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Course 18: Mathematics |
 | | | 18.01-18.499 | | | 18.50-18.THG | | |  |
Logic18.504 Seminar in Logic
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Prereq: (18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q) Units: 3-0-9 
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. Staff 18.510 Introduction to Mathematical Logic and Set Theory
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Prereq: None Units: 3-0-9
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. Staff 18.515 Mathematical Logic
 ()Not offered regularly; consult department Prereq: Permission of instructor Units: 3-0-9
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. Staff Probability and Statistics18.600 Probability and Random Variables
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Prereq: Calculus II (GIR) Units: 4-0-8 Credit cannot also be received for 6.3700, 6.3702  Lecture: MWF1 (54-100) Recitation: R10 (2-147) or R11 (2-147) or R12 (2-147) or R1 (2-147) +final
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. S. Sheffield No textbook information available 18.604 Seminar In Probability Theory
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Prereq: 18.05 or 18.600 Units: 3-0-9 Lecture: MW11-12.30 (2-146)
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. J. Borga No required or recommended textbooks 18.615 Introduction to Stochastic Processes
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Prereq: 6.3700 or 18.600 Units: 3-0-9 Lecture: MW9.30-11 (4-261)
Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion. E. Mossel No required or recommended textbooks 18.619[J] Discrete Probability and Stochastic Processes
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(Same subject as 6.7720[J], 15.070[J]) Prereq: 6.3702, 6.7700, 18.100A, 18.100B, or 18.100Q Units: 3-0-9
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. K. Liu 18.642 Topics in Mathematics with Applications in Finance
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Prereq: 18.03, 18.06, and (18.05 or 18.600) Units: 3-0-9 Lecture: TR2.30-4 (37-212)
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. P. Kempthorne, V. Strela, J. Xia No required or recommended textbooks 18.650[J] Fundamentals of Statistics
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(Same subject as IDS.014[J]) (Subject meets with 18.6501) Prereq: 6.3700 or 18.600 Units: 4-0-8 Lecture: MWF10 (2-190) Recitation: R11 (4-149) or R12 (4-149) or R2 (4-149) +final
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. P. Rigollet Textbooks (Fall 2025) 18.6501 Fundamentals of Statistics
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(Subject meets with 18.650[J], IDS.014[J]) Prereq: 6.3700 or 18.600 Units: 4-0-8 Lecture: MWF10 (2-190) Recitation: R11 (4-149) or R12 (4-149) or R2 (4-149) +final
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. Fall: P. Rigollet Spring: A. Katsevich No textbook information available 18.655 Mathematical Statistics
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Prereq: (18.650 and (18.100A, 18.100A, 18.100P, or 18.100Q)) or permission of instructor Units: 3-0-9
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. P. Kempthorne 18.656[J] Mathematical Statistics: a Non-Asymptotic Approach
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(Same subject as 9.521[J], IDS.160[J]) Prereq: (6.7700, 18.06, and 18.6501) or permission of instructor Units: 3-0-9
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 S. Rakhlin, P. Rigollet 18.657 Topics in Statistics
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Prereq: Permission of instructor Units: 3-0-9
Topics vary from term to term. P. Rigollet 18.675 Theory of Probability
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Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q Units: 3-0-9 Lecture: TR2.30-4 (4-163) +final
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. K. Kavvadias No required or recommended textbooks 18.676 Stochastic Calculus
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Prereq: 18.675 Units: 3-0-9
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. Staff 18.677 Topics in Stochastic Processes
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Prereq: 18.675 Units: 3-0-9
Topics vary from year to year. Staff For additional related subjects in Statistics, see: Civil and Environmental Engineering: 1.202, 1.203J, and 1.205 Electrical Engineering and Computer Science: 6.0002, 6.041, 6.231, 6.245, 6.262, 6.431, 6.434J, 6.435, 6.436J, 6.437, and 6.438 Management: 15.034, 15.060, 15.070J, 15.071, 15.075J, 15.077J, 15.098, and 15.456 IDSS: IDS.012J, IDS.013J, IDS.014J, IDS.136J, and IDS.700J Economics: 14.30, 14.310, 14.36, 14.381, 14.382, 14.384, and 14.386 Algebra and Number Theory18.700 Linear Algebra
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Prereq: Calculus II (GIR) Units: 3-0-9 Credit cannot also be received for 6.C06, 18.06, 18.C06, ES.1806 Lecture: MW9.30-11 (6-120) +final
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. J. Kim Textbooks (Fall 2025) 18.701 Algebra I
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Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, 18.090, or permission of instructor Units: 3-0-9 Lecture: MW2.30-4 (34-101)
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. H. Cohn No textbook information available 18.702 Algebra II
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Prereq: 18.701 Units: 3-0-9
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. Staff 18.703 Modern Algebra
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Prereq: Calculus II (GIR) Units: 3-0-9
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. V. G. Kac 18.704 Seminar in Algebra
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Prereq: 18.701, (18.06 and 18.703), or (18.700 and 18.703) Units: 3-0-9 Lecture: TR2.30-4 (2-132)
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. D. Maulik No textbook information available 18.705 Commutative Algebra
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Prereq: 18.702 Units: 3-0-9 Lecture: MW2.30-4 (4-145)
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. P. Etingof No textbook information available 18.706 Noncommutative Algebra
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Prereq: 18.702 Units: 3-0-9
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. Staff 18.708 Topics in Algebra
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Prereq: 18.705 Units: 3-0-9
Topics vary from year to year. Staff 18.715 Introduction to Representation Theory
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Prereq: 18.702 or 18.703 Units: 3-0-9
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. Staff 18.721 Introduction to Algebraic Geometry
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Prereq: 18.702 and 18.901 Units: 3-0-9
Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology. Staff 18.725 Algebraic Geometry I
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Prereq: None. Coreq: 18.705 Units: 3-0-9 Lecture: MW9.30-11 (2-139) +final
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. Z. Yun Textbooks (Fall 2025) 18.726 Algebraic Geometry II
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Prereq: 18.725 Units: 3-0-9
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. Staff 18.727 Topics in Algebraic Geometry
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Prereq: 18.725 Units: 3-0-9
Topics vary from year to year. Staff 18.737 Algebraic Groups
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Prereq: 18.705 Units: 3-0-9
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. Staff 18.745 Lie Groups and Lie Algebras I
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Prereq: (18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or 18.100Q) Units: 3-0-9 Lecture: TR2.30-4 (2-146)
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. V. Kac No textbook information available 18.747 Infinite-dimensional Lie Algebras
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Prereq: 18.745 Units: 3-0-9
Topics vary from year to year. Staff 18.748 Topics in Lie Theory
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Prereq: Permission of instructor Units: 3-0-9
Topics vary from year to year. Staff 18.755 Lie Groups and Lie Algebras II
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Prereq: 18.745 or permission of instructor Units: 3-0-9
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. Staff 18.757 Representations of Lie Groups
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Prereq: 18.745 or 18.755 Units: 3-0-9 Lecture: TR9.30-11 (2-146)
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. R. Bezrukavnikov No textbook information available 18.758 Methods of Representation Theory
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Prereq: 18.745 and (18.737 or 18.755) Units: 3-0-9
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. Staff 18.781 Theory of Numbers
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Prereq: None Units: 3-0-9
An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions. Staff 18.782 Introduction to Arithmetic Geometry
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Prereq: 18.702 Units: 3-0-9
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. Staff 18.783 Elliptic Curves
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(Subject meets with 18.7831) Prereq: 18.702, 18.703, or permission of instructor Units: 3-0-9 Lecture: TR9.30-11 (2-136)
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. A. Sutherland No required or recommended textbooks 18.7831 Elliptic Curves
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(Subject meets with 18.783) Prereq: 18.702, 18.703, or permission of instructor Units: 3-0-9 Lecture: TR9.30-11 (2-136)
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. A. Sutherland No textbook information available 18.784 Seminar in Number Theory
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Prereq: 18.701 or (18.703 and (18.06 or 18.700)) Units: 3-0-9
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited. Staff 18.785 Number Theory I
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Prereq: None. Coreq: 18.705 Units: 3-0-9 Lecture: TR11-12.30 (2-139)
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. A. Sutherland No required or recommended textbooks 18.786 Number Theory II
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Prereq: 18.785 Units: 3-0-9
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. Staff 18.787 Topics in Number Theory
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Prereq: Permission of instructor Units: 3-0-9 Lecture: TR1-2.30 (2-146)
Topics vary from year to year. W. Zhang No textbook information available Mathematics Laboratory18.821 Project Laboratory in Mathematics
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Prereq: Two mathematics subjects numbered 18.100 or above Units: 3-6-3 URL: http://math.mit.edu/classes/18.821/ Lecture: MW9-11 (2-131) Lab: TBA
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. Fall: T. Mrowka Spring: R. Bezrukavnikov No textbook information available 18.896[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part I: Advancing Your Professional Strategies and Skills
(); second half of term
(Same subject as 5.961[J], 8.396[J], 9.980[J], 12.396[J]) Prereq: None Units: 2-0-1 [P/D/F]
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. A. Frebel 18.897[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies
(); first half of term
(Same subject as 5.962[J], 8.397[J], 9.981[J], 12.397[J]) Prereq: None Units: 2-0-1 [P/D/F]
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. D. Rigos 18.899 Internship in Mathematics
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Prereq: None Units arranged [P/D/F] TBA.
Provides academic credit for students pursuing internships to gain practical experience applications of mathematical concepts and methods as related to their field of research. T. Cummings No required or recommended textbooks Topology and Geometry18.900 Geometry and Topology in the Plane
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Prereq: 18.03 or 18.06 Units: 3-0-9
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. Staff 18.901 Introduction to Topology
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(Subject meets with 18.9011) Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor Units: 3-0-9 Lecture: TR11-12.30 (6-120) +final
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. P. Seidel Textbooks (Fall 2025) 18.9011 Introduction to Topology
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(Subject meets with 18.901) Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor Units: 3-0-9 Lecture: TR11-12.30 (6-120) +final
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. Fall: A. Pieloch Spring: R. Jiang No textbook information available 18.904 Seminar in Topology
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Prereq: 18.901 Units: 3-0-9
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited. Staff 18.905 Algebraic Topology I
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Prereq: 18.901 and (18.701 or 18.703) Units: 3-0-9
Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality. Staff 18.906 Algebraic Topology II
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Prereq: 18.905 Units: 3-0-9 Lecture: TR11-12.30 (2-136)
Continues the development of algebraic topology, with a focus on homotopical questions and computational tools. Topics include basic homotopy theory, classifying spaces, spectral sequences, and cohomology operations. Usually, only one of 18.906 or 18.916 is offered in a given academic year. J. Hahn No textbook information available 18.916 Geometric Methods in Algebraic Topology
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Subject Cancelled
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