chaseharmon/MIT_Course_Catalog · Datasets at Hugging Face

Class Number stringlengths 4 15	Name stringlengths 4 124	Description stringlengths 23 1.14k	Offered bool 2 classes	Term stringclasses 97 values	Level stringclasses 2 values	Units stringclasses 194 values	Prerequisites stringlengths 4 127 ⌀	Equivalents stringlengths 7 63 ⌀	Lab bool 2 classes	Partial Lab bool 2 classes	REST bool 2 classes	GIR stringclasses 7 values	HASS stringclasses 5 values	CI / CI-HW stringclasses 3 values
18.338	Eigenvalues of Random Matrices	Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of Julia helpful, but not required.	true	Fall	Graduate	3-0-9	18.701 or permission of instructor	null	false	false	false	False	False	False
18.352[J]	Nonlinear Dynamics: The Natural Environment	Analyzes cooperative processes that shape the natural environment, now and in the geologic past. Emphasizes the development of theoretical models that relate the physical and biological worlds, the comparison of theory to observational data, and associated mathematical methods. Topics include carbon cycle dynamics; ecosystem structure, stability and complexity; mass extinctions; biosphere-geosphere coevolution; and climate change. Employs techniques such as stability analysis; scaling; null model construction; time series and network analysis.	true	Fall	Undergraduate	3-0-9	Calculus II (GIR) and Physics I (GIR); Coreq: 18.03	12.009[J]	false	false	false	False	False	False
18.353[J]	Nonlinear Dynamics: Chaos	Introduction to nonlinear dynamics and chaos in dissipative systems. Forced and parametric oscillators. Phase space. Periodic, quasiperiodic, and aperiodic flows. Sensitivity to initial conditions and strange attractors. Lorenz attractor. Period doubling, intermittency, and quasiperiodicity. Scaling and universality. Analysis of experimental data: Fourier transforms, Poincare sections, fractal dimension, and Lyapunov exponents. Applications to mechanical systems, fluid dynamics, physics, geophysics, and chemistry. See 12.207J/18.354J for Nonlinear Dynamics: Continuum Systems.	true	Fall	Undergraduate	3-0-9	Physics II (GIR) and (18.03 or 18.032)	2.050[J], 12.006[J]	false	false	false	False	False	False
18.354[J]	Nonlinear Dynamics: Continuum Systems	General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology.	true	Spring	Undergraduate	3-0-9	Physics II (GIR) and (18.03 or 18.032)	1.062[J], 12.207[J]	false	false	false	False	False	False
18.3541	Nonlinear Dynamics: Continuum Systems	General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology. Students in Courses 1, 12, and 18 must register for undergraduate version, 18.354.	true	Spring	Graduate	3-0-9	Physics II (GIR) and (18.03 or 18.032)	null	false	false	false	False	False	False
18.355	Fluid Mechanics	Topics include the development of Navier-Stokes equations, inviscid flows, boundary layers, lubrication theory, Stokes flows, and surface tension. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes classroom and laboratory demonstrations.	true	Spring	Graduate	3-0-9	2.25, 12.800, or 18.354	null	false	false	false	False	False	False
18.357	Interfacial Phenomena	Fluid systems dominated by the influence of interfacial tension. Elucidates the roles of curvature pressure and Marangoni stress in a variety of hydrodynamic settings. Particular attention to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments are accompanied by classroom demonstrations. Highlights the role of surface tension in biology.	false	Spring	Graduate	3-0-9	2.25, 12.800, 18.354, 18.355, or permission of instructor	null	false	false	false	False	False	False
18.358[J]	Nonlinear Dynamics and Turbulence	Reviews theoretical notions of nonlinear dynamics, instabilities, and waves with applications in fluid dynamics. Discusses hydrodynamic instabilities leading to flow destabilization and transition to turbulence. Focuses on physical turbulence and mixing from homogeneous isotropic turbulence. Also covers topics such as rotating and stratified flows as they arise in the environment, wave-turbulence, and point source turbulent flows. Laboratory activities integrate theoretical concepts covered in lectures and problem sets. Students taking graduate version complete additional assignments.	true	Spring	Graduate	3-2-7	1.060A	1.686[J], 2.033[J]	false	false	false	False	False	False
18.367	Waves and Imaging	The mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. Suitable for graduate students from all departments who have affinities with applied mathematics. Topics include acoustic, elastic, electromagnetic wave equations; geometrical optics; scattering series and inversion; migration and backprojection; adjoint-state methods; Radon and curvilinear Radon transforms; microlocal analysis of imaging; optimization, regularization, and sparse regression.	false	Fall	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.369[J]	Mathematical Methods in Nanophotonics	High-level approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, and coupled-mode theories.	true	Spring	Graduate	3-0-9	8.07, 18.303, or permission of instructor	8.315[J]	false	false	false	False	False	False
18.376[J]	Wave Propagation	Theoretical concepts and analysis of wave problems in science and engineering with examples chosen from elasticity, acoustics, geophysics, hydrodynamics, blood flow, nondestructive evaluation, and other applications. Progressive waves, group velocity and dispersion, energy density and transport. Reflection, refraction and transmission of plane waves by an interface. Mode conversion in elastic waves. Rayleigh waves. Waves due to a moving load. Scattering by a two-dimensional obstacle. Reciprocity theorems. Parabolic approximation. Waves on the sea surface. Capillary-gravity waves. Wave resistance. Radiation of surface waves. Internal waves in stratified fluids. Waves in rotating media. Waves in random media.	true	Spring	Graduate	3-0-9	2.003 and 18.075	1.138[J], 2.062[J]	false	false	false	False	False	False
18.377[J]	Nonlinear Dynamics and Waves	A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; self-excited oscillations; lock-in phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year.	true	Spring	Graduate	3-0-9	Permission of instructor	1.685[J], 2.034[J]	false	false	false	False	False	False
18.384	Undergraduate Seminar in Physical Mathematics	Covers the mathematical modeling of physical systems, with emphasis on the reading and presentation of papers. Addresses a broad range of topics, with particular focus on macroscopic physics and continuum systems: fluid dynamics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited.	true	Fall	Undergraduate	3-0-9	12.006, 18.300, 18.354, or permission of instructor	null	false	false	false	False	False	False
18.385[J]	Nonlinear Dynamics and Chaos	Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, Poincare-Bendixson theory. Floquet theory. Poincare maps. Averaging. Near-equilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment.	true	Spring	Graduate	3-0-9	18.03 or 18.032	2.036[J]	false	false	false	False	False	False
18.397	Mathematical Methods in Physics	Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects.	true	Fall	Graduate	3-0-9	18.745 or some familiarity with Lie theory	null	false	false	false	False	False	False
18.400[J]	Computability and Complexity Theory	Mathematical introduction to the theory of computing. Rigorously explores what kinds of tasks can be efficiently solved with computers by way of finite automata, circuits, Turing machines, and communication complexity, introducing students to some major open problems in mathematics. Builds skills in classifying computational tasks in terms of their difficulty. Discusses other fundamental issues in computing, including the Halting Problem, the Church-Turing Thesis, the P versus NP problem, and the power of randomness.	true	Spring	Undergraduate	4-0-8	(6.1200 and 6.1210) or permission of instructor	6.1400[J]	false	false	false	False	False	False
18.404	Theory of Computation	A more extensive and theoretical treatment of the material in 6.1400J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.	true	Fall	Undergraduate	4-0-8	6.1200 or 18.200	null	false	false	false	False	False	False
18.4041[J]	Theory of Computation	A more extensive and theoretical treatment of the material in 6.1400J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Students in Course 18 must register for the undergraduate version, 18.404.	true	Fall	Graduate	4-0-8	6.1200 or 18.200	6.5400[J]	false	false	false	False	False	False
18.405[J]	Advanced Complexity Theory	Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomial-time hierarchy. Interactive proof systems. Relativization. Definitions of randomness. Pseudo-randomness and derandomizations. Interactive proof systems and probabilistically checkable proofs.	true	Spring	Graduate	3-0-9	18.404	6.5410[J]	false	false	false	False	False	False
18.408	Topics in Theoretical Computer Science	Study of areas of current interest in theoretical computer science. Topics vary from term to term.	false	Fall	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.410[J]	Design and Analysis of Algorithms	Techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics include sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; greedy algorithms; amortized analysis; graph algorithms; and shortest paths. Advanced topics may include network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.	true	Fall, Spring	Undergraduate	4-0-8	6.1200 and 6.1210	6.1220[J]	false	false	false	False	False	False
18.413	Introduction to Computational Molecular Biology	Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.	true	Spring	Undergraduate	3-0-9	6.1210 or permission of instructor	null	false	false	false	False	False	False
18.415[J]	Advanced Algorithms	First-year graduate subject in algorithms. Emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Surveys a variety of computational models and the algorithms for them. Data structures, network flows, linear programming, computational geometry, approximation algorithms, online algorithms, parallel algorithms, external memory, streaming algorithms.	true	Fall	Graduate	5-0-7	6.1220 and (6.1200, 6.3700, or 18.600)	6.5210[J]	false	false	false	False	False	False
18.416[J]	Randomized Algorithms	Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.	true	Fall	Graduate	5-0-7	(6.1200 or 6.3700) and (6.1220 or 6.5210)	6.5220[J]	false	false	false	False	False	False
18.417	Introduction to Computational Molecular Biology	Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.	true	Spring	Graduate	3-0-9	6.1210 or permission of instructor	null	false	false	false	False	False	False
18.418[J]	Topics in Computational Molecular Biology	Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the International Conference on Computational Molecular Biology (RECOMB) and the Conference on Intelligent Systems for Molecular Biology (ISMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, biological networks, drug discovery, and privacy. Recent research by course participants also covered. Participants will be expected to present individual projects to the class.	true	Fall	Graduate	3-0-9	6.8701, 18.417, or permission of instructor	HST.504[J]	false	false	false	False	False	False
18.424	Seminar in Information Theory	Considers various topics in information theory, including data compression, Shannon's Theorems, and error-correcting codes. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.	true	Fall	Undergraduate	3-0-9	(6.3700, 18.05, or 18.600) and (18.06, 18.700, or 18.701)	null	false	false	false	False	False	False
18.425[J]	Foundations of Cryptography	A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives such as public-key encryption, digital signatures, and pseudo-random number generation, as well as advanced cryptographic primitives such as zero-knowledge proofs, homomorphic encryption, and secure multiparty computation.	true	Fall	Graduate	3-0-9	6.1220, 6.1400, or 18.4041	6.5620[J]	false	false	false	False	False	False
18.434	Seminar in Theoretical Computer Science	Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.	true	Fall	Undergraduate	3-0-9	6.1220	null	false	false	false	False	False	False
18.435[J]	Quantum Computation	Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.	true	Fall	Graduate	3-0-9	8.05, 18.06, 18.700, 18.701, or 18.C06	2.111[J], 6.6410[J], 8.370[J]	false	false	false	False	False	False
18.436[J]	Quantum Information Science	Examines quantum computation and quantum information. Topics include quantum circuits, the quantum Fourier transform and search algorithms, the quantum operations formalism, quantum error correction, Calderbank-Shor-Steane and stabilizer codes, fault tolerant quantum computation, quantum data compression, quantum entanglement, capacity of quantum channels, and quantum cryptography and the proof of its security. Prior knowledge of quantum mechanics required.	true	Spring	Graduate	3-0-9	18.435	6.6420[J], 8.371[J]	false	false	false	False	False	False
18.437[J]	Distributed Algorithms	Design and analysis of algorithms, emphasizing those suitable for use in distributed networks. Covers various topics including distributed graph algorithms, locality constraints, bandwidth limitations and communication complexity, process synchronization, allocation of computational resources, fault tolerance, and asynchrony. No background in distributed systems required.	true	Spring	Graduate	3-0-9	6.1220	6.5250[J]	false	false	false	False	False	False
18.453	Combinatorial Optimization	Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as 18.200) helpful.	true	Spring	Undergraduate	3-0-9	18.06, 18.700, or 18.701	null	false	false	false	False	False	False
18.4531	Combinatorial Optimization	Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as 18.200) helpful. Students in Course 18 must register for the undergraduate version, 18.453.	true	Spring	Graduate	3-0-9	18.06, 18.700, or 18.701	null	false	false	false	False	False	False
18.455	Advanced Combinatorial Optimization	Advanced treatment of combinatorial optimization with an emphasis on combinatorial aspects. Non-bipartite matchings, submodular functions, matroid intersection/union, matroid matching, submodular flows, multicommodity flows, packing and connectivity problems, and other recent developments.	true	Spring	Graduate	3-0-9	18.453 or permission of instructor	null	false	false	false	False	False	False
18.456[J]	Algebraic Techniques and Semidefinite Optimization	Theory and computational techniques for optimization problems involving polynomial equations and inequalities with particular, emphasis on the connections with semidefinite optimization. Develops algebraic and numerical approaches of general applicability, with a view towards methods that simultaneously incorporate both elements, stressing convexity-based ideas, complexity results, and efficient implementations. Examples from several engineering areas, in particular systems and control applications. Topics include semidefinite programming, resultants/discriminants, hyperbolic polynomials, Groebner bases, quantifier elimination, and sum of squares.	true	Spring	Graduate	3-0-9	6.7210 or 15.093	6.7230[J]	false	false	false	False	False	False
18.504	Seminar in Logic	Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.	true	Fall	Undergraduate	3-0-9	(18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)	null	false	false	false	False	False	False
18.510	Introduction to Mathematical Logic and Set Theory	Propositional and predicate logic. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Godel's incompleteness theorem.	true	Fall	Undergraduate	3-0-9	null	null	false	false	false	False	False	False
18.515	Mathematical Logic	More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and completeness. Compactness and its consequences. Quantifier elimination. Recursive sets and functions. Incompleteness and undecidability. Ordinals and cardinals. Set-theoretic formalization of mathematics.	true	Spring	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.600	Probability and Random Variables	Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem. Credit cannot also be received for 6.041A or 6.041B.	true	Fall, Spring	Undergraduate	4-0-8	Calculus II (GIR)	null	false	false	true	False	False	False
18.604	Seminar In Probability Theory (New)	Students work on group presentations on topics selected by students from a provided list of suggestions. Topics may include Benford's law, random walks and electrical networks, and Brownian motions. Assignments include three group presentations, two individual presentations, and a final individual term paper. Instruction in oral and written communication provided to effectively communicate about probability theory. Limited to 16.	true	Spring	Undergraduate	3-0-9	18.05 or 18.600	null	false	false	false	False	False	False
18.615	Introduction to Stochastic Processes	Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion.	true	Spring	Graduate	3-0-9	6.3700 or 18.600	null	false	false	false	False	False	False
18.619[J]	Discrete Probability and Stochastic Processes	Provides an introduction to tools used for probabilistic reasoning in the context of discrete systems and processes. Tools such as the probabilistic method, first and second moment method, martingales, concentration and correlation inequalities, theory of random graphs, weak convergence, random walks and Brownian motion, branching processes, Markov chains, Markov random fields, correlation decay method, isoperimetry, coupling, influences and other basic tools of modern research in probability will be presented. Algorithmic aspects and connections to statistics and machine learning will be emphasized.	true	Spring	Graduate	3-0-9	6.3702, 6.7700, 18.100A, 18.100B, or 18.100Q	6.7720[J], 15.070[J]	false	false	false	False	False	False
18.642	Topics in Mathematics with Applications in Finance	Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required.	true	Fall	Undergraduate	3-0-9	18.03, 18.06, and (18.05 or 18.600)	null	false	false	false	False	False	False
18.650[J]	Fundamentals of Statistics	A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification.	true	Fall, Spring	Undergraduate	4-0-8	6.3700 or 18.600	IDS.014[J]	false	false	false	False	False	False
18.6501	Fundamentals of Statistics	A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification. Students in Course 18 must register for the undergraduate version, 18.650.	true	Fall, Spring	Graduate	4-0-8	6.3700 or 18.600	null	false	false	false	False	False	False
18.655	Mathematical Statistics	Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis. Prior exposure to both probability and statistics at the university level is assumed.	true	Spring	Graduate	3-0-9	(18.650 and (18.100A, 18.100A, 18.100P, or 18.100Q)) or permission of instructor	null	false	false	false	False	False	False
18.656[J]	Mathematical Statistics: a Non-Asymptotic Approach	Introduces students to modern non-asymptotic statistical analysis. Topics include high-dimensional models, nonparametric regression, covariance estimation, principal component analysis, oracle inequalities, prediction and margin analysis for classification. Develops a rigorous probabilistic toolkit, including tail bounds and a basic theory of empirical processes	true	Spring	Graduate	3-0-9	(6.7700, 18.06, and 18.6501) or permission of instructor	9.521[J], IDS.160[J]	false	false	false	False	False	False
18.657	Topics in Statistics	Topics vary from term to term.	true	Spring	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.675	Theory of Probability	Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. Prior exposure to probability (e.g., 18.600) recommended.	true	Fall	Graduate	3-0-9	18.100A, 18.100B, 18.100P, or 18.100Q	null	false	false	false	False	False	False
18.676	Stochastic Calculus	Introduction to stochastic processes, building on the fundamental example of Brownian motion. Topics include Brownian motion, continuous parameter martingales, Ito's theory of stochastic differential equations, Markov processes and partial differential equations, and may also include local time and excursion theory. Students should have familiarity with Lebesgue integration and its application to probability.	true	Spring	Graduate	3-0-9	18.675	null	false	false	false	False	False	False
18.677	Topics in Stochastic Processes	Topics vary from year to year.	true	Fall	Graduate	3-0-9	18.675	null	false	false	false	False	False	False
18.700	Linear Algebra	Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06.	true	Fall	Undergraduate	3-0-9	Calculus II (GIR)	null	false	false	true	False	False	False
18.701	Algebra I	18.701-18.702 is more extensive and theoretical than the 18.700-18.703 sequence. Experience with proofs necessary. 18.701 focuses on group theory, geometry, and linear algebra.	true	Fall	Undergraduate	3-0-9	18.100A, 18.100B, 18.100P, 18.100Q, 18.090, or permission of instructor	null	false	false	false	False	False	False
18.702	Algebra II	Continuation of 18.701. Focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.	true	Spring	Undergraduate	3-0-9	18.701	null	false	false	false	False	False	False
18.703	Modern Algebra	Focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals and unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. 18.700 and 18.703 together form a standard algebra sequence.	true	Spring	Undergraduate	3-0-9	Calculus II (GIR)	null	false	false	false	False	False	False
18.704	Seminar in Algebra	Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Some experience with proofs required. Enrollment limited.	true	Fall	Undergraduate	3-0-9	18.701, (18.06 and 18.703), or (18.700 and 18.703)	null	false	false	false	False	False	False
18.705	Commutative Algebra	Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.	true	Fall	Graduate	3-0-9	18.702	null	false	false	false	False	False	False
18.706	Noncommutative Algebra	Topics may include Wedderburn theory and structure of Artinian rings, Morita equivalence and elements of category theory, localization and Goldie's theorem, central simple algebras and the Brauer group, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension.	true	Spring	Graduate	3-0-9	18.702	null	false	false	false	False	False	False
18.708	Topics in Algebra	Topics vary from year to year.	true	Fall	Graduate	3-0-9	18.705	null	false	false	false	False	False	False
18.715	Introduction to Representation Theory	Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.	false	Spring	Graduate	3-0-9	18.702 or 18.703	null	false	false	false	False	False	False
18.721	Introduction to Algebraic Geometry	Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology.	true	Spring	Undergraduate	3-0-9	18.702 and 18.901	null	false	false	false	False	False	False
18.725	Algebraic Geometry I	Introduces the basic notions and techniques of modern algebraic geometry. Covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. Introduction to the language of schemes and properties of morphisms. Knowledge of elementary algebraic topology, elementary differential geometry recommended, but not required.	true	Fall	Graduate	3-0-9	None. Coreq: 18.705	null	false	false	false	False	False	False
18.726	Algebraic Geometry II	Continuation of the introduction to algebraic geometry given in 18.725. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology.	true	Spring	Graduate	3-0-9	18.725	null	false	false	false	False	False	False
18.727	Topics in Algebraic Geometry	Topics vary from year to year.	true	Spring	Graduate	3-0-9	18.725	null	false	false	false	False	False	False
18.737	Algebraic Groups	Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof.	false	Spring	Graduate	3-0-9	18.705	null	false	false	false	False	False	False
18.745	Lie Groups and Lie Algebras I	Covers fundamentals of the theory of Lie algebras and related groups. Topics may include theorems of Engel and Lie; enveloping algebra, Poincare-Birkhoff-Witt theorem; classification and construction of semisimple Lie algebras; the center of their enveloping algebras; elements of representation theory; compact Lie groups and/or finite Chevalley groups.	true	Fall	Graduate	3-0-9	(18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or 18.100Q)	null	false	false	false	False	False	False
18.747	Infinite-dimensional Lie Algebras	Topics vary from year to year.	true	Fall	Graduate	3-0-9	18.745	null	false	false	false	False	False	False
18.748	Topics in Lie Theory	Topics vary from year to year.	true	Spring	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.755	Lie Groups and Lie Algebras II	A more in-depth treatment of Lie groups and Lie algebras. Topics may include homogeneous spaces and groups of automorphisms; representations of compact groups and their geometric realizations, Peter-Weyl theorem; invariant differential forms and cohomology of Lie groups and homogeneous spaces; complex reductive Lie groups, classification of real reductive groups.	true	Spring	Graduate	3-0-9	18.745 or permission of instructor	null	false	false	false	False	False	False
18.757	Representations of Lie Groups	Covers representations of locally compact groups, with emphasis on compact groups and abelian groups. Includes Peter-Weyl theorem and Cartan-Weyl highest weight theory for compact Lie groups.	true	Fall	Graduate	3-0-9	18.745 or 18.755	null	false	false	false	False	False	False
18.758	Methods of Representation Theory	Devoted to contemporary methods in representation theory of Lie groups, algebraic groups, and their generalizations. Topics may include: Springer correspondence, highest weight modules and Harish-Chandra bimodules, quantum groups and their representations, modular representations of algebraic groups and relation to quantum groups at a root of unity, representations of p-adic group, introduction to automorphic forms and Langlands duality, and representations of finite Chevalley groups.	true	Spring	Graduate	3-0-9	18.745 and (18.737 or 18.755)	null	false	false	false	False	False	False
18.781	Theory of Numbers	An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.	true	Spring	Undergraduate	3-0-9	null	null	false	false	false	False	False	False
18.782	Introduction to Arithmetic Geometry	Exposes students to arithmetic geometry, motivated by the problem of finding rational points on curves. Includes an introduction to p-adic numbers and some fundamental results from number theory and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include Mordell's theorem, the Weil conjectures, and Jacobian varieties.	true	Spring	Undergraduate	3-0-9	18.702	null	false	false	false	False	False	False
18.783	Elliptic Curves	Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem.	true	Fall	Undergraduate	3-0-9	18.702, 18.703, or permission of instructor	null	false	false	false	False	False	False
18.7831	Elliptic Curves	Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. Students in Course 18 must register for the undergraduate version, 18.783.	true	Fall	Graduate	3-0-9	18.702, 18.703, or permission of instructor	null	false	false	false	False	False	False
18.784	Seminar in Number Theory	Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.	true	Spring	Undergraduate	3-0-9	18.701 or (18.703 and (18.06 or 18.700))	null	false	false	false	False	False	False
18.785	Number Theory I	Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Local fields, ramification, discriminants. Zeta and L-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem.	true	Fall	Graduate	3-0-9	None. Coreq: 18.705	null	false	false	false	False	False	False
18.786	Number Theory II	Continuation of 18.785. More advanced topics in number theory, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, or quadratic forms.	true	Spring	Graduate	3-0-9	18.785	null	false	false	false	False	False	False
18.787	Topics in Number Theory	Topics vary from year to year.	true	Fall	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.821	Project Laboratory in Mathematics	Guided research in mathematics, employing the scientific method. Students confront puzzling and complex mathematical situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain them mathematically. Students choose three projects from a large collection of options. Each project results in a laboratory report subject to revision; oral presentation on one or two projects. Projects drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and probability. Enrollment limited.	true	Fall, Spring	Undergraduate	3-6-3	Two mathematics subjects numbered 18.100 or above	null	true	false	false	False	False	False
18.896[J]	Leadership and Professional Strategies & Skills Training (LEAPS), Part I: Advancing Your Professional Strategies and Skills	Part I (of two parts) of the LEAPS graduate career development and training series. Topics include: navigating and charting an academic career with confidence; convincing an audience with clear writing and arguments; mastering public speaking and communications; networking at conferences and building a brand; identifying transferable skills; preparing for a successful job application package and job interviews; understanding group dynamics and different leadership styles; leading a group or team with purpose and confidence. Postdocs encouraged to attend as non-registered participants. Limited to 80.	true	Spring	Graduate	2-0-1 [P/D/F]	null	5.961[J], 8.396[J], 9.980[J], 12.396[J]	false	false	false	False	False	False
18.897[J]	Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies	Part II (of two parts) of the LEAPS graduate career development and training series. Topics covered include gaining self awareness and awareness of others, and communicating with different personality types; learning about team building practices; strategies for recognizing and resolving conflict and bias; advocating for diversity and inclusion; becoming organizationally savvy; having the courage to be an ethical leader; coaching, mentoring, and developing others; championing, accepting, and implementing change. Postdocs encouraged to attend as non-registered participants. Limited to 80.	true	Spring	Graduate	2-0-1 [P/D/F]	null	5.962[J], 8.397[J], 9.981[J], 12.397[J]	false	false	false	False	False	False
18.899	Internship in Mathematics (New)	Provides academic credit for students pursuing internships to gain practical experience applications of mathematical concepts and methods as related to their field of research.	true	Fall, Spring, Summer	Graduate	rranged [P/D/F]	null	null	false	false	false	False	False	False
18.900	Geometry and Topology in the Plane	Introduction to selected aspects of geometry and topology, using concepts that can be visualized easily. Mixes geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). Suitable for students with no prior exposure to differential geometry or topology.	true	Spring	Undergraduate	3-0-9	18.03 or 18.06	null	false	false	false	False	False	False
18.901	Introduction to Topology	Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group.	true	Fall, Spring	Undergraduate	3-0-9	18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor	null	false	false	false	False	False	False
18.9011	Introduction to Topology	Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group. Students in Course 18 must register for the undergraduate version, 18.901.	true	Fall, Spring	Graduate	3-0-9	18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor	null	false	false	false	False	False	False
18.904	Seminar in Topology	Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.	true	Spring	Undergraduate	3-0-9	18.901	null	false	false	false	False	False	False
18.905	Algebraic Topology I	Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.	true	Fall	Graduate	3-0-9	18.901 and (18.701 or 18.703)	null	false	false	false	False	False	False
18.906	Algebraic Topology II	Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.	true	Spring	Graduate	3-0-9	18.905 and (18.101 or 18.965)	null	false	false	false	False	False	False
18.917	Topics in Algebraic Topology	Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.	true	Spring	Graduate	3-0-9	18.906	null	false	false	false	False	False	False
18.919	Graduate Topology Seminar	Study and discussion of important original papers in the various parts of topology. Open to all students who have taken 18.906 or the equivalent, not only prospective topologists.	true	Spring	Graduate	3-0-9	18.906	null	false	false	false	False	False	False
18.937	Topics in Geometric Topology	Content varies from year to year. Introduces new and significant developments in geometric topology.	false	Fall, Spring	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False
18.950	Differential Geometry	Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space.	true	Fall	Undergraduate	3-0-9	(18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)	null	false	false	false	False	False	False
18.9501	Differential Geometry	Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space. Students in Course 18 must register for the undergraduate version, 18.950.	true	Fall	Graduate	3-0-9	(18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)	null	false	false	false	False	False	False
18.952	Theory of Differential Forms	Multilinear algebra: tensors and exterior forms. Differential forms on Rn: exterior differentiation, the pull-back operation and the Poincaré lemma. Applications to physics: Maxwell's equations from the differential form perspective. Integration of forms on open sets of Rn. The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes' theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology.	false	Spring	Undergraduate	3-0-9	18.101 and (18.700 or 18.701)	null	false	false	false	False	False	False
18.965	Geometry of Manifolds I	Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a continuation of 18.965 and focuses more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. Prior exposure to calculus on manifolds, as in 18.952, recommended.	true	Fall	Graduate	3-0-9	18.101, 18.950, or 18.952	null	false	false	false	False	False	False
18.966	Geometry of Manifolds II	Continuation of 18.965, focusing more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology.	true	Spring	Graduate	3-0-9	18.965	null	false	false	false	False	False	False
18.968	Topics in Geometry	Content varies from year to year.	true	Spring	Graduate	3-0-9	18.965	null	false	false	false	False	False	False
18.979	Graduate Geometry Seminar	Content varies from year to year. Study of classical papers in geometry and in applications of analysis to geometry and topology.	true	Spring	Graduate	3-0-9	Permission of instructor	null	false	false	false	False	False	False

18.338

Eigenvalues of Random Matrices

Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of Julia helpful, but not required.

true

Fall

Graduate

3-0-9

18.701 or permission of instructor

null

false

False

18.352[J]

Nonlinear Dynamics: The Natural Environment

Analyzes cooperative processes that shape the natural environment, now and in the geologic past. Emphasizes the development of theoretical models that relate the physical and biological worlds, the comparison of theory to observational data, and associated mathematical methods. Topics include carbon cycle dynamics; ecosystem structure, stability and complexity; mass extinctions; biosphere-geosphere coevolution; and climate change. Employs techniques such as stability analysis; scaling; null model construction; time series and network analysis.

true

Fall

Undergraduate

3-0-9

Calculus II (GIR) and Physics I (GIR); Coreq: 18.03

12.009[J]

false

False

18.353[J]

Nonlinear Dynamics: Chaos

Introduction to nonlinear dynamics and chaos in dissipative systems. Forced and parametric oscillators. Phase space. Periodic, quasiperiodic, and aperiodic flows. Sensitivity to initial conditions and strange attractors. Lorenz attractor. Period doubling, intermittency, and quasiperiodicity. Scaling and universality. Analysis of experimental data: Fourier transforms, Poincare sections, fractal dimension, and Lyapunov exponents. Applications to mechanical systems, fluid dynamics, physics, geophysics, and chemistry. See 12.207J/18.354J for Nonlinear Dynamics: Continuum Systems.

true

Fall

Undergraduate

3-0-9

Physics II (GIR) and (18.03 or 18.032)

2.050[J], 12.006[J]

false

False

18.354[J]

Nonlinear Dynamics: Continuum Systems

General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology.

true

Spring

Undergraduate

3-0-9

Physics II (GIR) and (18.03 or 18.032)

1.062[J], 12.207[J]

false

False

18.3541

Nonlinear Dynamics: Continuum Systems

General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology. Students in Courses 1, 12, and 18 must register for undergraduate version, 18.354.

true

Spring

Graduate

3-0-9

Physics II (GIR) and (18.03 or 18.032)

null

false

False

18.355

Fluid Mechanics

Topics include the development of Navier-Stokes equations, inviscid flows, boundary layers, lubrication theory, Stokes flows, and surface tension. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes classroom and laboratory demonstrations.

true

Spring

Graduate

3-0-9

2.25, 12.800, or 18.354

null

false

False

18.357

Interfacial Phenomena

Fluid systems dominated by the influence of interfacial tension. Elucidates the roles of curvature pressure and Marangoni stress in a variety of hydrodynamic settings. Particular attention to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments are accompanied by classroom demonstrations. Highlights the role of surface tension in biology.

false

Spring

Graduate

3-0-9

2.25, 12.800, 18.354, 18.355, or permission of instructor

null

false

False

18.358[J]

Nonlinear Dynamics and Turbulence

Reviews theoretical notions of nonlinear dynamics, instabilities, and waves with applications in fluid dynamics. Discusses hydrodynamic instabilities leading to flow destabilization and transition to turbulence. Focuses on physical turbulence and mixing from homogeneous isotropic turbulence. Also covers topics such as rotating and stratified flows as they arise in the environment, wave-turbulence, and point source turbulent flows. Laboratory activities integrate theoretical concepts covered in lectures and problem sets. Students taking graduate version complete additional assignments.

true

Spring

Graduate

3-2-7

1.060A

1.686[J], 2.033[J]

false

False

18.367

Waves and Imaging

The mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. Suitable for graduate students from all departments who have affinities with applied mathematics. Topics include acoustic, elastic, electromagnetic wave equations; geometrical optics; scattering series and inversion; migration and backprojection; adjoint-state methods; Radon and curvilinear Radon transforms; microlocal analysis of imaging; optimization, regularization, and sparse regression.

false

Fall

Graduate

3-0-9

Permission of instructor

null

false

False

18.369[J]

Mathematical Methods in Nanophotonics

High-level approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, and coupled-mode theories.

true

Spring

Graduate

3-0-9

8.07, 18.303, or permission of instructor

8.315[J]

false

False

18.376[J]

Wave Propagation

Theoretical concepts and analysis of wave problems in science and engineering with examples chosen from elasticity, acoustics, geophysics, hydrodynamics, blood flow, nondestructive evaluation, and other applications. Progressive waves, group velocity and dispersion, energy density and transport. Reflection, refraction and transmission of plane waves by an interface. Mode conversion in elastic waves. Rayleigh waves. Waves due to a moving load. Scattering by a two-dimensional obstacle. Reciprocity theorems. Parabolic approximation. Waves on the sea surface. Capillary-gravity waves. Wave resistance. Radiation of surface waves. Internal waves in stratified fluids. Waves in rotating media. Waves in random media.

true

Spring

Graduate

3-0-9

2.003 and 18.075

1.138[J], 2.062[J]

false

False

18.377[J]

Nonlinear Dynamics and Waves

A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; self-excited oscillations; lock-in phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year.

true

Spring

Graduate

3-0-9

Permission of instructor

1.685[J], 2.034[J]

false

False

18.384

Undergraduate Seminar in Physical Mathematics

Covers the mathematical modeling of physical systems, with emphasis on the reading and presentation of papers. Addresses a broad range of topics, with particular focus on macroscopic physics and continuum systems: fluid dynamics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited.

true

Fall

Undergraduate

3-0-9

12.006, 18.300, 18.354, or permission of instructor

null

false

False

18.385[J]

Nonlinear Dynamics and Chaos

Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, Poincare-Bendixson theory. Floquet theory. Poincare maps. Averaging. Near-equilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment.

true

Spring

Graduate

3-0-9

18.03 or 18.032

2.036[J]

false

False

18.397

Mathematical Methods in Physics

Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects.

true

Fall

Graduate

3-0-9

18.745 or some familiarity with Lie theory

null

false

False

18.400[J]

Computability and Complexity Theory

Mathematical introduction to the theory of computing. Rigorously explores what kinds of tasks can be efficiently solved with computers by way of finite automata, circuits, Turing machines, and communication complexity, introducing students to some major open problems in mathematics. Builds skills in classifying computational tasks in terms of their difficulty. Discusses other fundamental issues in computing, including the Halting Problem, the Church-Turing Thesis, the P versus NP problem, and the power of randomness.

true

Spring

Undergraduate

4-0-8

(6.1200 and 6.1210) or permission of instructor

6.1400[J]

false

False

18.404

Theory of Computation

A more extensive and theoretical treatment of the material in 6.1400J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.

true

Fall

Undergraduate

4-0-8

6.1200 or 18.200

null

false

False

18.4041[J]

Theory of Computation

A more extensive and theoretical treatment of the material in 6.1400J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Students in Course 18 must register for the undergraduate version, 18.404.

true

Fall

Graduate

4-0-8

6.1200 or 18.200

6.5400[J]

false

False

18.405[J]

Advanced Complexity Theory

Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomial-time hierarchy. Interactive proof systems. Relativization. Definitions of randomness. Pseudo-randomness and derandomizations. Interactive proof systems and probabilistically checkable proofs.

true

Spring

Graduate

3-0-9

18.404

6.5410[J]

false

False

18.408

Topics in Theoretical Computer Science

Study of areas of current interest in theoretical computer science. Topics vary from term to term.

false

Fall

Graduate

3-0-9

Permission of instructor

null

false

False

18.410[J]

Design and Analysis of Algorithms

Techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics include sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; greedy algorithms; amortized analysis; graph algorithms; and shortest paths. Advanced topics may include network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.

true

Fall, Spring

Undergraduate

4-0-8

6.1200 and 6.1210

6.1220[J]

false

False

18.413

Introduction to Computational Molecular Biology

Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.

true

Spring

Undergraduate

3-0-9

6.1210 or permission of instructor

null

false

False

18.415[J]

Advanced Algorithms

First-year graduate subject in algorithms. Emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Surveys a variety of computational models and the algorithms for them. Data structures, network flows, linear programming, computational geometry, approximation algorithms, online algorithms, parallel algorithms, external memory, streaming algorithms.

true

Fall

Graduate

5-0-7

6.1220 and (6.1200, 6.3700, or 18.600)

6.5210[J]

false

False

18.416[J]

Randomized Algorithms

Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

true

Fall

Graduate

5-0-7

(6.1200 or 6.3700) and (6.1220 or 6.5210)

6.5220[J]

false

False

18.417

Introduction to Computational Molecular Biology

Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.

true

Spring

Graduate

3-0-9

6.1210 or permission of instructor

null

false

False

18.418[J]

Topics in Computational Molecular Biology

Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the International Conference on Computational Molecular Biology (RECOMB) and the Conference on Intelligent Systems for Molecular Biology (ISMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, biological networks, drug discovery, and privacy. Recent research by course participants also covered. Participants will be expected to present individual projects to the class.

true

Fall

Graduate

3-0-9

6.8701, 18.417, or permission of instructor

HST.504[J]

false

False

18.424

Seminar in Information Theory

Considers various topics in information theory, including data compression, Shannon's Theorems, and error-correcting codes. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

true

Fall

Undergraduate

3-0-9

(6.3700, 18.05, or 18.600) and (18.06, 18.700, or 18.701)

null

false

False

18.425[J]

Foundations of Cryptography

A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives such as public-key encryption, digital signatures, and pseudo-random number generation, as well as advanced cryptographic primitives such as zero-knowledge proofs, homomorphic encryption, and secure multiparty computation.

true

Fall

Graduate

3-0-9

6.1220, 6.1400, or 18.4041

6.5620[J]

false

False

18.434

Seminar in Theoretical Computer Science

Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

true

Fall

Undergraduate

3-0-9

6.1220

null

false

False

18.435[J]

Quantum Computation

Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.

true

Fall

Graduate

3-0-9

8.05, 18.06, 18.700, 18.701, or 18.C06

2.111[J], 6.6410[J], 8.370[J]

false

False

18.436[J]

Quantum Information Science

Examines quantum computation and quantum information. Topics include quantum circuits, the quantum Fourier transform and search algorithms, the quantum operations formalism, quantum error correction, Calderbank-Shor-Steane and stabilizer codes, fault tolerant quantum computation, quantum data compression, quantum entanglement, capacity of quantum channels, and quantum cryptography and the proof of its security. Prior knowledge of quantum mechanics required.

true

Spring

Graduate

3-0-9

18.435

6.6420[J], 8.371[J]

false

False

18.437[J]

Distributed Algorithms

Design and analysis of algorithms, emphasizing those suitable for use in distributed networks. Covers various topics including distributed graph algorithms, locality constraints, bandwidth limitations and communication complexity, process synchronization, allocation of computational resources, fault tolerance, and asynchrony. No background in distributed systems required.

true

Spring

Graduate

3-0-9

6.1220

6.5250[J]

false

False

18.453

Combinatorial Optimization

Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as 18.200) helpful.

true

Spring

Undergraduate

3-0-9

18.06, 18.700, or 18.701

null

false

False

18.4531

Combinatorial Optimization

Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as 18.200) helpful. Students in Course 18 must register for the undergraduate version, 18.453.

true

Spring

Graduate

3-0-9

18.06, 18.700, or 18.701

null

false

False

18.455

Advanced Combinatorial Optimization

Advanced treatment of combinatorial optimization with an emphasis on combinatorial aspects. Non-bipartite matchings, submodular functions, matroid intersection/union, matroid matching, submodular flows, multicommodity flows, packing and connectivity problems, and other recent developments.

true

Spring

Graduate

3-0-9

18.453 or permission of instructor

null

false

False

18.456[J]

Algebraic Techniques and Semidefinite Optimization

Theory and computational techniques for optimization problems involving polynomial equations and inequalities with particular, emphasis on the connections with semidefinite optimization. Develops algebraic and numerical approaches of general applicability, with a view towards methods that simultaneously incorporate both elements, stressing convexity-based ideas, complexity results, and efficient implementations. Examples from several engineering areas, in particular systems and control applications. Topics include semidefinite programming, resultants/discriminants, hyperbolic polynomials, Groebner bases, quantifier elimination, and sum of squares.

true

Spring

Graduate

3-0-9

6.7210 or 15.093

6.7230[J]

false

False

18.504

Seminar in Logic

Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

true

Fall

Undergraduate

3-0-9

(18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)

null

false

False

18.510

Introduction to Mathematical Logic and Set Theory

Propositional and predicate logic. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Godel's incompleteness theorem.

true

Fall

Undergraduate

3-0-9

null

false

False

18.515

Mathematical Logic

More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and completeness. Compactness and its consequences. Quantifier elimination. Recursive sets and functions. Incompleteness and undecidability. Ordinals and cardinals. Set-theoretic formalization of mathematics.

true

Spring

Graduate

3-0-9

Permission of instructor

null

false

False

18.600

Probability and Random Variables

Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem. Credit cannot also be received for 6.041A or 6.041B.

true

Fall, Spring

Undergraduate

4-0-8

Calculus II (GIR)

null

false

true

False

18.604

Seminar In Probability Theory (New)

Students work on group presentations on topics selected by students from a provided list of suggestions. Topics may include Benford's law, random walks and electrical networks, and Brownian motions. Assignments include three group presentations, two individual presentations, and a final individual term paper. Instruction in oral and written communication provided to effectively communicate about probability theory. Limited to 16.

true

Spring

Undergraduate

3-0-9

18.05 or 18.600

null

false

False

18.615

Introduction to Stochastic Processes

Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion.

true

Spring

Graduate

3-0-9

6.3700 or 18.600

null

false

False

18.619[J]

Discrete Probability and Stochastic Processes

Provides an introduction to tools used for probabilistic reasoning in the context of discrete systems and processes. Tools such as the probabilistic method, first and second moment method, martingales, concentration and correlation inequalities, theory of random graphs, weak convergence, random walks and Brownian motion, branching processes, Markov chains, Markov random fields, correlation decay method, isoperimetry, coupling, influences and other basic tools of modern research in probability will be presented. Algorithmic aspects and connections to statistics and machine learning will be emphasized.

true

Spring

Graduate

3-0-9

6.3702, 6.7700, 18.100A, 18.100B, or 18.100Q

6.7720[J], 15.070[J]

false

False

18.642

Topics in Mathematics with Applications in Finance

Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required.

true

Fall

Undergraduate

3-0-9

18.03, 18.06, and (18.05 or 18.600)

null

false

False

18.650[J]

Fundamentals of Statistics

A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification.

true

Fall, Spring

Undergraduate

4-0-8

6.3700 or 18.600

IDS.014[J]

false

False

18.6501

Fundamentals of Statistics

A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification. Students in Course 18 must register for the undergraduate version, 18.650.

true

Fall, Spring

Graduate

4-0-8

6.3700 or 18.600

null

false

False

18.655

Mathematical Statistics

Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis. Prior exposure to both probability and statistics at the university level is assumed.

true

Spring

Graduate

3-0-9

(18.650 and (18.100A, 18.100A, 18.100P, or 18.100Q)) or permission of instructor

null

false

False

18.656[J]

Mathematical Statistics: a Non-Asymptotic Approach

Introduces students to modern non-asymptotic statistical analysis. Topics include high-dimensional models, nonparametric regression, covariance estimation, principal component analysis, oracle inequalities, prediction and margin analysis for classification. Develops a rigorous probabilistic toolkit, including tail bounds and a basic theory of empirical processes

true

Spring

Graduate

3-0-9

(6.7700, 18.06, and 18.6501) or permission of instructor

9.521[J], IDS.160[J]

false

False

18.657

Topics in Statistics

Topics vary from term to term.

true

Spring

Graduate

3-0-9

Permission of instructor

null

false

False

18.675

Theory of Probability

Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. Prior exposure to probability (e.g., 18.600) recommended.

true

Fall

Graduate

3-0-9

18.100A, 18.100B, 18.100P, or 18.100Q

null

false

False

18.676

Stochastic Calculus

Introduction to stochastic processes, building on the fundamental example of Brownian motion. Topics include Brownian motion, continuous parameter martingales, Ito's theory of stochastic differential equations, Markov processes and partial differential equations, and may also include local time and excursion theory. Students should have familiarity with Lebesgue integration and its application to probability.

true

Spring

Graduate

3-0-9

18.675

null

false

False

18.677

Topics in Stochastic Processes

Topics vary from year to year.

true

Fall

Graduate

3-0-9

18.675

null

false

False

18.700

Linear Algebra

Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06.

true

Fall

Undergraduate

3-0-9

Calculus II (GIR)

null

false

true

False

18.701

Algebra I

18.701-18.702 is more extensive and theoretical than the 18.700-18.703 sequence. Experience with proofs necessary. 18.701 focuses on group theory, geometry, and linear algebra.

true

Fall

Undergraduate

3-0-9

18.100A, 18.100B, 18.100P, 18.100Q, 18.090, or permission of instructor

null

false

False

18.702

Algebra II

Continuation of 18.701. Focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

true

Spring

Undergraduate

3-0-9

18.701

null

false

False

18.703

Modern Algebra

Focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals and unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. 18.700 and 18.703 together form a standard algebra sequence.

true

Spring

Undergraduate

3-0-9

Calculus II (GIR)

null

false

False

18.704

Seminar in Algebra

Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Some experience with proofs required. Enrollment limited.

true

Fall

Undergraduate

3-0-9

18.701, (18.06 and 18.703), or (18.700 and 18.703)

null

false

False

18.705

Commutative Algebra

Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.

true

Fall

Graduate

3-0-9

18.702

null

false

False

18.706

Noncommutative Algebra

Topics may include Wedderburn theory and structure of Artinian rings, Morita equivalence and elements of category theory, localization and Goldie's theorem, central simple algebras and the Brauer group, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension.

true

Spring

Graduate

3-0-9

18.702

null

false

False

18.708

Topics in Algebra

Topics vary from year to year.

true

Fall

Graduate

3-0-9

18.705

null

false

False

18.715

Introduction to Representation Theory

Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.

false

Spring

Graduate

3-0-9

18.702 or 18.703

null

false

False

18.721

Introduction to Algebraic Geometry

Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology.

true

Spring

Undergraduate

3-0-9

18.702 and 18.901

null

false

False

18.725

Algebraic Geometry I

Introduces the basic notions and techniques of modern algebraic geometry. Covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. Introduction to the language of schemes and properties of morphisms. Knowledge of elementary algebraic topology, elementary differential geometry recommended, but not required.

true

Fall

Graduate

3-0-9

None. Coreq: 18.705

null

false

False

18.726

Algebraic Geometry II

Continuation of the introduction to algebraic geometry given in 18.725. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology.

true

Spring

Graduate

3-0-9

18.725

null

false

False

18.727

Topics in Algebraic Geometry

Topics vary from year to year.

true

Spring

Graduate

3-0-9

18.725

null

false

False

18.737

Algebraic Groups

Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof.

false

Spring

Graduate

3-0-9

18.705

null

false

False

18.745

Lie Groups and Lie Algebras I

Covers fundamentals of the theory of Lie algebras and related groups. Topics may include theorems of Engel and Lie; enveloping algebra, Poincare-Birkhoff-Witt theorem; classification and construction of semisimple Lie algebras; the center of their enveloping algebras; elements of representation theory; compact Lie groups and/or finite Chevalley groups.

true

Fall

Graduate

3-0-9

(18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or 18.100Q)

null

false

False

18.747

Infinite-dimensional Lie Algebras

Topics vary from year to year.

true

Fall

Graduate

3-0-9

18.745

null

false

False

18.748

Topics in Lie Theory

Topics vary from year to year.

true

Spring

Graduate

3-0-9

Permission of instructor

null

false

False

18.755

Lie Groups and Lie Algebras II

A more in-depth treatment of Lie groups and Lie algebras. Topics may include homogeneous spaces and groups of automorphisms; representations of compact groups and their geometric realizations, Peter-Weyl theorem; invariant differential forms and cohomology of Lie groups and homogeneous spaces; complex reductive Lie groups, classification of real reductive groups.

true

Spring

Graduate

3-0-9

18.745 or permission of instructor

null

false

False

18.757

Representations of Lie Groups

Covers representations of locally compact groups, with emphasis on compact groups and abelian groups. Includes Peter-Weyl theorem and Cartan-Weyl highest weight theory for compact Lie groups.

true

Fall

Graduate

3-0-9

18.745 or 18.755

null

false

False

18.758

Methods of Representation Theory

Devoted to contemporary methods in representation theory of Lie groups, algebraic groups, and their generalizations. Topics may include: Springer correspondence, highest weight modules and Harish-Chandra bimodules, quantum groups and their representations, modular representations of algebraic groups and relation to quantum groups at a root of unity, representations of p-adic group, introduction to automorphic forms and Langlands duality, and representations of finite Chevalley groups.

true

Spring

Graduate

3-0-9

18.745 and (18.737 or 18.755)

null

false

False

18.781

Theory of Numbers

An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.

true

Spring

Undergraduate

3-0-9

null

false

False

18.782

Introduction to Arithmetic Geometry

Exposes students to arithmetic geometry, motivated by the problem of finding rational points on curves. Includes an introduction to p-adic numbers and some fundamental results from number theory and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include Mordell's theorem, the Weil conjectures, and Jacobian varieties.

true

Spring

Undergraduate

3-0-9

18.702

null

false

False

18.783

Elliptic Curves

Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem.

true

Fall

Undergraduate

3-0-9

18.702, 18.703, or permission of instructor

null

false

False

18.7831

Elliptic Curves

Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. Students in Course 18 must register for the undergraduate version, 18.783.

true

Fall

Graduate

3-0-9

18.702, 18.703, or permission of instructor

null

false

False

18.784

Seminar in Number Theory

Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

true

Spring

Undergraduate

3-0-9

18.701 or (18.703 and (18.06 or 18.700))

null

false

False

18.785

Number Theory I

Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Local fields, ramification, discriminants. Zeta and L-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem.

true

Fall

Graduate

3-0-9

None. Coreq: 18.705

null

false

False

18.786

Number Theory II

Continuation of 18.785. More advanced topics in number theory, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, or quadratic forms.

true

Spring

Graduate

3-0-9

18.785

null

false

False

18.787

Topics in Number Theory

Topics vary from year to year.

true

Fall

Graduate

3-0-9

Permission of instructor

null

false

False

18.821

Project Laboratory in Mathematics

Guided research in mathematics, employing the scientific method. Students confront puzzling and complex mathematical situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain them mathematically. Students choose three projects from a large collection of options. Each project results in a laboratory report subject to revision; oral presentation on one or two projects. Projects drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and probability. Enrollment limited.

true

Fall, Spring

Undergraduate

3-6-3

Two mathematics subjects numbered 18.100 or above

null

true

false

False

18.896[J]

Leadership and Professional Strategies & Skills Training (LEAPS), Part I: Advancing Your Professional Strategies and Skills

Part I (of two parts) of the LEAPS graduate career development and training series. Topics include: navigating and charting an academic career with confidence; convincing an audience with clear writing and arguments; mastering public speaking and communications; networking at conferences and building a brand; identifying transferable skills; preparing for a successful job application package and job interviews; understanding group dynamics and different leadership styles; leading a group or team with purpose and confidence. Postdocs encouraged to attend as non-registered participants. Limited to 80.

true

Spring

Graduate

2-0-1 [P/D/F]

null

5.961[J], 8.396[J], 9.980[J], 12.396[J]

false

False

18.897[J]

Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies

Part II (of two parts) of the LEAPS graduate career development and training series. Topics covered include gaining self awareness and awareness of others, and communicating with different personality types; learning about team building practices; strategies for recognizing and resolving conflict and bias; advocating for diversity and inclusion; becoming organizationally savvy; having the courage to be an ethical leader; coaching, mentoring, and developing others; championing, accepting, and implementing change. Postdocs encouraged to attend as non-registered participants. Limited to 80.

true

Spring

Graduate

2-0-1 [P/D/F]

null

5.962[J], 8.397[J], 9.981[J], 12.397[J]

false

False

18.899

Internship in Mathematics (New)

Provides academic credit for students pursuing internships to gain practical experience applications of mathematical concepts and methods as related to their field of research.

true

Fall, Spring, Summer

Graduate

rranged [P/D/F]

null

false

False

18.900

Geometry and Topology in the Plane

Introduction to selected aspects of geometry and topology, using concepts that can be visualized easily. Mixes geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). Suitable for students with no prior exposure to differential geometry or topology.

true

Spring

Undergraduate

3-0-9

18.03 or 18.06

null

false

False

18.901

Introduction to Topology

Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group.

true

Fall, Spring

Undergraduate

3-0-9

18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor

null

false

False

18.9011

Introduction to Topology

Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group. Students in Course 18 must register for the undergraduate version, 18.901.

true

Fall, Spring

Graduate

3-0-9

18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor

null

false

False

18.904

Seminar in Topology

Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

true

Spring

Undergraduate

3-0-9

18.901

null

false

False

18.905

Algebraic Topology I

Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.

true

Fall

Graduate

3-0-9

18.901 and (18.701 or 18.703)

null

false

False

18.906

Algebraic Topology II

Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.

true

Spring

Graduate

3-0-9

18.905 and (18.101 or 18.965)

null

false

False

18.917

Topics in Algebraic Topology

Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.

true

Spring

Graduate

3-0-9

18.906

null

false

False

18.919

Graduate Topology Seminar

Study and discussion of important original papers in the various parts of topology. Open to all students who have taken 18.906 or the equivalent, not only prospective topologists.

true

Spring

Graduate

3-0-9

18.906

null

false

False

18.937

Topics in Geometric Topology

Content varies from year to year. Introduces new and significant developments in geometric topology.

false

Fall, Spring

Graduate

3-0-9

Permission of instructor

null

false

False

18.950

Differential Geometry

Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space.

true

Fall

Undergraduate

3-0-9

(18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)

null

false

False

18.9501

Differential Geometry

Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space. Students in Course 18 must register for the undergraduate version, 18.950.

true

Fall

Graduate

3-0-9

(18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)

null

false

False

18.952

Theory of Differential Forms

Multilinear algebra: tensors and exterior forms. Differential forms on Rn: exterior differentiation, the pull-back operation and the Poincaré lemma. Applications to physics: Maxwell's equations from the differential form perspective. Integration of forms on open sets of Rn. The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes' theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology.

false

Spring

Undergraduate

3-0-9

18.101 and (18.700 or 18.701)

null

false

False

18.965

Geometry of Manifolds I

Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a continuation of 18.965 and focuses more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. Prior exposure to calculus on manifolds, as in 18.952, recommended.

true

Fall

Graduate

3-0-9

18.101, 18.950, or 18.952

null

false

False

18.966

Geometry of Manifolds II

Continuation of 18.965, focusing more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology.

true

Spring

Graduate

3-0-9

18.965

null

false

False

18.968

Topics in Geometry

Content varies from year to year.

true

Spring

Graduate

3-0-9

18.965

null

false

False

18.979

Graduate Geometry Seminar

Content varies from year to year. Study of classical papers in geometry and in applications of analysis to geometry and topology.

true

Spring

Graduate

3-0-9

Permission of instructor

null

false

False