Department of Mathematics
The Department of Mathematics offers training at the undergraduate, graduate, and postgraduate levels. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, and theoretical physics.
Course 18 includes two undergraduate degrees: a Bachelor of Science in Mathematics and a Bachelor of Science in Mathematics with Computer Science. Undergraduate students may choose one of three options leading to the Bachelor of Science in Mathematics: applied mathematics, pure mathematics, or general mathematics. The general mathematics option provides a great deal of flexibility and allows students to design their own programs in conjunction with their advisors. The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program.
At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics.
The CLE Moore instructorships and Applied Mathematics instructorships bring mathematicians at the postdoctoral level to MIT and provide them with training in research and teaching.
Bachelor of Science in Mathematics (Course 18)
Bachelor of Science in Mathematics with Computer Science (Course 18-C)
Undergraduate Study
An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such fields as finance, business, or consulting. Students' programs are arranged through consultation with their faculty advisors.
Undergraduates in mathematics are encouraged to elect an undergraduate seminar during their junior or senior year. The experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students planning to pursue graduate work as well as for those going on to other careers. These seminars also provide training in the verbal and written communication of mathematics and may be used to fulfill the Communication Requirement.
Many mathematics majors take 18.821 Project Laboratory in Mathematics, which fulfills the Institute's Laboratory Requirement and counts toward the Communication Requirement.
Bachelor of Science in Mathematics (Course 18)
General Mathematics Option
In addition to the General Institute Requirements, the requirements consist of Differential Equations, plus eight additional 12-unit subjects in Course 18 of essentially different content, including at least six advanced subjects (first decimal digit one or higher). One of these eight subjects must be Linear Algebra. This leaves available 84 units of unrestricted electives. The requirements are flexible in order to accommodate students who pursue programs that combine mathematics with a related field (such as physics, economics, or management) as well as students who are interested in both pure and applied mathematics. More details can be found on the degree chart.
Applied Mathematics Option
Applied mathematics focuses on the mathematical concepts and techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.
Sophomores interested in applied mathematics typically enroll in 18.200 Principles of Discrete Applied Mathematics and 18.300 Principles of Continuum Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of applied mathematics and may be taken concurrently with 18.03 Differential Equations. Subject 18.300, offered in the spring term, is devoted to continuous aspects and makes considerable use of differential equations.
The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.
Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.
More detail on the Applied Mathematics option can be found on the degree chart.
Pure Mathematics Option
Pure (or "theoretical") mathematics is the study of the basic concepts and structure of mathematics. Its goal is to arrive at a deeper understanding and an expanded knowledge of mathematics itself.
Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.
The subjects 18.701 Algebra I and 18.901 Introduction to Topology are more advanced and should not be elected until a student has had experience with proofs, as in Real Analysis (18.100A, 18.100B, 18.100P or 18.100Q) or 18.700 Linear Algebra.
For more details, see the degree chart.
Bachelor of Science in Mathematics with Computer Science (Course 18-C)
Mathematics and computer science are closely related fields. Problems in computer science are often formalized and solved with mathematical methods. It is likely that many important problems currently facing computer scientists will be solved by researchers skilled in algebra, analysis, combinatorics, logic and/or probability theory, as well as computer science.
The purpose of this program is to allow students to study a combination of these mathematical areas and potential areas of application in computer science. Required subjects include linear algebra (18.06 or 18.700) because it is so broadly used, and discrete mathematics (18.062[J] or 18.200) to give experience with proofs and the necessary tools for analyzing algorithms. The required subjects covering complexity (18.404 Theory of Computation or 18.400[J] Automata, Computability, and Complexity) and algorithms (18.410[J] Design and Analysis of Algorithms) provide an introduction to the most theoretical aspects of computer science. We also require exposure to other areas of computer science (6.031, 6.033, 6.034, or 6.036) where mathematical issues may also arise. More details can be found on the degree chart.
Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject 18.701 Algebra I for 18.06 Linear Algebra, and, if they already have strong theorem-proving skills, may substitute 18.211 Combinatorial Analysis or 18.212 Algebraic Combinatorics for 18.062[J] Mathematics for Computer Science or 18.200 Principles of Discrete Applied Mathematics.
Minor in Mathematics
The requirements for a Minor in Mathematics are as follows: six 12-unit subjects in mathematics, beyond the Institute's Mathematics Requirement, of essentially different content, including at least three advanced subjects (first decimal digit one or higher).
See the Undergraduate Section for a general description of the minor program.
Inquiries
For further information, see the department's website or contact Math Academic Services, 617-253-2416.
Graduate Study
The Mathematics Department offers programs covering a broad range of topics leading to the Doctor of Philosophy or Doctor of Science degree. Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120-130 doctoral students, about two thirds are in Pure Mathematics, one third in Applied Mathematics.
The programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, theoretical physics, and the theory of computation. In addition, many mathematically oriented subjects are offered by other departments. Students in Applied Mathematics are especially encouraged to take subjects in engineering and scientific subjects related to their research.
All students pursue research under the supervision of the faculty and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.
Doctor of Philosophy or Doctor of Science
The requirements for these degrees are described on the department's website. In outline, they consist of a language requirement, an oral qualifying examination, a thesis proposal, completion of a minimum of 132 units (11 graduate subjects), and a thesis containing original research in mathematics.
Interdisciplinary Programs
Computational Science and Engineering
Students with primary interest in computational science may also consider applying to the interdisciplinary Computational Science and Engineering (CSE) program, with which the Mathematics Department is affiliated. For more information, see the CSE website.
Mathematics and Statistics
The Interdisciplinary Doctoral Program in Statistics provides training in statistics, including classical statistics and probability as well as computation and data analysis, to students who wish to integrate these valuable skills into their primary academic program. The program is administered jointly by the departments of Aeronautics and Astronautics, Economics, Mathematics, and Political Science, and the Statistics and Data Science Center within the Institute for Data, Systems, and Society. It is open to current doctoral students in participating departments, who may apply to enroll in the program at any time after the end of their first year. For more information, see the full program description under Interdisciplinary Graduate Programs.
Financial Support
Financial support is guaranteed for up to five years to students making satisfactory academic progress. Financial aid after the first year is usually in the form of a teaching or research assistantship.
Inquiries
For further information, see the department's website or contact Math Academic Services, 617-253-2416.
Faculty and Teaching Staff
Michel X. Goemans, PhD
Professor of Applied Mathematics
Head, Department of Mathematics
William Minicozzi, PhD
Professor of Mathematics
Associate Head, Department of Mathematics
Professors
Michael Artin, PhD
Professor Post-Tenure of Mathematics
Martin Z. Bazant, PhD
Professor of Chemical Engineering
Professor of Mathematics
Executive Officer, Department of Chemical Engineering
Bonnie Berger, PhD
Simons Professor of Mathematics
Professor of Computer Science
Member, Health Sciences and Technology Faculty
Roman Bezrukavnikov, PhD
Professor of Mathematics
Alexei Borodin, PhD
Professor of Mathematics
John W. M. Bush, PhD
Professor of Mathematics
Hung Cheng, PhD
Professor of Mathematics
Tobias Colding, PhD
Cecil and Ida Green Distinguished Professor
Professor of Mathematics
Alan Edelman, PhD
Professor of Mathematics
Pavel I. Etingof, PhD
Professor of Mathematics
Victor W. Guillemin, PhD
Professor of Mathematics
(On leave, spring)
Lawrence Guth, PhD
Professor of Mathematics
(On leave, fall)
Anette E. Hosoi, PhD
Neil and Jane Pappalardo Professor
Professor of Mechanical Engineering
Professor of Mathematics
Associate Dean, School of Engineering
David S. Jerison, PhD
Professor of Mathematics
(On leave, fall)
Steven G. Johnson, PhD
Professor of Mathematics
Professor of Physics
Victor Kac, PhD
Professor of Mathematics
Jonathan Adam Kelner, PhD
Mark Hyman, Jr. Career Development Professor
Professor of Mathematics
Ju-Lee Kim, PhD
Professor of Mathematics
Frank Thomson Leighton, PhD
Professor of Mathematics
(On leave)
George Lusztig, PhD
Edward A. Abdun-Nur (1924) Professor of Mathematics
Arthur P. Mattuck, PhD
Professor Post-Tenure of Mathematics
Davesh Maulik, PhD
Professor of Mathematics
Richard B. Melrose, PhD
Professor of Mathematics
(On leave, fall)
Haynes R. Miller, PhD
Professor of Mathematics
Elchanan Mossel, PhD
Professor of Mathematics
Member, Institute for Data, Systems, and Society
(On leave, fall)
Tomasz S. Mrowka, PhD
Professor of Mathematics
Bjorn Poonen, PhD
Claude E. Shannon (1940) Professor of Mathematics
Alexander Postnikov, PhD
Professor of Mathematics
Rodolfo R. Rosales, PhD
Professor of Mathematics
Paul Seidel, PhD
Levinson Professor in Mathematics
Scott Roger Sheffield, PhD
Leighton Family Professor of Mathematics
Member, Institute for Data, Systems, and Society
Peter W. Shor, PhD
Henry Adams Morss and Henry Adams Morss, Jr. (1934) Professor
Professor of Mathematics
(On leave, fall)
Michael Sipser, PhD
Donner Professor of Mathematics
Dean, School of Science
Gigliola Staffilani, PhD
Abby Rockefeller Mauzé Professor of Mathematics
Gilbert Strang, PhD
MathWorks Professor of Mathematics
Member, Institute for Data, Systems, and Society
Daniel W. Stroock, PhD
Professor Post-Tenure of Mathematics
David A. Vogan, PhD
Norbert Wiener Professor of Mathematics
(On leave, fall)
Chenyang Xu, PhD
Professor of Mathematics
Zhiwei Yun, PhD
Professor of Mathematics
Wei Zhang, PhD
Professor of Mathematics
Associate Professors
Laurent Demanet, PhD
Associate Professor of Mathematics
Associate Professor of Earth, Atmospheric and Planetary Sciences
Joern Dunkel, PhD
Associate Professor of Mathematics
Semyon Dyatlov, PhD
Associate Professor of Mathematics
(On leave)
Ankur Moitra, PhD
Rockwell International Career Development Professor
Associate Professor of Mathematics
Philippe Rigollet, PhD
Associate Professor of Mathematics
Member, Institute for Data, Systems, and Society
(On leave, spring)
Nike Sun, PhD
Associate Professor of Mathematics
Gonçalo Jorge Trigo Neri Tabuada, PhD
Associate Professor of Mathematics
Assistant Professors
Tristan Collins, PhD
Assistant Professor of Mathematics
Vadim Gorin, PhD
Assistant Professor of Mathematics
Peter Hintz, PhD
Assistant Professor of Mathematics
Andrew Lawrie, PhD
Assistant Professor of Mathematics
(On leave, fall)
Andrei Negut, PhD
Assistant Professor of Mathematics
(On leave, fall)
Aaron Pixton, PhD
Class of 1957 Career Development Professor
Assistant Professor of Mathematics
Giulia Sacca, PhD
Assistant Professor of Mathematics
(On leave)
Yufei Zhao, PhD
Assistant Professor of Mathematics
Visiting Professors
Olivier Bernardi, PhD
Visiting Professor of Mathematics
(Spring)
Alissa Crans, PhD
Visiting Professor of Mathematics
Kyo Nishiyama, PhD
Visiting Professor of Mathematics
Kasso Okoudjou, PhD
Visiting Professor of Mathematics
Bruce Sagan, PhD
Visiting Professor of Mathematics
Guoliang Wang, PhD
Visiting Professor of Mathematics
Visiting Associate Professors
Sebastian Cioaba, PhD
Visiting Associate Professor of Mathematics
Tamas Kalman, PhD
Visiting Associate Professor of Mathematics
Rodrigo Ristow Montes, PhD
Visiting Associate Professor of Mathematics
Visiting Assistant Professors
Jonathan Bloom, PhD
Visiting Assistant Professor of Mathematics
Jeffrey Oishi, PhD
Visiting Assistant Professor of Mathematics
(Fall)
Adjunct Professors
Henry Cohn, PhD
Adjunct Professor of Mathematics
Lecturers
Jennifer French, PhD
Lecturer in Mathematics
Slava Gerovitch, PhD
Lecturer in Mathematics
Peter J. Kempthorne, PhD
Lecturer in Mathematics
Tanya Khovanova, PhD
Lecturer in Mathematics
CLE Moore Instructors
Tom Bachmann, PhD
CLE Moore Instructor of Mathematics
Stéphane Benoist, PhD
CLE Moore Instructor of Mathematics
Kyeongsu Choi, PhD
CLE Moore Instructor of Mathematics
Nicholas Edelen, PhD
CLE Moore Instructor of Mathematics
Max D. Engelstein, PhD
CLE Moore Instructor of Mathematics
Pei-Ken Hung, PhD
CLE Moore Instructor of Mathematics
Benjamin Landon, PhD
CLE Moore Instructor of Mathematics
Jianfeng Lin, PhD
CLE Moore Instructor of Mathematics
Christos Mantoulidis, PhD
CLE Moore Instructor of Mathematics
Yu Pan, PhD
CLE Moore Instructor of Mathematics
Druv Ranganathan, PhD
CLE Moore Instructor of Mathematics
Casey Rodriguez, PhD
CLE Moore Instructor of Mathematics
Ananth Shankar, PhD
CLE Moore Instructor of Mathematics
Junliang Shen, PhD
CLE Moore Instructor of Mathematics
Luca Spolaor, PhD
CLE Moore Instructor of Mathematics
Matthew Stoffregen, PhD
CLE Moore Instructor of Mathematics
Chen Wan, PhD
CLE Moore Instructor of Mathematics
Junho Whang, PhD
CLE Moore Instructor of Mathematics
Zhouli Xu, PhD
CLE Moore Instructor of Mathematics
Xueying Yu, PhD
CLE Moore Instructor of Mathematics
Yiming Zhao, PhD
CLE Moore Instructor of Mathematics
Instructors of Applied Mathematics
Diego Cifuentes, PhD
Instructor of Applied Mathematics
Yash Deshpande, PhD
Schramm Fellow Instructor
Matthew Durey, PhD
Instructor of Applied Mathematics
Asaf Ferber, PhD
Instructor of Applied Mathematics
Zilin Jiang, PhD
Instructor of Applied Mathematics
Laszlo Lovasz, PhD
Instructor of Applied Mathematics
Tyler Maunu, PhD
Instructor of Applied Mathematics
Philip Pearce, PhD
Instructor of Applied Mathematics
Christopher Rackauckas, PhD
Instructor of Applied Mathematics
(On leave, spring)
Andrej Risteski, PhD
Norber Wiener Fellow
Elina Robeva, PhD
Instructor of Statistics
Henrik Ronellenfitsch, PhD
Instructor of Applied Mathematics
Pedro Sáenz Hervías, PhD
Instructor of Applied Mathematics
Subhabrata Sen, PhD
Schramm Fellow Instructor
Stuart Thomson, PhD
Instructor of Applied Mathematics
Instructors of Pure Mathematics
Zarathustra Brady, PhD
Instructor of Pure Mathematics
Erin Compaan, PhD
Instructor of Pure Mathematics
Research Staff
Principal Research Scientists
Andrew Victor Sutherland II, PhD
Principal Research Scientist of Mathematics
Research Scientists
Edgar Costa, PhD
Research Scientist of Mathematics
Maarten Derickx, PhD
Research Scientist of Mathematics
Dohyeong Kim, PhD
Research Scientist of Mathematics
David Roe, PhD
Research Scientist of Mathematics
David I. Spivak, PhD
Research Scientist of Mathematics
Professors Emeriti
Richard Dudley, PhD
Professor Emeritus of Mathematics
Daniel Z. Freedman, PhD
Professor Emeritus of Mathematics
Professor Emeritus of Physics
Harvey P. Greenspan, PhD
Professor Emeritus of Mathematics
Sigurdur Helgason, PhD
Professor Emeritus of Mathematics
Steven L. Kleiman, PhD
Professor Emeritus of Mathematics
Daniel J. Kleitman, PhD
Professor Emeritus of Mathematics
James R. Munkres, PhD
Professor Emeritus of Mathematics
Gerald E. Sacks, PhD
Professor Emeritus of Mathematics
Isadore Manuel Singer, PhD
Institute Professor Emeritus
Professor Emeritus of Mathematics
Richard P. Stanley, PhD
Professor Emeritus of Mathematics
Harold Stark, PhD
Professor Emeritus of Mathematics
Alar Toomre, PhD
Professor Emeritus of Mathematics
General Mathematics
18.01 Calculus
Prereq: None
U (Fall, Spring)
5-0-7 units. CALC I
Credit cannot also be received for 18.01A, CC.181A, ES.1801, ES.181A
Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions.
Fall: P. Seidel. Spring: Information: J. W. Bush
18.01A Calculus
Prereq: Knowledge of differentiation and elementary integration
U (Fall; first half of term)
5-0-7 units. CALC I
Credit cannot also be received for 18.01, CC.181A, ES.1801, ES.181A
Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Prerequisites: one year of high-school calculus or the equivalent, with a score of 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the 18.01 advanced standing exam.
D. Maulik
18.02 Calculus
Prereq: Calculus I (GIR)
U (Fall, Spring)
5-0-7 units. CALC II
Credit cannot also be received for 18.022, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A
Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.
Fall: W. Minicozzi. Spring: L. Guth
18.02A Calculus
Prereq: Calculus I (GIR)
U (Fall, IAP, Spring)
5-0-7 units. CALC II
Credit cannot also be received for 18.02, 18.022, CC.1802, CC.182A, ES.1802, ES.182A
First half is taught during the last six weeks of the Fall term; covers material in the first half of 18.02 (through double integrals). Second half of 18.02A can be taken either during IAP (daily lectures) or during the second half of the Spring term; it covers the remaining material in 18.02.
D. Maulik
18.022 Calculus
Prereq: Calculus I (GIR)
U (Fall)
5-0-7 units. CALC II
Credit cannot also be received for 18.02, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A
Calculus of several variables. Topics as in 18.02 but with more focus on mathematical concepts. Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions.
A. Borodin
18.03 Differential Equations
Prereq: None. Coreq: Calculus II (GIR)
U (Fall, Spring)
5-0-7 units. REST
Credit cannot also be received for 18.032, CC.1803, ES.1803
Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams.
Fall: J. Dunkel. Spring: D. Jerison
18.031 System Functions and the Laplace Transform
Prereq: None. Coreq: 18.03
U (IAP, Spring; second half of term)
1-0-2 units
Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering.
P. Pearce
18.032 Differential Equations
Prereq: None. Coreq: Calculus II (GIR)
U (Spring)
5-0-7 units. REST
Credit cannot also be received for 18.03, CC.1803, ES.1803
Covers much of the same material as 18.03 with more emphasis on theory. The point of view is rigorous and results are proven. Local existence and uniqueness of solutions.
T. Colding
18.04 Complex Variables with Applications
Prereq: Calculus II (GIR) and (18.03 or 18.032)
U (Spring)
4-0-8 units
Credit cannot also be received for 18.075, 18.0751
Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations.
J. Dunkel
18.05 Introduction to Probability and Statistics
Prereq: Calculus II (GIR)
U (Spring)
4-0-8 units. REST
Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.
J. Orloff
18.06 Linear Algebra
Prereq: Calculus II (GIR)
U (Fall, Spring)
4-0-8 units. REST
Credit cannot also be received for 18.700
Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses linear algebra software. Compared with 18.700, more emphasis on matrix algorithms and many applications.
S. G. Johnson, A. Edelman
18.062[J] Mathematics for Computer Science
Same subject as 6.042[J]
Prereq: Calculus I (GIR)
U (Fall, Spring)
5-0-7 units. REST
See description under subject 6.042[J].
F. T. Leighton, Z. R. Abel, A. Moitra
18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
Subject meets with 18.0651
Prereq: 18.06
U (Spring)
3-0-9 units
Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices.
G. Strang
18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
Subject meets with 18.065
Prereq: 18.06
G (Spring)
3-0-9 units
Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices. Students in Course 18 must register for the undergraduate version, 18.065.
G. Strang
18.075 Methods for Scientists and Engineers
Subject meets with 18.0751
Prereq: Calculus II (GIR) and 18.03
U (Spring)
3-0-9 units
Credit cannot also be received for 18.04
Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations.
H. Cheng
18.0751 Methods for Scientists and Engineers
Subject meets with 18.075
Prereq: Calculus II (GIR) and 18.03
G (Spring)
3-0-9 units
Credit cannot also be received for 18.04
Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations. Students in Courses 6, 8, 12, 18, and 22 must register for undergraduate version, 18.075.
H. Cheng
18.085 Computational Science and Engineering I
Subject meets with 18.0851
Prereq: Calculus II (GIR) and (18.03 or 18.032)
U (Fall, Spring, Summer)
3-0-9 units
Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.
Fall: W. G. Strang. Spring: L. Demanet
18.0851 Computational Science and Engineering I
Subject meets with 18.085
Prereq: Calculus II (GIR) and (18.03 or 18.032)
G (Fall, Spring, Summer)
3-0-9 units
Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications. Students in Course 18 must register for the undergraduate version, 18.085.
Fall: W.G. Strang. Spring: L. Demanet
18.086 Computational Science and Engineering II
Subject meets with 18.0861
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: U (Spring)
3-0-9 units
Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods.
Information: W. G. Strang
18.0861 Computational Science and Engineering II
Subject meets with 18.086
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods. Students in Course 18 must register for the undergraduate version, 18.086.
Information: W. G. Strang
18.089 Review of Mathematics
Prereq: Permission of instructor
G (Summer)
5-0-7 units
One-week review of one-variable calculus (18.01), followed by concentrated study covering multivariable calculus (18.02), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances.
Information: J. W. Bush
18.094[J] Teaching College-Level Science and Engineering
Same subject as 1.95[J], 5.95[J], 7.59[J], 8.395[J]
Subject meets with 2.978
Prereq: None
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
2-0-2 units
See description under subject 5.95[J].
J. Rankin
18.095 Mathematics Lecture Series
Prereq: Calculus I (GIR)
U (IAP)
2-0-4 units
Can be repeated for credit.
Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session.
Information: J. W. Bush
18.098 Internship in Mathematics
Prereq: Permission of instructor
U (Fall, IAP, Spring, Summer)
Units arranged [P/D/F]
Can be repeated for credit.
Provides academic credit for students pursuing internships to gain practical experience in the applications of mathematical concepts and methods.
Information: J. W. Bush
18.099 Independent Study
Prereq: Permission of instructor
U (Fall, IAP, Spring, Summer)
Units arranged
Can be repeated for credit.
Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval.
Information: J. W. Bush
Analysis
18.1001 Real Analysis
Subject meets with 18.100A
Prereq: Calculus II (GIR)
G (Fall, Spring)
3-0-9 units
Credit cannot also be received for 18.100B, 18.100P, 18.100Q
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line. Students in Course 18 must register for undergraduate version 18.100A.
Fall: A. P. Mattuck, Spring: L. Spolaor
18.1002 Real Analysis
Subject meets with 18.100B
Prereq: Calculus II (GIR)
G (Fall, Spring)
3-0-9 units
Credit cannot also be received for 18.100A, 18.100Q
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Students in Course 18 must register for undergraduate version 18.100B.
Fall: J.-L. Kim, Spring: C. Rodriguez
18.100A Real Analysis
Subject meets with 18.1001
Prereq: Calculus II (GIR)
U (Fall, Spring)
3-0-9 units
Credit cannot also be received for 18.100B, 18.100P, 18.100Q
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line.
Fall: A. P. Mattuck. Spring: L. Spolaor
18.100B Real Analysis
Subject meets with 18.1002
Prereq: Calculus II (GIR)
U (Fall, Spring)
3-0-9 units
Credit cannot also be received for 18.100A, 18.100Q
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space.
Fall: J.-L. Kim. Spring: C. Rodriguez
18.100P Real Analysis
Prereq: Calculus II (GIR)
U (Spring)
4-0-11 units
Credit cannot also be received for 18.1001, 18.100A, 18.100B, 18.100Q
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line. Includes instruction and practice in written communication. Enrollment limited.
K. Okoudjou
18.100Q Real Analysis
Prereq: Calculus II (GIR)
U (Fall)
4-0-11 units
Credit cannot also be received for 18.1001, 18.1002, 18.100A, 18.100B, 18.100P
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Includes instruction and practice in written communication. Enrollment limited.
M. Engelstein
18.101 Analysis and Manifolds
Subject meets with 18.1011
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Fall)
3-0-9 units
Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.901 helpful but not required.
V. W. Guillemin
18.1011 Analysis and Manifolds
Subject meets with 18.101
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Fall)
3-0-9 units
Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.9011 helpful but not required. Students in Course 18 must register for the undergraduate version, 18.101.
V. W. Guillemin
18.102 Introduction to Functional Analysis
Subject meets with 18.1021
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Spring)
3-0-9 units
Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem.
P. Etingof
18.1021 Introduction to Functional Analysis
Subject meets with 18.102
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Spring)
3-0-9 units
Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem. Students in Course 18 must register for the undergraduate version, 18.102.
P. Etingof
18.103 Fourier Analysis: Theory and Applications
Subject meets with 18.1031
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Fall)
3-0-9 units
Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals.
T. Collins
18.1031 Fourier Analysis: Theory and Applications
Subject meets with 18.103
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Fall)
3-0-9 units
Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. Students in Course 18 must register for the undergraduate version, 18.103.
T. Collins
18.104 Seminar in Analysis
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
U (Fall)
3-0-9 units
Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.
G. Staffilani
18.112 Functions of a Complex Variable
Subject meets with 18.1121
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Fall)
3-0-9 units
Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.
A. Borodin
18.1121 Functions of a Complex Variable
Subject meets with 18.112
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Fall)
3-0-9 units
Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions. Students in Course 18 must register for the undergraduate version, 18.112.
A. Borodin
18.116 Riemann Surfaces
Prereq: 18.112
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory.
Information: R. B. Melrose
18.117 Topics in Several Complex Variables
Prereq: 18.112 and 18.965
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Can be repeated for credit.
Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds.
B. Poonen
18.118 Topics in Analysis
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
L. Guth
18.125 Measure Theory and Analysis
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
G (Spring)
3-0-9 units
Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.
N. Sun
18.137 Topics in Geometric Partial Differential Equations
Prereq: Permission of instructor
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
T. Colding
18.152 Introduction to Partial Differential Equations
Subject meets with 18.1521
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Spring)
3-0-9 units
Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems.
K. Choi
18.1521 Introduction to Partial Differential Equations
Subject meets with 18.152
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Spring)
3-0-9 units
Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems. Students in Course 18 must register for the undergraduate version, 18.152.
K. Choi
18.155 Differential Analysis I
Prereq: 18.102 or 18.103
G (Fall)
3-0-9 units
First part of a two-subject sequence. Review of Lebesgue integration. Lp spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Recommended prerequisite: 18.112.
T. S. Mrowka
18.156 Differential Analysis II
Prereq: 18.155
G (Spring)
3-0-9 units
Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations.
G. Staffilani
18.157 Introduction to Microlocal Analysis
Prereq: 18.155
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
The semi-classical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semi-classical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject.
P. Hintz
18.158 Topics in Differential Equations
Prereq: 18.157
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
G. Staffilani
18.175 Theory of Probability
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
G (Fall)
3-0-9 units
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.
V. Gorin
18.176 Stochastic Calculus
Prereq: 18.175
G (Spring)
3-0-9 units
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.
D. W. Stroock
18.177 Topics in Stochastic Processes
Prereq: 18.175
G (Fall, Spring)
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
Fall: N. Sun, Spring: V. Gorin
18.199 Graduate Analysis Seminar
Prereq: Permission of instructor
G (Fall)
Not offered regularly; consult department
3-0-9 units
Can be repeated for credit.
Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance.
V. W. Guillemin
Discrete Applied Mathematics
18.200 Principles of Discrete Applied Mathematics
Prereq: None. Coreq: 18.06
U (Spring)
4-0-11 units
Credit cannot also be received for 18.200A
Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.
P. W. Shor, A. Ferber
18.200A Principles of Discrete Applied Mathematics
Prereq: None. Coreq: 18.06
U (Fall)
3-0-9 units
Credit cannot also be received for 18.200
Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming.
A. Risteski
18.204 Undergraduate Seminar in Discrete Mathematics
Prereq: ((6.042[J] or 18.200) and (18.06, 18.700, or 18.701)) or permission of instructor
U (Fall, Spring)
3-0-9 units
Seminar in combinatorics, graph theory, and discrete mathematics in general. Participants read and present papers from recent mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited.
A. Ferber, Z. Jiang, L. Lovasz
18.211 Combinatorial Analysis
Prereq: Calculus II (GIR) and (18.06, 18.700, or 18.701)
U (Fall)
3-0-9 units
Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful.
Y. Zhao
18.212 Algebraic Combinatorics
Prereq: 18.701 or 18.703
U (Spring)
3-0-9 units
Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.
A. Postnikov
18.217 Combinatorial Theory
Prereq: Permission of instructor
G (Fall)
3-0-9 units
Can be repeated for credit.
Content varies from year to year.
A. Postnikov
18.218 Topics in Combinatorics
Prereq: Permission of instructor
G (Spring)
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
Y. Zhao
18.219 Seminar in Combinatorics
Prereq: Permission of instructor
G (Fall)
Not offered regularly; consult department
3-0-9 units
Can be repeated for credit.
Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class.
Information: R. P. Stanley
Continuous Applied Mathematics
18.300 Principles of Continuum Applied Mathematics
Prereq: Calculus II (GIR) and (18.03 or 18.032)
U (Spring)
3-0-9 units
Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment.
S. Thomson
18.303 Linear Partial Differential Equations: Analysis and Numerics
Prereq: 18.06 or 18.700
U (Spring)
3-0-9 units
Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science and engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. Some programming required for homework and final project.
C. Rackauckas
18.305 Advanced Analytic Methods in Science and Engineering
Prereq: 18.04, 18.075, or 18.112
G (Fall)
3-0-9 units
Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques; WKB method on partial differential equations.
H. Cheng
18.306 Advanced Partial Differential Equations with Applications
Prereq: (18.03 or 18.032) and (18.04, 18.075, or 18.112)
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc.
R. R. Rosales
18.327 Topics in Applied Mathematics
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
L. Demanet
18.330 Introduction to Numerical Analysis
Prereq: Calculus II (GIR) and (18.03 or 18.032)
U (Spring)
3-0-9 units
Basic techniques for the efficient numerical solution of problems in science and engineering. Root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra. Knowledge of programming in a language such as MATLAB, Python, or Julia is helpful.
L. Demanet
18.335[J] Introduction to Numerical Methods
Same subject as 6.337[J]
Prereq: 18.06, 18.700, or 18.701
G (Spring)
3-0-9 units
Advanced introduction to numerical analysis: accuracy and efficiency of numerical algorithms. In-depth coverage of sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) may also be surveyed. Final project involves some programming.
S. Johnson
18.336[J] Fast Methods for Partial Differential and Integral Equations
Same subject as 6.335[J]
Prereq: 6.336[J], 16.920[J], 18.085, 18.335[J], or permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.
L. Demanet
18.337[J] Numerical Computing and Interactive Software
Same subject as 6.338[J]
Prereq: 18.06, 18.700, or 18.701
G (Fall)
3-0-9 units
Interdisciplinary introduction to computing with Julia. Covers scientific computing and data analysis problems. Combines knowledge from computer science and computational science illustrating Julia's approach to scientific computing. Sample scientific computing topics include dense and sparse linear algebra, Fourier transforms, data handling, machine learning, and N-body problems. Provides direct experience with the modern realities of programming supercomputers, GPUs, and multicores in a high-level language.
A. Edelman
18.338 Eigenvalues of Random Matrices
Prereq: 18.701 or permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
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 MATLAB hepful, but not required.
A. Edelman
18.352[J] Nonlinear Dynamics: The Natural Environment
Same subject as 12.009[J]
Prereq: Calculus II (GIR) and Physics I (GIR); Coreq: 18.03
U (Spring)
3-0-9 units
See description under subject 12.009[J].
D. H. Rothman
18.353[J] Nonlinear Dynamics: Chaos
Same subject as 2.050[J], 12.006[J]
Prereq: Physics II (GIR) and (18.03 or 18.032)
U (Fall)
3-0-9 units
See description under subject 12.006[J].
H. Ronellenfitsch
18.354[J] Nonlinear Dynamics: Continuum Systems
Same subject as 1.062[J], 12.207[J]
Subject meets with 18.3541
Prereq: Physics II (GIR) and (18.03 or 18.032)
U (Spring)
3-0-9 units
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.
P. Pearce
18.3541 Nonlinear Dynamics: Continuum Systems
Subject meets with 1.062[J], 12.207[J], 18.354[J]
Prereq: Physics II (GIR) and (18.03 or 18.032)
G (Spring)
3-0-9 units
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[J].
P. Pearce
18.355 Fluid Mechanics
Prereq: 2.25, 12.800, or 18.354[J]
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
J. W. Bush
18.357 Interfacial Phenomena
Prereq: 2.25, 12.800, 18.354[J], 18.355, or permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
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.
J. W. Bush
18.358[J] Nonlinear Dynamics and Turbulence
Same subject as 1.686[J], 2.033[J]
Subject meets with 1.068
Prereq: 1.060A
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-2-7 units
See description under subject 1.686[J].
L. Bourouiba
18.367 Waves and Imaging
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
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.
L. Demanet
18.369[J] Mathematical Methods in Nanophotonics
Same subject as 8.315[J]
Prereq: 8.07, 18.303, or permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
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.
S. G. Johnson
18.376[J] Wave Propagation
Same subject as 1.138[J], 2.062[J]
Prereq: 2.003[J] and 18.075
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
See description under subject 2.062[J].
T. R. Akylas, R. R. Rosales
18.377[J] Nonlinear Dynamics and Waves
Same subject as 1.685[J], 2.034[J]
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
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.
R. R. Rosales
18.384 Undergraduate Seminar in Physical Mathematics
Prereq: 12.006[J], 18.300, 18.354[J], or permission of instructor
U (Fall)
3-0-9 units
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.
P. Saenz
18.385[J] Nonlinear Dynamics and Chaos
Same subject as 2.036[J]
Prereq: 18.03 or 18.032
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
R. R. Rosales
18.397 Mathematical Methods in Physics
Prereq: 18.745 or some familiarity with Lie theory
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
Can be repeated for credit.
Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects.
V. G. Kac
Theoretical Computer Science
18.400[J] Automata, Computability, and Complexity
Same subject as 6.045[J]
Prereq: 6.042[J]
U (Spring)
4-0-8 units
See description under subject 6.045[J].
R. Williams, R. Rubinfeld
18.404 Theory of Computation
Subject meets with 6.840[J], 18.4041[J]
Prereq: 6.042[J] or 18.200
U (Fall)
4-0-8 units
A more extensive and theoretical treatment of the material in 6.045[J]/18.400[J], 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.
M. Sipser
18.4041[J] Theory of Computation
Same subject as 6.840[J]
Subject meets with 18.404
Prereq: 6.042[J] or 18.200
G (Fall)
4-0-8 units
A more extensive and theoretical treatment of the material in 6.045[J]/18.400[J], 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.
M. Sipser
18.405[J] Advanced Complexity Theory
Same subject as 6.841[J]
Prereq: 18.404
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
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.
R. Williams
18.408 Topics in Theoretical Computer Science
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Can be repeated for credit.
Study of areas of current interest in theoretical computer science. Topics vary from term to term.
A. Moitra, J. A. Kelner
18.410[J] Design and Analysis of Algorithms
Same subject as 6.046[J]
Prereq: 6.006
U (Fall, Spring)
4-0-8 units
See description under subject 6.046[J].
E. Demaine, M. Goemans
18.415[J] Advanced Algorithms
Same subject as 6.854[J]
Prereq: 6.046[J] and (6.041B, 6.042[J], or 18.600)
G (Fall)
5-0-7 units
See description under subject 6.854[J].
A. Moitra, D. R. Karger
18.416[J] Randomized Algorithms
Same subject as 6.856[J]
Prereq: 6.041B, 6.042[J], or 6.854[J]
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
5-0-7 units
See description under subject 6.856[J].
D. R. Karger
18.417 Introduction to Computational Molecular Biology
Prereq: 6.006, 6.01, or permission of instructor
G (Fall)
Not offered regularly; consult department
3-0-9 units
Introduces the basic computational methods used to model and predict the structure of biomolecules (proteins, DNA, RNA). Covers classical techniques in the field (molecular dynamics, Monte Carlo, dynamic programming) to more recent advances in analyzing and predicting RNA and protein structure, ranging from Hidden Markov Models and 3-D lattice models to attribute Grammars and tree Grammars.
Information: B. Berger
18.418[J] Topics in Computational Molecular Biology
Same subject as HST.504[J]
Prereq: 6.047, 18.417, or permission of instructor
G (Spring)
3-0-9 units
Can be repeated for credit.
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.
B. Berger
18.424 Seminar in Information Theory
Prereq: (6.041B, 18.05, or 18.600) and (18.06, 18.700, or 18.701)
U (Spring)
3-0-9 units
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.
S. Sen
18.425[J] Cryptography and Cryptanalysis
Same subject as 6.875[J]
Prereq: 6.046[J]
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
See description under subject 6.875[J].
S. Goldwasser, S. Micali, V. Vaikuntanathan
18.434 Seminar in Theoretical Computer Science
Prereq: 6.046[J]
U (Fall)
3-0-9 units
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
Z. Remscrim
18.435[J] Quantum Computation
Same subject as 2.111[J], 8.370[J]
Prereq: Permission of instructor
G (Fall)
3-0-9 units
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.
I. Chuang, A. Harrow, S. Lloyd, P. Shor
18.436[J] Quantum Information Science
Same subject as 6.443[J], 8.371[J]
Prereq: 18.435[J]
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
See description under subject 8.371[J].
I. Chuang, A. Harrow
18.437[J] Distributed Algorithms
Same subject as 6.852[J]
Prereq: 6.046[J]
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
See description under subject 6.852[J].
N. A. Lynch
18.453 Combinatorial Optimization
Subject meets with 18.4531
Prereq: 18.06, 18.700, or 18.701
Acad Year 2018-2019: U (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
Z. Brady
18.4531 Combinatorial Optimization
Subject meets with 18.453
Prereq: 18.06, 18.700, or 18.701
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
Z. Brady
18.455 Advanced Combinatorial Optimization
Prereq: 18.453 or permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
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.
M. X. Goemans
18.456[J] Algebraic Techniques and Semidefinite Optimization (New)
Same subject as 6.256[J]
Prereq: 6.251[J] or 15.093[J]
G (Spring)
3-0-9 units
See description under subject 6.256[J].
P. Parrilo
Logic
18.504 Seminar in Logic
Prereq: (18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Acad Year 2018-2019: U (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
H. Cohn
18.510 Introduction to Mathematical Logic and Set Theory
Prereq: None
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: U (Fall)
3-0-9 units
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.
H. Cohn
18.515 Mathematical Logic
Prereq: Permission of instructor
G (Spring)
Not offered regularly; consult department
3-0-9 units
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.
Information: B. Poonen
Probability and Statistics
18.600 Probability and Random Variables
Prereq: Calculus II (GIR)
U (Fall, Spring)
4-0-8 units. REST
Credit cannot also be received for 15.079, 15.0791
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.
J. A. Kelner, S. Sheffield
18.615 Introduction to Stochastic Processes
Prereq: 6.041B or 18.600
G (Spring)
3-0-9 units
Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion.
E. Mossel
18.642 Topics in Mathematics with Applications in Finance
Prereq: 18.03, 18.06, and (18.05 or 18.600)
U (Fall)
4-0-11 units
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. Instruction and practice in written communication provided. Limited to 30.
P. Kempthorne, V. Strela, J. Xia
18.650[J] Fundamentals of Statistics
Same subject as IDS.014[J]
Subject meets with 18.6501
Prereq: 6.041B or 18.600
U (Fall, Spring)
4-0-8 units
Credit cannot also be received for 15.075[J], IDS.013[J]
A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.
Fall: P. Rigollet. Spring: P. Kempthorne
18.6501 Fundamentals of Statistics
Subject meets with 18.650[J], IDS.014[J]
Prereq: 6.041B or 18.600
G (Fall, Spring)
4-0-8 units
Credit cannot also be received for 15.075[J], IDS.013[J]
A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics. Students in Course 18 must register for the undergraduate version, 18.650[J].
Fall: P. Rigollet. Spring: P. Kempthorne
18.655 Mathematical Statistics
Prereq: Permission of instructor
G (Spring)
3-0-9 units
Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis.
Y. Deshpande
18.657 Topics in Statistics
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Can be repeated for credit.
Topics vary from term to term.
P. Rigollet
Algebra and Number Theory
18.700 Linear Algebra
Prereq: Calculus II (GIR)
U (Fall)
3-0-9 units. REST
Credit cannot also be received for 18.06
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.
S. Benoist
18.701 Algebra I
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor
U (Fall)
3-0-9 units
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.
M. Artin
18.702 Algebra II
Prereq: 18.701
U (Spring)
3-0-9 units
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.
M. Artin
18.703 Modern Algebra
Prereq: Calculus II (GIR)
U (Spring)
3-0-9 units
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
Prereq: 18.701, (18.06 and 18.703), or (18.700 and 18.703)
U (Fall)
3-0-9 units
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.
B. Poonen
18.705 Commutative Algebra
Prereq: 18.702
G (Fall)
3-0-9 units
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.
C. Xu
18.706 Noncommutative Algebra
Prereq: 18.702
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
R. Bezrukavnikov
18.708 Topics in Algebra
Prereq: 18.705
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
Z. Yun
18.715 Introduction to Representation Theory
Prereq: 18.702 or 18.703
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
G. Lusztig
18.721 Introduction to Algebraic Geometry
Prereq: 18.702 and 18.901
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: U (Spring)
3-0-9 units
Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology.
M. Artin
18.725 Algebraic Geometry I
Prereq: None. Coreq: 18.705
G (Fall)
3-0-9 units
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.
A. Pixton
18.726 Algebraic Geometry II
Prereq: 18.725
G (Spring)
3-0-9 units
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.
A. Pixton
18.727 Topics in Algebraic Geometry
Prereq: 18.725
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
C. Xu
18.737 Algebraic Groups
Prereq: 18.705
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
B. Poonen
18.745 Introduction to Lie Algebras
Prereq: 18.701 or 18.703
G (Fall)
3-0-9 units
Topics may include structure of finite-dimensional Lie algebras; theorems of Engel and Lie; Cartan subalgebras and regular elements; trace form and Cartan's criterion; Chevalley's conjugacy theorem; classification and construction of semisimple Lie algebras; Weyl group; universal enveloping algebra and the Casimir operator; Weyl's complete reducibility theorem, Levi and Maltsev theorems; Verma modules; classification of irreducible finite-dimensional representations of semisimple Lie algebras; Weyl's character and dimension formulas.
V. G. Kac
18.747 Infinite-dimensional Lie Algebras
Prereq: 18.745
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
Topics vary from year to year.
R. Bezrukavnikov
18.748 Topics in Lie Theory
Prereq: Permission of instructor
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
P. I. Etingof
18.755 Introduction to Lie Groups
Prereq: (18.700 or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Fall)
3-0-9 units
A general introduction to manifolds and Lie groups. The role of Lie groups in mathematics and physics. Exponential mapping. Correspondence with Lie algebras. Homogeneous spaces and transformation groups. Adjoint representation. Covering groups. Automorphism groups. Invariant differential forms and cohomology of Lie groups and homogeneous spaces. 18.101 recommended but not required.
P. I. Etingof
18.757 Representations of Lie Groups
Prereq: 18.745 or 18.755
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
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.
Information: R. Bezrukavnikov
18.781 Theory of Numbers
Prereq: None
U (Spring)
3-0-9 units
An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.
D. A. Vogan
18.782 Introduction to Arithmetic Geometry
Prereq: 18.702
Acad Year 2018-2019: U (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
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.
A. Shankar
18.783 Elliptic Curves
Subject meets with 18.7831
Prereq: 18.703; or Coreq: 18.702; or permission of instructor
Acad Year 2018-2019: U (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
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
18.7831 Elliptic Curves
Subject meets with 18.783
Prereq: 18.703; or Coreq: 18.702; or permission of instructor
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
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
18.784 Seminar in Number Theory
Prereq: 18.701 or (18.703 and (18.06 or 18.700))
U (Spring)
3-0-9 units
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
J.-L. Kim
18.785 Number Theory I
Prereq: None. Coreq: 18.705
G (Fall)
3-0-9 units
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
18.786 Number Theory II
Prereq: 18.785
G (Spring)
3-0-9 units
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.
W. Zhang
18.787 Topics in Number Theory
Prereq: Permission of instructor
Acad Year 2018-2019: G (Fall)
Acad Year 2019-2020: Not offered
3-0-9 units
Can be repeated for credit.
Topics vary from year to year.
W. Zhang
Mathematics Laboratory
18.821 Project Laboratory in Mathematics
Prereq: Two mathematics subjects numbered 18.10 or above
U (Fall, Spring)
3-6-3 units. Institute LAB
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: H. Cohn. Spring: R. Bezrukavnikov
Topology and Geometry
18.900 Geometry and Topology in the Plane (New)
Prereq: 18.03 or 18.06
U (Spring)
3-0-9 units
Covers selected topics in geometry and topology, which can be visualized in the two-dimensional plane. Polygonal paths. Polygonal complexes and triangulations. Billiards. Plane curves. Winding number and rotation number. Hyperbolic geometry. Continuous curves and Jordan's theorem. Other topics may be included as time permits.
P. Seidel
18.901 Introduction to Topology
Subject meets with 18.9011
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor
U (Fall, Spring)
3-0-9 units
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.
Fall: Y. Pan. Spring: G. Lusztig
18.9011 Introduction to Topology
Subject meets with 18.901
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor
G (Fall, Spring)
3-0-9 units
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: Y. Pan. Spring: G. Lusztig
18.904 Seminar in Topology
Prereq: 18.901
U (Spring)
3-0-9 units
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
G. Tabuada
18.905 Algebraic Topology I
Prereq: 18.901 and (18.701 or 18.703)
G (Fall)
3-0-9 units
Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.
G. Tabuada
18.906 Algebraic Topology II
Prereq: 18.905
G (Spring)
3-0-9 units
Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.
T. Bachmann
18.917 Topics in Algebraic Topology
Prereq: 18.906
G (Spring)
3-0-9 units
Can be repeated for credit.
Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.
Z. Xu
18.919 Graduate Topology Seminar
Prereq: 18.906
G (Fall)
3-0-9 units
Study and discussion of important original papers in the various parts of algebraic topology. Open to all students who have taken 18.906 or the equivalent, not only prospective topologists.
H. R. Miller
18.937 Topics in Geometric Topology
Prereq: Permission of instructor
Acad Year 2018-2019: G (Spring)
Acad Year 2019-2020: Not offered
3-0-9 units
Can be repeated for credit.
Content varies from year to year. Introduces new and significant developments in geometric topology.
T. S. Mrowka
18.950 Differential Geometry
Subject meets with 18.9501
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Fall)
3-0-9 units
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.
N. Edelen
18.9501 Differential Geometry
Subject meets with 18.950
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
G (Fall)
3-0-9 units
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.
N. Edelen
18.952 Theory of Differential Forms
Prereq: 18.101 and (18.700 or 18.701)
U (Spring)
3-0-9 units
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.
V. W. Guillemin
18.965 Geometry of Manifolds I
Prereq: 18.101, 18.950, or 18.952
G (Fall)
3-0-9 units
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.
C. Mantoulidis
18.966 Geometry of Manifolds II
Prereq: 18.965
G (Spring)
3-0-9 units
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.
T. Collins
18.968 Topics in Geometry
Prereq: 18.965
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
3-0-9 units
Can be repeated for credit.
Content varies from year to year.
T. Colding
18.979 Graduate Geometry Seminar
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
3-0-9 units
Can be repeated for credit.
Content varies from year to year. Study of classical papers in geometry and in applications of analysis to geometry and topology.
T. Mrowka
18.994 Seminar in Geometry
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
U (Fall)
3-0-9 units
Students present and discuss 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.
J. Lin
18.999 Research in Mathematics
Prereq: Permission of instructor
G (Fall, Spring, Summer)
Units arranged
Can be repeated for credit.
Opportunity for study of graduate-level topics in mathematics under the supervision of a member of the department. For graduate students desiring advanced work not provided in regular subjects.
Information: W. Minicozzi
18.UR Undergraduate Research
Prereq: Permission of instructor
U (Fall, IAP, Spring, Summer)
Units arranged [P/D/F]
Can be repeated for credit.
Undergraduate research opportunities in mathematics. Permission required in advance to register for this subject. For further information, consult the departmental coordinator.
Information: J. W. Bush
18.THG Graduate Thesis
Prereq: Permission of instructor
G (Fall, IAP, Spring, Summer)
Units arranged
Can be repeated for credit.
Program of research leading to the writing of a Ph.D. thesis; to be arranged by the student and an appropriate MIT faculty member.
Information: W. Minicozzi
18.S096 Special Subject in Mathematics
Prereq: Permission of instructor
U (Fall)
Units arranged
Can be repeated for credit.
Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. 18.S097 is graded P/D/F.
Y. Deshpande
18.S097 Special Subject in Mathematics
Prereq: Permission of instructor
U (IAP)
Units arranged [P/D/F]
Can be repeated for credit.
Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. 18.S097 is graded P/D/F.
Information: J. W. Bush
18.S995 Special Subject in Mathematics
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Fall)
Units arranged
Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval.
J. Dunkel
18.S996 Special Subject in Mathematics
Prereq: Permission of instructor
G (Spring)
Units arranged
Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.
A. Negut
18.S997 Special Subject in Mathematics
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
Units arranged
Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.
Y. Zhao
18.S998 Special Subject in Mathematics
Prereq: Permission of instructor
Acad Year 2018-2019: Not offered
Acad Year 2019-2020: G (Spring)
Units arranged
Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.
Information: J. W. Bush