lex
completion."
|
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|
3,559 |
As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as answers so they may be voted up and down with the rest. This is a community-wiki question: One colloquial statement and its mathematical meaning per answer please!
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A drunk man will find his way home, but a drunk bird may get lost forever. This encodes the fact that a 2-dimensional random walk is recurrent (appropriately defined for either the discrete or continuous case) whereas in higher dimensions random walks are not. More details can be found for instance in this enjoyable blog post by Michael Lugo.
|
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|
3,591 |
It is well-known that many great mathematicians were prodigies. Were there any great mathematicians who started off later in life?
|
Joan Birman went back to grad school in math in her forties, and is now one of the top researchers in knot theory.
|
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|
3,596 |
It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 mod 4 and its number of divisors which are congruent to 3 mod 4. Alternatively, there are no such representations if the prime factorization of N contains any prime of form 4k+3 an odd number of times. If the prime factorization of N contains all such primes an even number of times, then we have r 2 (N) = 4(b 1 + 1)(b 2 + 1)...(b r +1) where b 1 , ..., b r are the exponents of the primes congruent to 1 mod 4 in the factorization of N. For example, 325 = 5 2 × 13 can be written in 4(2+1)(1+1) = 24 ways as a sum of squares. These are 18 2 + 1 2 , 17 2 + 6 2 , 15 2 + 10 2 , and the representations obtained from these by changing signs and/or permuting. Is there an analogous formula in the three-square case? I know that an integer can be written as the sum of three squares if and only if it is not of the form 4 m (8n+7). There is a simple argument that shows that the number of ways to write all integers up to N as a sum of three squares is asymptotically 4πN 3/2 /3 -- representations of an integer less than N as a sum of three squares can be identified with points in the ball in R 3 centered at the origin with radius N 1/2 . Differentiating, a "typical" integer near N should have about 2πN 1/2 representations as a sum of three squares. From playing around with some data it looks like lim n → ∞ #{ k ≤ n and r 3 (k)/k 1/2 ≤ x} / n might be a nonzero constant. That is, for each positive real x, the probability that a random integer k can be written in no more than x k 1/2 ways approaches some constant in the open interval (0, 1) as k → ∞. One way to prove this (if it is in fact true) would be if there were some formula for r 3 (k), in terms of the prime factorization, which is why I'm curious. (I apologize if this is something that is well-known to number theorists, although I'd appreciate a pointer if it is. I am not a number theorist, I just play around with this sort of thing every so often and generate amusing conjectures.)
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I just can't stop myself from putting up the following, from the MOTD on the Berkeley server: Oct 2: Warning: Due to a known bug, the default Linux document viewer
evince prints N*N copies of a PDF file when N copies requested.
As a workaround, use Adobe Reader acroread for printing multiple
copies of PDF documents, or use the fact that every natural number
is a sum of at most four squares.
|
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|
3,656 |
Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old at the time) seemed to know what he was talking about and has had a not too unsuccessful career since. So my (quite connected) questions are 1) Why do so few books use this approach ( I only know Massey's) ? 2) What are the pro's and con's of both approaches ? 3) Does it matter ? After all the homology groups obtained are the same.
|
Others have mentioned the advantages to cubical sets and so I don't want to say much on those; I'll just mention some facts about the other direction. The main disadvantage that comes solely from the point of view of homology is that you have this irritating normalization procedure: you have to take the cubical chains on X and take the quotient by degenerate cubes. (You can do the same for simplicial chains and you get the same answer as the original.) The cubical theory gets niceties that others have stated. The main advantages of simplices are mostly apparent when you move a little further along into homotopy theory. The homotopy theory of simplicial sets is in some senses simply easier than the cubical analogue. For example, the cartesian product of two simplicial sets has as geometric realization the product of the geometric realizations of its factors. On the other hand, the standard "cubical interval" I, which realizes to [0,1], has a self-product I^2, whose geometric realization has fundamental group ℤ. Simplices in some ways play more nicely with degeneracies than cubes do. Simplices are also more closely tied to categories via the simplicial nerve functor. For example, there are the simplicial constructions of classifying spaces coming from categories, and these give you nice constructions of group cohomology. (Perhaps I just don't know the cubical analogues.) There is also the ubiquitous "bar construction" which is unreasonably useful in algebraic topology, and which comes most naturally from a simplicial construction. We use this to resolve modules, show equivalences between algebras over different operads, and more applications that are almost too numerous to name. For example, May used it quite heavily in his proofs that spaces X which are algebras over an E ∞ -operad have infinitely many deloopings BX, B 2 X, ... There has been a lot of work done on cubical sets, however, but I'm not the best authority on it. Try here for a starting point.
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|
3,764 |
Let f be a complex-valued function of one real variable, continuous and compactly supported. Can it have a Fourier transform that is not Lebesgue integrable?
|
Here's a sketch of what I think is an example of the sort you want. Consider a trapezoidal function T δ , supported on [-1,1], which is 1 on [-1+δ , 1-δ ] and is defined on the remaining intervals by interpolation in the obvious way. Then as δ tends to zero, the Fourier transform of T δ is going to tend to infinity in the L 1 (R)-norm -- I can't remember the details of the proof, but since T δ is a linear combination of Fourier transforms of Féjer kernels one can probably do a fairly direct computation. Of course, the supremum norm of each T δ is always 1. So the idea is to now stack scaled copies of these together, so as to obtain a function on [-1,1] which will be continuous (by uniform convergence) but whose Fourier transform is not integrable because its the limit of things with increasing L 1 -norm. To be a little more precise: suppose that for each n we can find δ(n) such that T δ(n) has a Fourier transform with L 1 -norm equal to n 2 3 n . Put S m = Σ j=1 m m -2 T δ(m) and note that the sequence (S m ) converges uniformly to a continuous function S which is supported on [-1,1]. The Fourier transform of S certainly makes sense as an L 2 function. On the other hand, the L 1 -norm of the Fourier transform of S m is bounded below by 3 m - (3 m-1 + ... + 3 + 1) ~ 3 m /2 which suggests that the Fourier transform of S ought to have infinite L 1 -norm -- at the moment lack of sleep prevents me from remembering how to finish this off. Alternatively, one could argue as follows. Consider the Banach space C of all continuous functions on [-1,1] which vanish at the endpoints, equipped with the supremum norm. If the Fourier transform mapped C into L 1 , then by an application of the closed graph theorem it would have to do so continuously, and hence boundedly. That means there would exists a constant M >0, such that the Fourier transform of every norm-one function in C has L 1 -norm at most M. But the functions T δ show this is impossible.
|
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|
3,776 |
Many well-known theorems have lots of "different" proofs. Often new proofs of a theorem arise surprisingly from other branches of mathematics than the theorem itself. When are two proofs really the same proof? What I mean is this. Suppose two different proofs of theorem are first presented formally and then expanded out so that the formal proofs are presented starting from first principles, that is, starting from the axioms. Then in some sense two proofs are the same if there are trivial operations on the sequence of steps of the first formal proof to transform that proof into the second formal proof. (I'm not sure what I mean by "trivial")
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You've hit on an area of research that's picking up some momentum at the moment. It involves connections between proof theory, homotopy theory and higher categories. The idea is that a proof or deduction is something like a path (from the premiss to the conclusion), and when you "deform" one proof into another by a sequence of trivial steps, that's something like a homotopy between paths. Or, in the language of higher-dimensional categories, a deduction is a 1-morphism, and a deformation of deductions is a 2-morphism. You can keep going to higher deductions. There are close connections with type theory too. If you have the right kind of background, the following papers might be helpful: Awodey and Warren, Homotopy theoretic models of identity types, http://arxiv.org/abs/0709.0248 Van den Berg and Garner, Types are weak omega-groupoids, http://arxiv.org/abs/0812.0298
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|
3,787 |
Has anyone seen the new MSC 2010? I was browsing around and to my suprise there is another revision of MSC. Has anyone noticed any major changes in there? Do major journals already accept papers with MSC 2010 classification?
|
You've hit on an area of research that's picking up some momentum at the moment. It involves connections between proof theory, homotopy theory and higher categories. The idea is that a proof or deduction is something like a path (from the premiss to the conclusion), and when you "deform" one proof into another by a sequence of trivial steps, that's something like a homotopy between paths. Or, in the language of higher-dimensional categories, a deduction is a 1-morphism, and a deformation of deductions is a 2-morphism. You can keep going to higher deductions. There are close connections with type theory too. If you have the right kind of background, the following papers might be helpful: Awodey and Warren, Homotopy theoretic models of identity types, http://arxiv.org/abs/0709.0248 Van den Berg and Garner, Types are weak omega-groupoids, http://arxiv.org/abs/0812.0298
|
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|
3,819 |
Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
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Well, real-valued analytic functions are just as rigid as their complex-valued counterparts. The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic. As Qiaochu says, one answer is elliptic regularity: complex differentiable functions obey a non-trivial equation (the Cauchy-Riemann equation) which implies a integral representation (the Cauchy integral formula) which then implies analyticity (Taylor expansion of the Cauchy kernel); the ellipticity of the Cauchy-Riemann equation is what gives the analyticity of its fundamental solution, the Cauchy kernel. Real differentiable functions obey no such equation. Another approach is via Cauchy's theorem. In both the real and complex setting, differentiability implies that the integral over a closed (or more precisely, exact) contour is zero. But in the real case this conclusion has trivial content because all closed contours are degenerate in one (topological) dimension. In the complex case we have non-trivial closed contours, and this makes all the difference. EDIT: Actually, the above two answers are basically equivalent; the latter is basically the integral form of the former (Morera's theorem). Also, to be truly nitpicky, "differentiable" should be "continuously differentiable" in the above discussion.
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|
3,820 |
I asked a related question on this mathoverflow thread . That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered. So quoting myself from that thread: How hard is it to compute the number of prime factors of a given integer? This can't be as hard as factoring, since you already know this value for semi-primes, and this information doesn't seem to help at all. Also, determining whether the number of prime factors is 1 or greater than 1 can be done efficiently using Primality Testing.
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There is a folklore observation that if one was able to quickly count the number of prime factors of an integer n, then one would likely be able to quickly factor n completely. So the counting-prime-factors problem is believed to have comparable difficulty to factoring itself. The reason for this is that we expect any factoring-type algorithm that works over the integers, to also work over other number fields (ignoring for now the issue of unique factorisation, which in principle can be understood using class field theory). Thus, for instance, we should also be able to count the number of prime factors over the Gaussian integers, which would eventually reveal how many of the (rational) prime factors of the original number n were equal to 1 mod 4 or to 3 mod 4. Using more and more number fields (but one should only need polylog(n) such fields) and using various reciprocity relations, one would get more and more congruence relations on the various prime factors, and pretty soon one should be able to use the Chinese remainder theorem to pin down the prime factors completely. More generally, the moment one has a way of extracting even one non-trivial useful bit of information about the factors of a number, it is likely that one can vary this procedure over various number fields (or by changing other parameters, e.g. twisting everything by a Dirichlet character) and soon extract out enough bits of information to pin down the factors completely. The hard part is to first get that one useful bit... [EDIT: The above principle seems to have one exception, namely the parity bit of various number-theoretic functions. For instance, in the (now stalled) Polymath4 project to find primes, we found a quick way to compute the parity of the prime counting function $\pi(x)$, but it has proven stubbornly difficult to perturb this parity bit computation to find other useful pieces of information about this prime counting function.]
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|
3,858 |
Can someone indicate me a good introductory text on geometric group theory?
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de la Harpe's book is quite nice and has an amazing bibliography, but it doesn't really prove any deep theorems (though it certainly discusses them!). Some other sources. 1) Bridson and Haefliger's book "Metric Spaces of Non-Positive Curvature". Very easy to read and covers a lot of ground. 2) Ghys and de la Harpe's book on hyperbolic groups. Another classic, but in French. If you look around the web, you can find English translations. 3) Cannon's survey "Geometric Group Theory" in the Handbook of Geometric Topology is very nice. 4) Bowditch's survey "A course on geometric group theory" is also very nice. 5) Bridson has written two beautiful surveys entitled "Non-Positive Curvature in Group Theory" and "The Geometry of the Word Problem". The latter was one of the first things I read in any depth. 6) Geoghegan's "Topological Methods in Group Theory" is very nice, with a more topological approach. 7) Mike Davis's "The Geometry and Topology of Coxeter Groups" is a bit specific, but covers a lot of important material in a nice way. 8) John Meier's book "Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups" is well-written and pretty gentle.
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3,871 |
For a compact space $K$ , the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions vanishing at that point). Now take $K=\mathbb{R}$ . Is there a useful characterization of the set of maximal ideals of $C(\mathbb{R})$ , the ring of continuous functions on $\mathbb{R}$ ? Note that I'm not imposing any boundedness conditions at infinity (if one does, I think the answer has to do with the Stone–Čech compactification of $\mathbb{R}$ — but I can't say I'm totally clear on that part either). Is this ring too large to allow a reasonable description of its maximal ideals?
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Peter Johnstone's book Stone Spaces (p. 144) proves that for any X, maximal ideals in $C(X)$ are the same as maximal ideals in $C_b(X)$ (bounded functions), i.e. the Stone-Cech compactification $\beta X$ . Indeed, if I is a maximal ideal, let Z(I) be the set of all zero sets of elements of I; this is a filter on the lattice of all closed sets that are zero sets of functions. Then I is contained in J(Z(I)), the set of functions whose zero sets are in Z(I), so by maximality they are equal. But also, by maximality, Z(I) must be a maximal filter on the lattice of zero sets, and we get a bijection between maximal filters of zero sets and maximal ideals in $C(X)$ . Now the exact same discussion applies to $C_b(X)$ to give a bijection between maximal filters of zero sets and maximal ideals of $C(X)$ (since the possible zero sets of bounded functions are the same as the possible zero sets of all functions). But the maximal ideals of $C_b(X)$ are just $\beta X$ . The difference between $C_b(X)$ and $C(X)$ is that for $C_b(X)$ , the residue fields for all of these maximal ideals are just $C$ , while for $C(X)$ you can get more exotic things. Indeed, if a maximal ideal in $C(X)$ has residue field $C$ , then every function on X must automatically extend continuously to the corresponding point of $\beta X$ . This can actually happen for noncompact X, e.g. the ordinal $\omega_1$ . Section IV.3 of Johnstone's book has a pretty thorough discussion of this stuff if you want more details.
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|
3,939 |
Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the characteristic polynomial of a non-negative integer matrix? (The motivation here is that I want $r_1, ... r_n$ to be the eigenvalues of a directed multigraph.) Edit: If that condition isn't strong enough, how about the additional condition that
$$\frac{1}{n} \sum_{d | n} \mu(d) p_{n/d}$$ is a non-negative integer for all d?
|
This question is completely answered, and the result is that the condition involving
the Moebius inversion you mention is both necessary and sufficient! See K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series , J. Amer. Math. Soc. 13 (2000) 773–806. https://doi.org/10.1090/S0894-0347-00-00342-8 This is really a remarkable and
beautiful theorem.
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|
3,951 |
I always have trouble memorizing theorems. Does anybody have any good tips?
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As far as possible, you should turn yourself into the kind of person who does not have to remember the theorem in question. To get to that stage, the best way I know is simply to attempt to prove the theorem yourself. If you've tried sufficiently hard at that and got stuck, then have a quick look at the proof -- just enough to find out what the point is that you are missing. That should give you an Aha! feeling that will make the step far easier to remember in the future than if you had just passively read it.
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|
3,973 |
What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post. Ex: Just to throw some ideas out there
Mathematical Physics (for math students, not for physics students)
Complexity Theory
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Programming. I think it varies a lot from department to department but some places seem to do a bad job of teaching programming and it can be a really important skill.
|
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|
4,011 |
Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with a well-ordering) which is "bi-translation invariant": a < b should imply cad < cbd. Does anyone know any examples? Totally ordered abelian groups are easy to come up with: any direct product of subgroups of the reals, with the lexicographic ordering, will do. Knowing some non-abelian ones would help reveal what aspects of totally ordered abelian groups really depend on them being abelian... Edit: Via Andy Putman's answer below, I found this great summary of results about ordered and bi-ordered groups (i.e. groups with bi-translation invariant orderings) on Dale Rolfsen's site: Lecture notes on Ordered Groups and Topology He shows numerous examples of non-abelian bi-orderable groups, including a bi-ordering (bi-translation invariant ordering) on the free group with two generators. As well, he mentions, due to Rhemtulla, that a left-orderable group is abelian iff every left-ordering is a bi-ordering, which I think really highlights the relationship between ordering and abelianity.
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This concept is usually called biorderability (there is also left- and right-orderability). There are many examples, such as free groups and surface groups. Most spectacularly, the pure braid groups are biorderable, while the full braid groups are left orderable but not biorderable. The left ordering on the braid groups is usually attributed to Dehornoy, though it was discovered even earlier by Thurston (but not published). Dale Rolfsen has several nice surveys of material related to this on his webpage here . In particular, there is the complete text of a nice book called "Why are braids orderable?" that he wrote with Patrick Dehornoy, Ivan Dynnikov, and Bert Wiest. I believe that a new and much expanded edition of this book was just published. EDIT 1 : I just found the website for the much-expanded version of Rolfsen et al's book here . EDIT 2 : Thurston's construction of a left-ordering on the braid groups (which, of course, uses hyperbolic geometry) is very beautiful. It is explained very nicely in the first few pages of the paper "Orderings of mapping class groups after Thurston" by Short and Wiest, which is available on the arXiv here . The intro sections of this paper also contain a brief but enlightening account of the general theory of group orderings. Also, I have not read it, but there is a book entitled "Orderable Groups" by Rehmtulla and Mura. However, it is from 1977 and will thus omit a lot of recent work.
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|
4,023 |
Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
|
Stephen Abbott, Understanding Analysis Strongly recommended to students who are ony getting to grips with abstraction in mathematics. Find a review here .
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|
4,062 |
Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\_{T,gp}(G,H)$ Theorem (SGA 3, expose XXIV, 7.3.1(a)): Suppose G is reductive. Then
$\textbf{Hom}_{S,gp}(G,H)$ is representable by a scheme. Can this fail if $G$ is not reductive? I worked out a few example with $G = \mathbb{G}_a$, but they were representable.
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$\operatorname{Hom}(\mathbb{G}_a, \mathbb{G}_m)$ is not representable. Let $R$ be a $\mathbb{Q}$ -algebra. I claim that $\operatorname{Hom}(\mathbb{G}_a, \mathbb{G}_m)(\operatorname{Spec} R)$ is {Nilpotent elements of $R$ }. Intuitively, all homs are of the form $x\mapsto e^{nx}$ with $n$ nilpotent. More precisely, the schemes underlying $\mathbb{G}_a$ and $\mathbb{G}_m$ are $\operatorname{Spec} R[x]$ and $\operatorname{Spec} R[y, y^{-1}]$ respectively. Any hom of schemes is of the form $y \mapsto \sum f_i x^i $ for some $f_i$ in $R$ . The condition that this be a hom of groups says that $\sum f_k (x_1+x_2)^k = (\sum f_i x_1^i)(\sum f_j x_2^j)$ . Expanding this, $f_{i+j}/(i+j)! = f_i/i! f_j/j!$ . So every hom is of the form $f_i = n^i/i!$ , and n must be nilpotent so that the sum will be finite. Now, let's see that this isn't representable. For any positive integer $k$ , let $R_k = C[t]/t^k$ . The map $x \mapsto e^{tx}$ is in $\operatorname{Hom}(\mathbb{G}_a, \mathbb{G}_m)(\operatorname{Spec} R_k)$ for every $k$ . However, if $R$ is the inverse limit of the $R_k$ , there is no corresponding map in $\operatorname{Hom}(\mathbb{G}_a, \mathbb{G}_m)(\operatorname{Spec} R)$ . So the functor is not representable.
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|
4,075 |
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed links. How would that be with Spec Z? Then, topologists have things like virtual 3-manifolds, has that analogies in arithmetics? Edit: New MFO report : "At the moment the topic of most active interaction between topologists and number theorists are quantum invariants of 3-manifolds and their asymptotics. This year’s meeting showed significant progress in the field." Edit: "What is the analogy of quantum invariants in arithmetic topology?", "If a prime number is a knot, what is a crossing?" asks this old report . An other such question:
Minhyong Kim stresses the special complexity of number theory: "To our present day understanding, number fields display exactly the kind of order ‘at the edge of chaos’ that arithmeticians find so tantalizing, and which might have repulsed Grothendieck." Probably a feeling of such a special complexity makes one initially interested in NT. Knot theory is an other case inducing a similar impression. Could both cases be connected by the analogy above? How could a precise description of such special complexity look like and would it cover both cases? Taking that analogy, I'm inclined to answer Minhyong's question with the contrast between low-dimensional (= messy) and high-dimensional (= harmonized) geometry. Then I wonder, if "harmonizing by increasing dimensions"-analogies in number theory or the Langlands program exist. Minhyong hints in a mail to "the study of moduli spaces of bundles over rings of integers and over three manifolds as possible common ground between the two situations". A google search produces an old article by Rapoport "Analogien zwischen den Modulräumen von Vektorbündeln und von Flaggen" (Analogies between moduli spaces of vector bundles and flags) (p. 24 here , MR ). There, Rapoport describes the cohomology of such analogous moduli spaces, inspired by a similarity of vector bundles on Riemann surfaces and filtered isocrystals from p-adic cohomologies, "beautifull areas of mathematics connected by entirely mysterious analogies". ( book by R., Orlik, Dat) As interesting as that sounds, I wonder if google's hint relates to the initial theme. What do you think about it? (And has the mystery Rapoport describes now been elucidated?) Edit: Lectures by Atiyah discussing the above analogies and induced questions of "quantum Weil conj.s" etc. This interesting essay by Gromov discusses the topic of "interestung structures" in a very general way. Acc. to him, "interesting structures" exist never in isolation, but only as "examples of structurally organized classes of structured objects", Z only because of e.g. algebraic integers as "surrounding" similar structures. That would fit to the guesses above, but not why numbers were perceived as esp. fascinating as early as greek antiquity, when the "surrounding structures" Gromov mentions were unknown. Perhaps Mochizuki has with his "inter-universal geometry" a kind of substitute in mind? Edit: Hidekazu Furusho : "Lots of analogies between algebraic number theory and 3-dimensional topology are suggested in arithmetic topology, however, as far as we know, no direct relationship seems to be known. Our attempt of this and subsequent papers is to give a direct one particularly between Galois groups and knots."
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The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields"
(arXiv:0107210): the author states the Poincare conjecture as "S 3 is the only closed 3-manifold with no unbranched covers." The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1. The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2. Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture. Edit: An article by B. Morin on Sikora's dictionary (and how it relates to Lichtenbaum's cohomology, p. 28): "he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology."
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4,125 |
Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of Dedekind domains, and I've been trying to understand the classical analogy between the two. As I understand it, $\operatorname{Spec} \mathcal{O}_k$ should be thought of as the curve / Riemann surface itself. Is there a good notion of integration in this setting? Is there any hope of recovering an analogue of the Cauchy integral formula? (If I am misunderstanding the point of the analogy or stretching it too far, please let me know.)
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For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $\operatorname{Frob}_x$ on the stalk of sheaf, the so-called naive local term. Note that $\operatorname{Frob}_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map
$$
H^{2d}_c(X,\mathbf{Q}_{\ell})\to\mathbf{Q}_{\ell}(-d),
$$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces. For the case of number fields, each closed point $v$ in $\operatorname{Spec} O_k$ still defines a "loop" $\operatorname{Frob}_v$ in $\pi_1(\operatorname{Spec} k)$ (let's allow ramified covers. One can take the image of $\operatorname{Frob}_v$ under $\pi_1(\operatorname{Spec} k)\to\pi_1(\operatorname{Spec} O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincaré duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused. So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
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4,214 |
Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech style (first by defining the Čech cohomology groups subordinate to an open cover, and then taking the direct limit of these groups over all covers). When exactly are these two definitions equivalent? I'm unhappy with the explanation given by Hartshorne. Are they the same for any paracompact Hausdorff space? Or a locally contractible space? And what is the relationship between these two sheaf cohomologies and singular cohomology? Any elaboration on this circle of ideas related to the relationship between all the different cohomology theories would be appreciated.
|
Dear Victoria: here is a summary of the main comparison results I know of between Grothendieck cohomology (which is usually just called cohomology and written $\newcommand{\F}{\mathcal F}H^i(X,\F)$ ) and other cohomologies. 1) If $X$ is locally contractible then the cohomology of a constant sheaf coincides with singular cohomology. [This is Eric's answer, but there is no need for his hypothesis that open subsets be acyclic] 2) Cartan's theorem: Given a topological space $X$ and a sheaf $\F$ , assume there exists a basis of open sets $\mathcal{U}$ , stable under finite intersections, such that the CECH cohomology groups for the sheaf $\F$ are trivial (in positive dimension) for every open $U$ in the basis: $H^i(U,\F)=0$ Then the Cech cohomology of $\F$ on $X$ coincides with (Grothendieck) cohomology 3) Leray's Theorem: Given a topological space $X$ and a sheaf $\F$ , assume that for some covering $(U_i)$ of $X$ we know that the (Grothendieck!) cohomology in positive dimensions of $\F$ vanishes on every finite intersection of the $U_i$ 's.
Then the cohomology of $\F$ is already calculated by the Cech cohomology OF THE COVERING $(U_i)$ : no need to pass to the inductive limit on all covers.
This contains Dinakar's favourite example of a quasi-coherent sheaf on a separated scheme covered by affines. 4) If $X$ is paracompact and Hausdorff, Cech cohomology coincides with Grothendieck cohomology for ALL SHEAVES If you think this is too nice to be true, you can check Théorème 5.10.1 in Godement's book cited below
[So Eric's remark that no matter how nice the space is, Cech cohomology would probably not coincide with derived functor cohomology for arbitrary sheaves turns out to be too pessimistic] 5) Cohomology can be calculated by taking sections of any acyclic resolution of the studied sheaf: you don't need to take an injective resolution. This contains De Rham's theorem that singular cohomology can be calculated with differential forms on manifolds. 6) If you study sheaves of non-abelian groups, Cech cohomology is convenient: for example vector bundles on $X$ ( a topological space or manifold or scheme or...) are parametrized by $H^1(X, GL_r)$ . I don't know if there is a description of sheaf cohomology for non-abelian sheaves in the derived functor style. Good references are a) A classic: Godement, Théorie des faisceaux (in French, alas) b) S.Ramanan, Global Calculus,AMS graduate Studies in Mahematics, volume 65.
(An amazingly lucid book, in the best Indian tradition.) c) Torsten Wedhorn's quite detailed on-line notes , which prove 1) above (Theorem 9.16, p.92) and much, much more. By the way, @Wedhorn is one of the two authors of a great book on algebraic geometry . d) Ciboratu, Proposition 2.1 and Voisin's Hodge Theory and Complex Algebraic Geometry I, Theorem 4.47, page 109 , which both also prove 1) above.
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4,224 |
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am investigating this with regard to finding the normalized graph cut under general convex constraints. Any pointers will be very helpful.
|
The problem of describing the possible eigenvalues of the sum of two hermitian matrices in terms of the spectra of the summands leads into deep waters. The most complete description was conjectured by Horn, and has now been proved by work of Knutson and Tao (and others?) - for a good discussion, see the Notices AMS article by those two authors Depending on what you want, there should be simpler results giving estimates on the eigenvalues of the sum. A book like Bhatia's Matrix Analysis might have some helpful material.
|
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4,235 |
I'm wondering what the relation of category theory to programming language theory is. I've been reading some books on category theory and topos theory, but if someone happens to know what the connections and could tell me it'd be very useful, as that would give me reason to continue this endeavor strongly, and know where to look. Motivation: I'm currently researching undergraduate/graduate mathematics education, specifically teaching programming to mathematics grads/undergrads. I'm toying with the idea that if I play to mathematicians strengths I can better instruct them on programming and they will be better programmers, and what they learn will be useful to them. I'm in the process (very early stages) of writing a textbook on the subject.
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The most immediately obvious relation to category theory is that we have a category consisting of types as objects and functions as arrows. We have identity functions and can compose functions with the usual axioms holding (with various caveats). That's just the starting point. One place where it starts getting deeper is when you consider polymorphic functions. A polymorphic function is essentially a family of functions, parameterised by types. Or categorically, a family of arrows, parameterised by objects. This is similar to what a natural transformation is. By introducing some reasonable restrictions we find that a large class of polymorphic functions are in fact natural transformations and lots of category theory now applies. The standard examples to give here are the free theorems . Category theory also meshes nicely with the notion of an 'interface' in programming. Category theory encourages us not to look at what an object is made of, but how it interacts with other objects, and itself. By separating an interface from an implementation a programmer doesn't need to know anything about the implementation. Similarly category theory encourages us to think about objects up to isomorphism - it doesn't precisely proclaim which sets our groups comprise, it just matters what the operations on our groups are. Category theory precisely captures this notion of interface. There is also a beautiful relationship between pure typed lambda calculus and cartesian closed categories (CCC). Any expression in the lambda calculus can be interpreted as the composition of the standard functions that come with a CCC: like the projection onto the factors of a product, or the evaluation of a function. So lambda expressions can be interpreted as applying to any CCC. In other words, lambda calculus is an internal language for CCCs. This is explicated in Lambek and Scott. This means for instance that the theory of CCCs is deeply embedded in Haskell, because Haskell is essentially pure typed lambda calculus with a bunch of extensions. Another example is the way structural recursion over recursive datatypes can be nicely described in terms of initial objects in categories of F-algebras. You can find some details here . And one last example: dualising (in the categorical sense) definitions turns out to be very useful in the programming languages world. For example, in the previous paragraph I mentioned structural recursion. Dualising this gives the notions of F-coalgebras and guarded recursion and leads to a nice way to work with 'infinite' data types such as streams. Working with streams is tricky because how do you guard against inadvertently trying to walk the entire length of a stream causing an infinite loop? The appropriate dual of structural recursion leads to a powerful way to deal with streams that is guaranteed to be well behaved. Bart Jacobs, for example, has many nice papers in this area.
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4,276 |
It has been many years since I first read Categories for the Working Mathematician, but I still have a question about one of the first exercises. Question 5 in section 1.3 asks you to find two different functors $\mathsf{T}: \mathsf{Groups} \to \mathsf{Groups}$ with object function $\mathsf{T}(\mathsf{G}) = \mathsf{G}$ for every group $\mathsf{G}$. I have played with this for a long time, and none of the obvious choices end up working. Was this a mistake on Mac Lane's part, or am I just missing something very obvious? If it turns out there are no "obvious" choices, does anyone have an idea of how to prove that there are not two such functors?
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This is an " evil " question, which deserves an evil answer. Pick your favorite pair of an object G 0 of Groups and a nontrivial automorphism φ of G 0 . Define the functor T : Groups → Groups by T(G) = G, T(f) = [φ ∘] f [∘ φ -1 ] where we compose with φ if the target of f is G 0 and compose with φ -1 if the domain of f is G 0 . This T is clearly different from the identity functor (it acts by φ on Hom(Z, G 0 ) = the underlying set of G 0 ). This answer isn't very satisfying though because T is naturally isomorphic to the identity. I don't know whether one can find an example where T is not naturally isomorphic to the identity.
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4,329 |
During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has pure imaginary roots. If you plot the roots of truncations of $e^x - 1$ (or check out the ready-made plots in this Mathematica notebook , now available as a PDF ) you can see lots of cool features. I'd like to know where they come from! I know there's a vast literature on polynomials, but I'm a total beginner, and I don't know where to start. Here are a few specific questions: The roots of a high-degree truncation seem to fall into two categories: roots that lie very close to the imaginary axis, and roots that lie on a C-shaped curve. (Another interpretation is that all of the roots lie on a curve, which has a very sharp kink near the imaginary axis.) Can you write down an equation for the curve? If you put the roots of a lot of consecutive truncations together on the same plot, you'll see definite "stripes" to the right of the imaginary axis. Once a stripe appears, each higher-degree truncation sticks another root onto the end, making the stripe grow outward. Can you write down equations for the stripes? If $k$ is odd, the truncation of degree $k$ has no nonzero real roots. If k is even, the truncation of degree $k$ has one nonzero real root. The location of this root depends almost linearly on $k$. Why is the dependence so close to linear? Does it get more linear as $k$ increases, or less? Can roots be given identities that persist across time? That is, as $k$ increases, can you point to a sequence of roots and say, "those are all the same individual, which was born at $k$ = so-forth, is following such-and-such trajectory, and will grow up to become the root ($2\pi i\cdot$ whatever) of $e^x - 1$"?
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I finally got around to googling this a bit, and I immediately came up with http://www.mai.liu.se/~halun/complex/taylor/ which describes the same phenomenon for the exponential function itself. Briefly, if P n is the Taylor polynomial of e x , then the zeroes of P n (nx) pile up on the curve |ze 1-z |=1, |z|≤1. They credit this discovery to Szegő (1924). See also the paper On the Zeros of the Taylor Polynomials Associated with the Exponential Function by Brian Conrey and Amit Ghosh (The American Mathematical Monthly, 95 , No. 6 (1988), pp. 528-533), http://www.jstor.org/stable/2322757 if you have JSTOR access.
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4,331 |
Is the wedge product of two harmonic forms on a compact Riemannian manifold harmonic? I'm looking for a counter-example that the textbooks say exists. I would like to see a counter example that is on a complex manifold, Ricci-flat (or Einstein) manifold or both, if it is at all possible. In general, I'm trying to understand the interaction between the wedge product, Hodge star and the Laplacian on forms and it's eigen-vectors, references will be much appreciated.
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It is easy to construct examples on Riemann surfaces of genus $>1$. Take any surface like this. Let $A$ and $B$ be two harmonic $1$-forms, that are not proportional. Then $A \wedge B$ is non-zero, but it vanishes at some point, since both $A$ and $B$ have zeros. At the same time a harmonic $2$-form on a Riemann surface is constant.
Explicit examples of $1$-forms on Riemann surfaces can be obtained as real parts of holomorphic $1$-forms. Note of course that the above example is complex, and Einstein just take the standard metric of curvature $-1$. If you want an example on a Ricci flat manifold you should take a $K3$ surface. It is complex and admits a Ricci flat metric. Now, its second cohomology has dimension $22$. Now it should be possible to find two anti-self-dual two-forms whose wedge product vanishes at one point on $K3$ but is not identically zero. This is because the dimension of the space of self dual forms is 19 which is big enought to get vanishing at one point
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4,347 |
The question is about the function f(x) so that f(f(x))=exp (x)-1. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://scottaaronson.com/blog/?p=263 The growth rate of the function f (as x goes to infinity) is larger than linear (linear means O(x)), polynomial (meaning exp (O(log x))), quasi-polynomial (meaning exp(exp O(log log x))) quasi-quasi-polynomial etc. On the other hand the function f is subexponential (even in the CS sense f(x)=exp (o(x))), subsubexponential (f(x)=exp exp (o(log x))) subsubsub exponential and so on. What can be said about f(x) and about other functions with such an intermediate growth behavior? Can such an intermediate growth behavior be represented by analytic functions? Is this function f(x) or other functions with such an intermediate growth relevant to any interesting mathematics? (It appears that quite a few interesting mathematicians and other scientists thought about this function/growth-rate.) Related MO questions: solving $f(f(x))=g(x)$ How to solve $f(f(x)) = \cos(x)$? Does the exponential function has a square root Closed form functions with half-exponential growth $f\circ f=g$ revisited The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees The functional equation $f(f(x))=x+f(x)^2$ Rational functions with a common iterate Smoothness in Ecalle's method for fractional iterates .
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Let me see if I can summarize the conversation so far. If we want $f(f(z)) = e^z+z-1$, then there will be a solution, analytic in a neighborhood of the real axis. See either fedja's Banach space argument, or my sketchier iteration argument . The previous report of numerical counter-examples were in error; they came from computing $(k! f_k)^{1/k}$ instead of $f_k^{1/k}$. We do not know whether this function is entire. If it is, then there must be some place on the circle of radius $R$ where it is larger than $e^R$. (See fedja's comment here.) If we want $f(f(z)) = e^z-1$, there is no solution, even in an $\epsilon$-ball around $0$. According to mathscinet, this is proved in a paper of Irvine Noel Baker , Zusammensetzungen ganzer Funktionen , Math. Z. 69 (1958), 121--163. However, there are two
half-iterates (or associated Fatou coordinates $\alpha(e^z - 1) = \alpha(z) + 1$) that are holomorphic with very large domains. One is holomorphic on the complex
numbers without the ray $\left[ 0,\infty \right)$ along the positive real axis, the other is holomorphic on the complex numbers
without the ray $\left(- \infty,0\right]$ along the negative real axis. And both have the formal power series of the half-iterate $f(z)$ as asymptotic
series at 0. If we want $f(f(z))=e^z$, there are analytic solutions in a neighborhood of the real line, but they are known not to be entire. I'll make this answer community wiki. What else have I left out of my summary? Here is a related MO question . The answers to the new question contain further interesting information. Let me mention here a link with many references on "iterative roots and fractional iterations" one particular link on the iterative square root of exp (x) is here . The following two links mentioned in the old blog discussion may be helpful http://www.math.niu.edu/~rusin/known-math/97/sqrt.exp (outdated link) http://www.math.niu.edu/~rusin/known-math/99/sqrt_exp (outdated link) http://web.archive.org/web/20140521065943/http://www.math.niu.edu/~rusin/known-math/97/sqrt.exp http://web.archive.org/web/20140521065943/http://www.math.niu.edu/~rusin/known-math/99/sqrt_exp
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4,394 |
A similar question reminds me: When giving talks, I often want to refer to the work of Henry Crapo. I have asked several mathematicians, and none of them were sure how to pronounce his last name. Any help?
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Gordon Royle is right, I'm living in La Vacquerie. The reference to GWU is not correct: that is the workplace of my colleague Bill Schmitt. The US pronunciation is indeed "cray-poe", but in France it tends to become "crah-poe".
Henry
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4,434 |
Let g(z) be an entire function of a complex variable z. Does there exist an entire function f(z) such that f(z+1)-f(z)=g(z)? As I learned several years back, the answer to this is apparently 'yes' , but I have not felt satisfied with the proof because it goes beyond my expertise. I tried to find f using the power series expansion of g, for that works when g is a polynomial. But the results of partial inversions kept diverging. Representing g as an integral via Cauchy's formula, and doing inversion inside the integration led to similar problems. Perhaps I am overly optimistic, but a question this elementary should have equally elementary solution. Is there such a solution? If not, is there a reason to expect that no simple and elementary solution should exist?
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It took me some time to find a solution that satisfies both requirements: a) If should be based on the power series expansion b) It should use no tools heavier than contour integration. So, let $g(z)=\sum a_ k z^k$. We know that $a_ k$ decay faster than any geometric progression. We want analytic functions $F_ k(z)$ such that $F_ k(z+1)-F_ k(z)=z^k$ and $F_ k(z)$ grows not faster than a geometric progression as a function of $k$ in every disk. Then $f=\sum a_ k F_ k$ is what we want. So, just choose any odd multiples $r_ k\in(k,k+2\pi)$ of $\pi$ and put
$$
F_ k(z)=\frac{k!}{2\pi i}\oint_ {|z|=r_ k}\frac {e^{z\zeta}}{e^\zeta-1}\frac{d\zeta}{\zeta^{k+1}}
$$ The key is that $|e^\zeta-1|\ge \frac 12$ on the circle and $r_ k^k\ge k!$, so $|F_ k(z)|\le 2e^{|z|(k+2\pi)}$ on the plane.
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4,442 |
I have come across a bit of folklore(?) which goes something like "given any finite sequence of numbers, there is more than one 'valid' way of continuing the sequence". For example see here . I would like to know if this is actually stated and proved rigorously, and if so, where can I find a statement and proof? EDIT: The first couple of answers reflect my concerns exactly. But to quote from the book review I linked to, Wittgenstein's Finite Rule Paradox implies that any finite sequence of numbers can be a continued in a variety of different ways - some natural, others unexpected and surprising but equally valid. I didn't use the terms "Wittgenstein's Finite Rule Paradox" and "Wittgenstein on rule-following" before as googling them turns up results which look more like philosophy and linguistics than mathematics. My background in logic is nonexistent, I'm looking for any logicians out there who may have seen this before.
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Since you mentioned Wittgenstein's paradox, I thought someone should explain what that is. Warning: As you suspected, what follows is really philosophy and not mathematics. Nevertheless, it seems worth clarifying what people mean by Wittgenstein's paradox so that you don't go chasing wild geese. In Wittgenstein's Philosophical Investigations , he makes an argument about "private languages" that Saul Kripke later interpreted in a certain way. The basic point is that it is very difficult, if not impossible, to pin down what a "rule" is. Imagine that you are trying to teach a Martian the syntactic rule, "append a 1 to the end of a string." The Martian looks puzzled so you give some examples: 0 $\to$ 01 101 $\to$ 1011 0010 $\to$ 00101 and so forth. The Martian seems to get the idea, and does a few examples to confirm with you. The first few examples look good, but then all of a sudden the Martian comes up with 1111111 $\to$ 111111110 Say what? Somehow it seems that the Martian hasn't gotten the rule after all. Or maybe the Martian has extrapolated from your examples to a different rule? How do you make sure you communicate the rule you intend? If you have previously already agreed on some basic rules then you can build on those to define new rules, but how do you get started? It's hard to get more basic than "append a 1." Perhaps you could try building a physical device that optically scans its input and writes a 1 next to it. But any physical device will eventually fail to implement your intended "append a 1" rule when it reaches a certain physical limit, so the device doesn't unambiguously communicate your intended rule to the Martian either. No matter how you slice it, it seems that you can't guarantee that you have communicated your rule to the Martian, since any finite amount of interaction is consistent with infinitely many rules. Once we see this, we could take a more radical step and wonder, maybe I'm the Martian. Maybe all these years I've been assuming that I know what people mean when they specify syntactic rules, but actually I've just been lucky and haven't discovered the discrepancy between my understanding of what "append a 1" means and what everyone else means by it. (Here you can get a glimpse of where Wittgenstein's term "private language" comes into the discussion.) Even more radically, we could wonder whether the notion of a "rule" is incoherent. Perhaps there really is no such thing as a "rule" in the sense of some unambiguous finite description of something that applies to an infinite number of cases. Anyway, it is not my intent to start a debate here on MO about this issue, about which many philosophical papers have been written. It is just to confirm that Wittgenstein isn't who you're looking for, if you want a mathematical answer to your question.
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4,454 |
Is there a natural measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or not? By the word 'natural' I am trying to exclude measures defined in terms of the characteristic function of the set of true sentences.
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It seems to me that the probability that a statement is provable and that it is undecidable should both be bounded away from 0, for any reasonable probability distribution. Let $C_n$ be the number of grammatical statements of length $n$. For any statement $S$, the statement $S$, or $1=1$ is a theorem. So the number of provable statements of length $n$ is bounded below by $C_{n-k}$, where $k$ is the number of characters needed to tag on "or $1=1$". On the other hand, let $G$ be an undecidable sentence, and $S$ any sentence. Then Either $S$ and $1 \neq 1$, or else $G$ is undecidable. So the number of undecidable sentences of length $n$ is bounded below by $C_{n-\ell}$, for some constant $\ell$. For any reasonable grammar, the ratios $C_{n-k}/C_n$ and $C_{n- \ell}/C_n$ should both be bounded away from 0. I am currently trying to figure out why my computation is seemingly incompatible with the paper of Calude and Jurgensen cited by Konrad . I suspect that the answer is hidden in the definition of prefix free, on page 4, but I am trouble understanding it. Any help?
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4,468 |
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ : For example : Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ? Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$ , star-shaped ? Thank you for your answers and proofs
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Ad question 1): Yes, all open star-shaped subsets of $\mathbb{R}^n$ are diffeomorphic to $\mathbb{R}^n$. This is surprisingly little-known and there is a proof due to Stefan Born.
You can find this (fairly complicated) proof in Dirk Ferus's course notes http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf page 154, Satz 237 [The notes are alas in German] Added December 30, 2009: My excellent colleague Erwann Aubry informs me that this result is also proved more simply on page 60 of Gonnord & Tosel's book "Calcul Différentiel", ellipses,1998. [This book is in French, and moreover published by "ellipses" a valiant little publisher, completely unknown outside of France because it caters to the idiosyncratic French academic system] Kudos to any reference in honest English, rather than exotic foreign languages :)
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4,499 |
What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening and playing music can stimulate mathematical thinking. Any particular style that someone finds helpful? Other things that help? (In case you haven't figured it out, I've been stuck on some things lately.)
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Some blocks are caused by desiring a mathematical goal but being unable to achieve it. Some are caused by losing your desire to do mathematics. I think the second type of block happens more often than most people admit. It needn't be cataclysmic: it might just show up in a feeling of being burnt-out and tired. You've lost your appetite, your energy, your fizz. Everything's a burden. When that happens, it can be useful to remember that you don't have to spend your life doing mathematics. However great a mathematician you are, it will make next to no difference to the world at large if you drop math and become a postman. Seriously: you could do something else. You'd survive, and you might be perfectly happy. Once you've really internalized that, the feeling of heaviness should lift. You're doing mathematics because you want to, not because you have to. It's not an obligation. And if your appetite for mathematics never returns, that's probably a signal that you should do something else. This might sound like depressing advice, but it shouldn't be. I find it freeing and energizing.
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4,547 |
There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are these two concepts somehow related? I think I read somewhere that the Hecke algebra with functions includes Iwahori-Hecke algebras, is that correct? Is there a good motivation for introducing and studying either type of algebras? How much is known about their representations?
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A Hecke algebra describes the most reasonable way to convolve functions or measures on a homogeneous space. Suppose that you have seen the definition of convolution of functions on a vector space, or on a discrete group --- the latter is just the group algebra of the group or some completion. Then how could you reasonably define convolution on a sphere? There is no rotationally invariant way to convolve a general $f$ with a general $g$. However, if $f$ is symmetric around a reference point, say the north pole, then you can define the convolution $f * g$, even if $g$ is arbitrary. This is the basic idea of the Hecke algebra. The $(n-1)$-sphere is the homogeneous space $SO(n)/SO(n-1)$. A function $g$ on the sphere is a function on left cosets. A function $f$ on the sphere which is symmetric about a reference point is a function on double cosets. If $H \subseteq G$ is any pair of compact groups, if $f$ is any continuous function on $H\backslash G/H$, and if $g$ is any continuous function on $G/H$, then their product in the continuous group algebra is well-defined on $G/H$. The functions on double cosets make an algebra, the Hecke algebra, and the functions on left cosets are a bimodule of the Hecke algebra and the parental group $G$. It is important for the same reasons that any other kind of convolution is important. A particular case studied by Hecke Iwahori and others from before quantum algebra was the finite group $GL(n,q)$ and the upper triangular subgroup $B$. This is "the" Hecke algebra; it turns out that it is one algebra with a parameter $q$. Or as Ben says, this generalizes to the Iwahori-Hecke algebra of an algebraic group $G$ with a Borel subgroup $B$. The other place that the Hecke algebra arises is as an interesting deformation of the symmetric group, or rather as a deformation of its group algebra. It has a parameter $q$ and you obtain the symmetric group when $q=1$. As I said, it is also the Hecke algebra of $GL(n,q)/GL(n,q)^+$, where $B = GL(n,q)^+$ is the Borel subgroup of upper-triangular matrices (all of them, not just the unipotent ones). There is a second motivation for the Hecke algebra that I should have mentioned: It immediately gives you a representation of the braid group, and this representation reasonably quickly leads to the Jones polynomial and even the HOMFLY polynomial. When the Jones and HOMFLY were first discovered, it was simply a remark that the braid group representation was through the same Hecke algebra as the convolutional Hecke algebra for $GL(n,q)/B$ (or equivalently $SL(n,q)/B$). Even so, it's a really good question to confirm this "coincidence", as Arminius asks in the comment. Particularly because it is now a fundamental and useful relation and not a coincidence at all. As Ben explains in his blog post, the first model of the Hecke algebra is important for the categorification of the second model. The coset space of $GL(n,q)/B$ consists of flags in $\mathbb{F}_q^n$, and you can see these more easily using projective geometry. When $n=2$, there is an identity double coset 1 and another double coset $T$. A flag is just a point in $\mathbb{P}^1$, and the action of $T$ is to replace the point by the formal sum of the other $q$ points. Thus you immediately get $T^2 = (q-1)T + q.$ When $n=3$, a flag is a point and a line containing it in $\mathbb{P}^2$. The two smallest double cosets other than the identity are $T_1$ and $T_2$. $T_1$ acts by moving the point in the line; $T_2$ acts by moving the line containing the point. A little geometry then gives you that $T_1T_2T_1$ and $T_2T_1T_2$ both yield one copy of the largest double coset and nothing else. Thus they are equal; this is the braid relation of the Hecke algebra. When $n \ge 3$, the Hecke algebra is given by these same local relations, which must still hold.
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4,561 |
Y.I. Manin mentions in a recent interview the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I thought that is codified in e.g. Lurie’s articles. But I read only his survey on elliptic cohomology and some standard articles on symmetric spectra. Taking the quoted remark as indicator for me having missed to notice something, I’d like to read what others think about that, esp. what the intuition on “brave new algebra” is. Edit:
In view of Rognes' transfer of Galois theory into the context of "brave new rings" and his conference last year, I wonder if themes discussed in Kato's article (e.g. reciprocity laws) have "brave new variants". Edit: I found Greenlees' introductions ( 1 , 2 ) and Vogt's "Introduction to Algebra over Brave New Rings" for getting an idea of the topological background very helpful.
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I hardly know how to begin to reduce this subject to some kind of intuitive ideas, but here are some thoughts: * Adding homotopy to algebra allows for generalizations of familiar algebraic notions. For instance, a topological commutative ring is a commutative ring object in the category of spaces; it has addition and multiplication maps which satisfy the usual axioms such as associativity and commutativity. But instead, one might instead merely require that associativity and commutativity hold "up to all possible homotopies" (and we'll think of the homotopies as part of the structure). (It is hard to give the flavor of this if you haven't seen a definition of this sort.) This gives one possible definition of a "brave new commutative ring". * What is really being generalized is not algebraic objects, but derived categories of algebraic objects. So if you have a brave new ring R, you don't really want to study the category of R-modules; rather, the proper object of study is the derived category of R-modules. If your ring R is an ordinary (cowardly old?) ring, then the derived category of R-modules is equivalent to the classical derived category of R. If you want to generalize some classical algebraic notion to the new setting, you usually first have to figure out how to describe it in terms of derived notions; this can be pretty non-trivial in some cases, if not impossible. (For instance, I don't think there's any good notion of a subring of a brave new ring.) * As for Manin's remarks: the codification of these things has being an ongoing process for at least 40 years. It seems we've only now reached the point where these ideas are escaping homotopy theory and into the broad stream of mathematics. It will probably take a little while longer before things are so well codified that brave new rings get introduced in the grade school algebra curriculum, so the process certainly isn't over yet!
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4,562 |
Many people know that there is a (3×3) nine lemma in category theory. There is also apparently a sixteen lemma, as used in a paper on the arXiv (see page 24). There might be a twenty-five lemma, as it's mentioned satirically on Wikipedia's nine lemma page. Are the 4×4 and 5×5 lemmas true? Is there an n×n lemma? How about even more generally, if I have an infinity × infinity commutative diagram with all columns and all but one row exact, is the last row exact too? For all of these, if they are true, what are their exact statements, and if they are false, what are counterexamples? Note : There are a few possibilities for what infinity × infinity means -- e.g., it could be Z × Z indexed or N × N indexed. Also, in the N × N case, there are some possibilities on which way arrows point and which row is concluded to be exact.
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I hardly know how to begin to reduce this subject to some kind of intuitive ideas, but here are some thoughts: * Adding homotopy to algebra allows for generalizations of familiar algebraic notions. For instance, a topological commutative ring is a commutative ring object in the category of spaces; it has addition and multiplication maps which satisfy the usual axioms such as associativity and commutativity. But instead, one might instead merely require that associativity and commutativity hold "up to all possible homotopies" (and we'll think of the homotopies as part of the structure). (It is hard to give the flavor of this if you haven't seen a definition of this sort.) This gives one possible definition of a "brave new commutative ring". * What is really being generalized is not algebraic objects, but derived categories of algebraic objects. So if you have a brave new ring R, you don't really want to study the category of R-modules; rather, the proper object of study is the derived category of R-modules. If your ring R is an ordinary (cowardly old?) ring, then the derived category of R-modules is equivalent to the classical derived category of R. If you want to generalize some classical algebraic notion to the new setting, you usually first have to figure out how to describe it in terms of derived notions; this can be pretty non-trivial in some cases, if not impossible. (For instance, I don't think there's any good notion of a subring of a brave new ring.) * As for Manin's remarks: the codification of these things has being an ongoing process for at least 40 years. It seems we've only now reached the point where these ideas are escaping homotopy theory and into the broad stream of mathematics. It will probably take a little while longer before things are so well codified that brave new rings get introduced in the grade school algebra curriculum, so the process certainly isn't over yet!
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|
4,596 |
It is well-known that A: The series of the reciprocals of the primes diverges My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers. Property A tells us that the primes are a rather fat subset of $\mathbb{N}$. Is there a way to define a topology on $\mathbb{N}$ such that every dense subset of $\mathbb{N}$ (under this topology) corresponds to a fat subset of the natural numbers? What do you think about this?
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Yes, it's possible. Define the closed sets to be the sets the sum of whose reciprocals converges, together with $\mathbb{N}$. This collection of subsets is closed under arbitrary intersection and finite union, so it does form the closed sets of a topology. A subset of $\mathbb{N}$ is dense in this topology if its closure is $\mathbb{N}$, in other words, if it is not contained in any smaller closed set -- in other words, if it is not contained in any set the sum of whose reciprocals converges. This is equivalent to the sum of its reciprocals not converging.
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|
4,640 |
Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to the copies of the sign representation. This definition gives the usual morphisms, but it does not account for the braiding of the tensor product, which I've asked about previously . Since the monoidal structure on $\text{Rep}(G)$ comes from the comultiplication on the group Hopf algebra, which in this case is $H = \mathbb{C}[\mathbb{Z}/2\mathbb{Z}]$, it seems reasonable to guess that the unusual tensor product structure comes from replacing the usual comultiplication, which is $\Delta g \mapsto g \otimes g$, with something else, or alternately from placing some extra structure on $H$. What extra structure accounts for the braiding?
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The answer is yes, but comultiplication is not what you change. The symmetrizer (or braiding as you call it) is given by an $R$-element in $H \otimes H$ that makes $H$ into a triangular Hopf algebra . (The Wikipedia article says quasitriangular; triangular means that plus that the modified switching map $\widetilde{R}$ is an involution.) The $R$ element is the correction to the usual switching map $x \otimes y \mapsto y \otimes x$. You can solve for it directly in terms of the usual description of the switching map on supervector spaces, and worry later about what axioms it satisfies. Let $1$ and $a$ be the group elements of $H$ and let $\epsilon$ and $\sigma$ be the dual vectors on $H$ which are the trivial and sign representations. Then
we want
$$(\epsilon \otimes \epsilon)(R) = 1 \quad (\epsilon \otimes \sigma)(R) = 1 \quad (\sigma \otimes \epsilon)(R) = 1 \quad (\sigma \otimes \sigma)(R) = -1,$$
because that is the correction in the modified switching map $x \otimes y \mapsto (-1)^{|x||y|} y \otimes x$. You can solve the linear system of equation to obtain
$$R = \frac{1 \otimes 1 + a \otimes 1 + 1 \otimes a - a \otimes a}{2}.$$
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4,648 |
Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the diagonal elements, but many mathematicians would never consider such a definition since it presupposes a choice of basis. For someone working on algorithms, however, this might be a very natural perspective. What are the advantages and disadvantages to choosing a specific basis? Are there any situations where the "right" proof requires choosing a basis? (I mean a proof with the most clarity and insight -- this is subjective, of course.) What about the opposite situation, where the right proof never picks a basis? Or is it the case that one can very generally argue that any proof done in one manner can be easily translated to the other setting? Are there examples of proofs where the only known proof relies on choosing a basis?
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One answer to your question is already hinted at in the question. At the level of algorithms, basis-independent vector spaces don't really exist. If you want to compute a linear map $L:V \to W$, then you're not really computing anything unless both $V$ and $W$ have a basis. This is a useful reminder in our area, quantum computation, that has come up in discussion with one of my students. In that context, a quantum algorithm might compute $L$ as a unitary operator between Hilbert spaces $V$ and $W$. But the Hilbert spaces have to be implemented in qubits, which then imply a computational basis. So again, nothing is being computed unless both Hilbert spaces have distinguished orthonormal bases. The reminder is perhaps more useful quantumly than classically, since serious quantum computers don't yet exist. On the other hand, when proving a basis-independent theorem, it is almost never enlightening (for me at least) to choose bases for vector spaces. The reason has to do with data typing: It is better to write formulas in such a way that the two sides of an incorrect equation are unlikely to even be of the same type. In algebra, there is a trend towards using bases as sparingly as possible. For instance, there is widespread use of direct sum decompositions and tensor decompositions as a way to have partial bases. I think that your question about examples of proofs can't have an explicit answer. No basis-independent result needs a basis, and yet all of them do. If you have a reason to break down and choose a basis, it means that the basis-independent formalism is incomplete. On the other hand, anything that is used to build that formalism (like the definition of determinant and trace and the fact that they are basis-independent) needs a basis. There is an exception to the point about algorithms. A symbolic mathematics package can have a category-theoretic layer in which vector spaces don't have bases. In fact, defining objects in categories is a big part of the interest in modern symbolic math packages such as Magma and SAGE.
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4,665 |
Fix an integer n. Can you find two finite CW-complexes X and Y which * are both n connected, * are not homotopy equivalent, yet * $\pi_q X \approx \pi_q Y$ for all $q$. In Are there two non-homotopy equivalent spaces with equal homotopy groups? some solutions are given with n=0 or 1. Along the same lines, you can get an example with n=3, as follows. If $F\to E\to B$ is a fiber bundle of connected spaces such that the inclusion $F\to E$ is null homotopic, then there is a weak equivalence $\Omega B\approx F\times \Omega E$. Thus two such fibrations with the same $F$ and $E$ have base spaces with isomorphic homotopy groups. Let $E=S^{4m-1}\times S^{4n-1}$. Think of the spheres as unit spheres in the quaternionic vector spaces $\mathbb{H}^m, \mathbb{H}^n$, so that the group of unit quarternions $S^3\subset \mathbb{H}$ acts freely on both. Quotienting out by the action on one factor or another, we get fibrations
$$ S^3 \to E \to \mathbb{HP}^{m-1} \times S^{4n-1},\qquad S^3\to E\to S^{4m-1}\times \mathbb{HP}^{n-1}.$$
The inclusions of the fibers are null homotopic if $m,n>1$, so the base spaces have the same homotopy groups and are 3-connected, but aren't homotopy equivalent if $m\neq n$. There aren't any n-connected lie groups (or even finite loop spaces) for $n\geq 3$, so you can't push this trick any further. Is there any way to approach this problem? Or reduce it to some well-known hard problem? (Note: the finiteness condition is crucial; without it, you can easily build examples using fibrations of Eilenberg-MacLane spaces, for instance.)
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Here is a method for constructing examples. If a fiber bundle $F \to E \to B$ has a section, the associated long exact sequence of homotopy groups splits, so the homotopy groups of $E$ are the same as for the product $F \times B$, at least when these spaces are simply-connected so the homotopy groups are all abelian. To apply this idea we need an example using highly-connected finite complexes $F$ and $B$ where $E$ is not homotopy equivalent to $F \times B$ and the bundle has a section. The simplest thing to try would be to take $F$ and $B$ to be high-dimensional spheres. A sphere bundle with a section can be constructed by taking the fiberwise one-point compactification $E^\bullet$ of a vector bundle $E$. (This is equivalent to taking the unit sphere bundle in the direct sum of $E$ with a trivial line bundle.) The set of points added in the compactification then gives a section at infinity. For the case that the base is a sphere $S^k$, take a vector bundle $E_f$ with clutching function $f : S^{k-1} \to SO(n)$ so the fibers are $n$-dimensional. Suppose that the compactification $E_f^\bullet$ is homotopy equivalent to the product $S^n \times S^k$. If $n > k$, such a homotopy equivalence can be deformed to a homotopy equivalence between the pairs $(E_f^\bullet,S^k)$ and $(S^n \times S^k,S^k)$ where $S^k$ is identified with the image of a section in both cases. The quotients with the images of the section collapsed to a point are then homotopy equivalent. Taking the section at infinity in $E_f^\bullet$, this says that the Thom space $T(E_f)$ is homotopy equivalent to the wedge $S^n \vee S^{n+k}$. It is a classical elementary fact that $T(E_f)$ is the mapping cone of the image $Jf$ of $f$ under the $J$ homomorphism $\pi_{k-1}SO(n) \to \pi_{n+k-1}S^n$. The $J$ homomorphism is known to be nontrivial when $k = 4i$ and $n\gg k>0$. Choosing $f$ so that $Jf$ is nontrivial, it follows that the mapping cone of $Jf$ is not homotopy equivalent to the wedge $S^n \vee S^{n+k}$, using the fact that a complex obtained by attaching an $(n+k)$-cell to $S^n$ is homotopy equivalent to $S^n \vee S^{n+k}$ only when the attaching map is nullhomotopic (an exercise). Thus we have a contradiction to the assumption that $E_f^\bullet$ was homotopy equivalent to the product $S^n \times S^k$. This gives the desired examples since $k$ can be arbitrarily large. These examples use Bott periodicity and nontriviality of the $J$ homomorphism, so they are not as elementary as one might wish. (One can use complex vector bundles and then complex Bott periodicity suffices, which is easier than in the real case.) Perhaps there are simpler examples. It might be interesting to find examples where just homology suffices to distinguish the two spaces.
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|
4,669 |
When people work with infinite sets, there are some who (with good reason) don't like to use the Axiom of Choice. This is defensible, since the axiom is independent of the other axioms of ZF set theory. When people work with finite sets, there are still some people who don't like to use the "finite Axiom of Choice" -- i.e., they don't like to pick out a distinguished element of a set, or a distinguished isomorphism between a set with $n$ elements and $\{0, 1, \ldots, n-1\}$ (without some algorithm to pick it, that is). This is still an aesthetically defensible position, since oftentimes proofs that proceed that way don't give as much insight as proofs that don't use finite choice. But ZF set theory allows us to do this for finite sets! Is there a general framework in which we can disallow, if we so choose, the method of distinguished element? I have a hunch that the fact that, even for a finite-dimensional vector space $V$, $V$ and $V^\ast$ aren't naturally isomorphic is the first step towards the "right answer," but I don't really see where to go from there. (To be clear, as Ilya points out, I'm referring primarily to set theory; I know that/how category theory tells us about the non-naturality of the dual vector space isomorphism, in particular. My question is, is there something that either subsumes this or parallels it for more combinatorial constructions?)
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You might want topos theory. A topos is something like the category of sets, but the internal logic of a general topos is much weaker than ZF; it need not even be Boolean. An example of a topos is the category of sheaves of sets on a topological space. You can use topos theory to turn voodoo-sounding statements of constructive mathematics into ordinary mathematical statements which you can understand: for example "a nonempty set S might not have any elements" becomes "a sheaf S which is not the empty sheaf might not have any global sections" (or possibly "might not have local sections everywhere"), which is of course true. A canonical reference on this subject is Mac Lane & Moerdijk, Sheaves in Geometry and Logic.
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|
4,775 |
I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ where $\epsilon$ is about $0.75 \times 10^{-12}$. Call the integer here $N$, so $R = N - \epsilon$. So $R^2 = N^2 - 2N\epsilon + \epsilon^2$. It turns out that $N\epsilon$ is itself nearly an integer, namely $196884$, and so $R^2$ is again an almost-integer. More precisely, $$j(\tau) = 1/q + 744 + 196884q + 21493760q^2 + O(q^3)$$ where $q = \exp(2\pi i\tau)$. For $\tau = (1+\sqrt{-163})/2$, and hence $q = \exp(-\pi\sqrt{163})$, it's known that the left-hand side is an integer. Squaring both sides, $$j(\tau)^2 = 1/q^2 + 1488/q + 974304 + 335950912q + O(q^2).$$ To show that $1/q^2$ is nearly an integer, we can rearrange a bit to get $$j(\tau)^2 - 1/q^2 - 974304 = 1488/q + 335950912q + O(q^2)$$ and we want the left-hand side to be nearly zero. $1488/q$ is nearly an integer since $1/q$ is nearly an integer; since q is small the higher-order terms on the right-hand side are small. As noted by Mark Thomas in this question , $R^5$ is also very close to an integer -- but as I pointed out, that integer is not $N^5$. This isn't special to fifth powers. $R$, $R^2$, $R^3$, $R^4$, $R^5$, $R^6$, respectively differ from the nearest integer by less than $10^{-12}$, $10^{-9}$, $10^{-8}$, $10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-2}$. But the method of proof outlined above doesn't work for higher powers, since the coefficients of the $q$-expansion of $j(\tau)^5$ (for example) grow too quickly. Is there some explanation for the fact that these higher powers are almost integers?
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Another take on this: As David Speyer and FC's answer shows, this question can be answered without any additional deep theory. However, I'd like to explain a variant on their arguments that puts this in a little more context regarding modular forms. It also means we can use a technique which makes it easier to see how good these approximations are in terms of the growth rate of the coeffients of the j-function. The important fact here is that any modular function (for SL_2(Z)) with integer coefficients in its q-expansion takes on integer values at τ = (1+√(-163))/2 (and so q = exp(-π√163)). This fact is in fact a consequence of the integrality of the j-value here, since any such function can be expressed as a polynomial in j with integer coefficients (although similar things are true in other contexts, such as modular functions of higher level, where there is not a canonical generator for the ring of such modular invariants). This means that, just as we can use the integrality of $j(\tau) = q^{-1} + 744 + O(q) $ to get an integer approximation to $q^{-1}$ , if we have a modular function $f_n$ with power series of the form $f_n(\tau) = q^{-n} + integer + O(q)$ , we can get an integer approximation to $q^{-n}$ . How good this approximation is will depend upon the size of the coefficients of the power series for the $O(q)$ part. Fortunately for us, such a function $f_n$ always exists (and is unique up to adding integer constants). How can we construct it? One way is to take an appropriate polynomial in $j$ , that is, take an appropriate linear combination of $j, j^2, \cdots , j^n$ to get a function with the right principal part. This clearly works, and if one works out the details, it should turn out equivalent to FC's and David's approach. However, now that we're in the modular forms mindset, we have other tools at our disposal. In particular, another way to create new modular functions is to apply Hecke operators to existing modular functions, such as $j$ . This turns out to be an effective way to get modular functions of the type we need, since Hecke operators do predictable things to principal parts of q-series (for example, if $p$ is prime, $T_p j = q^{-p} + O(1)$ ). I'll just explain how this works for $n = 5$ , although the method should generalize immediately to any prime $n$ (composite $n$ might be a little trickier, but not much). The theory of Hecke operators tells us that the function $T_5 j$ defined by $$(T_5 j)(z) = j(5 z) + \sum_{i \ mod \ 5} j (\frac{z + i}{5})$$ is modular, with q-expansion given by $$(T_5 j)(\tau) = q^{-5} + \sum_{n = 0}^{\infty} (5 c_{5n} + c_{n/5}) q^n$$ . where the $c_n$ are the coefficients in $$j(\tau) = q^{-1} + \sum_{n = 0}^{\infty} c_n q^n$$ (and $c_n = 0$ if $n$ is not an integer). So if as before we set $q = e^{- \pi \sqrt{163}}$ , we find that $q^{-5} + 6 c_0 + 5 c_5 q + 5 c_{10} q^2 + 5 c_{15} q^3 + 5 c_{20} q^4 + (c_1 + 5 c_{25}) q^5 + \dots$ is an integer. Now, $q$ is roughly $4 \cdot 10^{-18}$ , and looking up the coefficients for $j(z)$ on OEIS , we find that $q^{-5} + 6 \cdot 744 + 5 \cdot (\sim 3\cdot 10^{11}) (\sim 4\cdot10^{-18}) + \text{clearly smaller terms}$ is an integer. Hence $q^{-5}$ should be off from an integer by roughly $6 \cdot 10^{-6}$ . This agrees pretty well with what Wolfram Alpha is giving me (it wouldn't be hard to get more digits here, but I'm feeling back-of-the-envelope right now and will call it a night :-)
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|
4,802 |
Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups are the higher Chow groups CH^i(X,m) (m=0 gives the classical ones). Does anyone have an intuition to share about these higher Chow groups? What do they measure/mean? If I pass from the simplicial group to a chain complex, what does it mean to be in the kernel/image of the differential? Could one say that the higher Chow groups keep track of in how many ways two cycles can be rationally equivalent (and which of these different ways are then equivalent etc.)? Finally: I don't see any reason why the definition shouldn't make sense over the integers or worse base schemes. Is this true? Does it maybe still make sense but lose its intended meaning?
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I believe Bloch's original insight was something like the following: First, if $X$ is a regular scheme, you can filter $K_0$ by ``codimension of support''; that is, view $K_0(X)$ as the Grothendieck group of the category of all finitely generated modules and let $F^iK_0(X)$ be the part generated by modules with codimension of support greater than or equal to $i$. Next, suppose you want to mimic this construction for $K_m$ instead of $K_0$. The first step is to notice that if you patch two copies of $\Delta^m_X$ together along their "boundary" (i.e. the union of the images of the various copies of $\Delta^{m-1}_X$) and call the result $S^m_X$, then Karoubi-Villamayor theory tells you that $K_m(X)$ is a direct summand of $K_0(S^m_X)$. (The complementary direct summand is $K_0(X)$.) So it suffices to find a "filtration by codimension of support" on $K_0(S^m_X)$. The usual constructions don't work because $S^m_X$ is not regular (so that in particular, not all modules correspond to $K$-theory classes.) But: a cycle in $z^i(X,m)$ has a positive part $z_+$ and a negative part $z_-$ which, (if it is homologically a cycle) must agree on the boundary. Therefore you can imagine taking $\Delta^m_X$-modules $M_+$ and $M_-$ supported on these positive and negative parts and patching them along the boundary to get a module on $S^m_X$. If this module has finite projective dimension (which it ``ought'' to because of all the proper-meeting conditions, and as long as it has no bad imbedded components), then it gives a class in $K_0(S^m_X)$, hence a class in $K_m(X)$, and we can take the $i-th$ part of the filtration to be generated by the classes that arise in this way. The Bloch-Lichtenbaum work largely bypasses this intuition, but this was (I think) the original intuition for why it ought to work.
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4,841 |
(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification ; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$? ; categorification-request .
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The Wikipedia answer is one answer that is commonly used: replace sets with categories, replace functions with functors, and replace identities among functions with natural transformations (or isomorphisms) among functors. One hopes for newer deeper results along the way. In the case of work of Lauda and Khovanov, they often start with an algebra (for example ${\bf C}[x]$ with operators $d (x^n)= n x^{n-1}$ and $x \cdot x^n = x^{n+1}$ subject to the relation $d \circ x = x \circ d +1$) and replace this with a category of projective $R$-modules and functors defined thereupon in such a way that the associated Grothendieck group is isomorphic to the original algebra. Khovanov's categorification of the Jones polynomial can be thought of in a different way even though, from his point of view, there is a central motivating idea between this paragraph and the preceding one. The Khovanov homology of a knot constructs from the set of $2^n$ Kauffman bracket smoothing of the diagram ($n$ is the crossing number) a homology theory whose graded Euler characteristic is the Jones polynomial. In this case, we can think of taking a polynomial formula and replacing it with a formula that inter-relates certain homology groups. Crane's original motivation was to define a Hopf category (which he did) as a generalization of a Hopf algebra in order to use this to define invariants of $4$-dimensional manifolds. The story gets a little complicated here, but goes roughly like this. Frobenius algebras give invariants of surfaces via TQFTs. More precisely, a TQFT on the $(1+1)$ cobordism category (e.g. three circles connected by a pair of pants) gives a Frobenius algebra. Hopf algebras give invariants of 3-manifolds. What algebraic structure gives rise to a $4$-dimensional manifold invariant, or a $4$-dimensional TQFT? Crane showed that a Hopf category was the underlying structure. So a goal from Crane's point of view, would be to construct interesting examples of Hopf categories. Similarly, in my question below, a goal is to give interesting examples of braided monoidal 2-categories with duals. In the last sense of categorification, we start from a category in which certain equalities hold. For example, a braided monoidal category has a set of axioms that mimic the braid relations. Then we replace those equalities by
$2$-morphisms that are isomorphisms and that satisfy certain coherence conditions. The resulting $2$-category may be structurally similar to another known entity. In this case, $2$-functors (objects to objects, morphisms to morphisms, and $2$-morphisms to $2$-morphisms in which equalities are preserved) can be shown to give invariants. The most important categorifications in terms of applications to date are
(in my own opinion) the Khovanov homology, Oszvath-Szabo's invariants of knots, and Crane's original insight. The former two items are important since they are giving new and interesting results.
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|
4,895 |
I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one often hears that they are two different ways of looking at the same thing. Can anybody give a precise statement of this relationship?
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The Wikipedia article is more technical than it should be, and for the reader in a hurry not all that well written. Here is a summary of the main points as best I understand them: Complex manifolds are analogous to smooth complex algebraic varieties, not to the singular ones. But that discrepancy is surmountable, because you can also have complex analytic varieties which can have similar singularities. Then the first and most important relation is that every complex algebraic variety is a complex analytic variety. Every Zariski open set is analytically open; analytic gluing maps are more general than algebraic ones; and the allowed analytic charts are more general than the allowed algebraic charts. Also every algebraic morphism is an analytic morphism, so you get a morphism between categories. But the connection is better than that because of the GAGA principle (globally analytic implies globally algebraic). My understanding of GAGA is very sketchy, but I think that the following is correct. Among other consequences of GAGA, a closed analytic subvariety of a proper (equivalent to compact) algebraic variety is algebraic. An analytic isomorphism between two proper algebraic varieties is algebraic. I would suppose that there is a similar principle for compact fibrations as well. So, if you make compact analytic varieties algebraically, you can't escape from the algebraic class some of the main constructions of complex manifolds do not escape from the algebraic class. (But not all: deformations and infinite group actions can escape.) All projective analytic varieties are algebraic, and in dimension 1 all compact curves are projective. Moreover, there are limited ways for a compact analytic manifold to avoid being projective, by Moishezon's theorem and Kodaira's theorem. In practice, then, most of the complex manifolds that people make are algebraic. Also, most of the analytic calculations on a proper algebraic variety are algebraic: Many global calculations are algebraic by GAGA, and many local calculations are algebraic just by truncating Taylor series. Contrast all this with real algebraic vs real analytic. It is still true that (the real points of) a smooth real algebraic variety is a real analytic manifold. More strongly than in the complex case, although it is highly non-trivial, every compact real analytic manifold is real algebraic. But the real algebraic structure is massively not unique, even for a circle, and that makes all the difference. The other answerers in this thread, who are more expert in this topic than I am, had more information about why a compact complex manifold might not be a smooth projective variety. Just for clarity, I will restrict attention to the compact, smooth case. Also, you say "proper" rather than "compact" in the algebraic category because every algebraic variety is "compact" in the extremely coarse Zariski topology. An algebraic variety is proper if and only if it is analytically compact. The main use of the word proper is to emphasize that it is more general than projective, which means given by polynomial equations in complex projective space. There are two very different initial reasons that an analytic complex manifold might not be projective. It might not be Moishezon: A complex $n$-manifold is Moishezon if it has $n$ algebraically independent meromorphic functions. (The number of algebraically independent elements or the transcendence degree of a field is called the Krull dimension. The meromorphic Krull dimension of a compact complex $n$-manifold is at most $n$.) Or it might not be Kähler: A complex $n$-manifold is Kähler if it has a Riemannian metric such that the covariant derivative of the complex structure vanishes. So to summarize what people said about compact complex manifolds (much of which is in the back of Hartshorne's book): projective ⇒ algebraic ⇒ Moishezon ⟺ bimeromorphically projective projective ⇒ Kähler ⇒ symplectic ⇒ non-zero $H^2$ algebraic ⇒ non-zero $H^2$ ( exposited by David Speyer) Moishezon and Kähler ⟺ projective (Moishezon) Kähler and integrally symplectic ⟺ projective (Kodaira) In addition, projective and algebraic structure and the Moishezon property are all unstable with respect to analytic deformation. And bimeromorphic equivalence preserves $\pi_1$. Taubes found compact complex manifolds that have the wrong $\pi_1$ to be Kähler; indeed they can have any $\pi_1$. Voisin found compact Kähler manifolds with the wrong homotopy type to be projective, disproving Kodaira's conjecture that every compact Kähler manifold can be deformed to projective. And, way out in left field, a left-invariant complex structure (LICS) on a compact simple Lie group is a compact complex manifold that has no $H^2$ and can be simply connected too. Still, despite these beautiful non-projective compact complex manifolds, it's generally easier to study projective examples. It's generally easier to sidestep analysis and do algebra instead.
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4,939 |
Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now. Is there a compact (Hausdorff, or even T1) (topological) group which is infinite, but has countable cardinality? The "obvious" choices don't work; for instance, $\mathbb{Q}/\mathbb{Z}$ (with the obvious induced topology) is non-compact, and I get the impression that profinite groups are all uncountable (although I might be wrong there). So does someone have an example, or a reference in the case that there are no such groups?
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No, there is no countably infinite compact Hausdorff topological group. Indeed such a group $G$ would have a left-invariant Haar measure $m$ with $m(G)=1$
and all points would have the same measure (since the group acts transitively on itself).
But then, by countable additivity of the measure $m$, the group itself would have measure $m(G)=0$ or $m(G)=\infty$ according as its points all had $m(p)=0$ or $m(p)>0$ . A contradiction in both cases to the fact that $m(G)=1$ .
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