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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), minor in mathematics, 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.

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) that are distributed over at least three distinct areas (at least three distinct first decimal digits). 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 .

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 , 18.C06[J] , 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] Computability and Complexity Theory ) 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 ( 6.1020 , 6.1800 , 6.4100 , or 6.3900 ) 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 .

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 .

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 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 an oral qualifying examination, a thesis proposal, completion of a minimum of 96 units (8 graduate subjects), experience in classroom teaching, and a thesis containing original research in mathematics.

Interdisciplinary Programs

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, Mechanical Engineering, Physics, 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. For more information, including department-specific requirements, 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.

For further information, see the department's website or contact Math Academic Services, 617-253-2416.

Faculty and Teaching Staff

Michel X. Goemans, PhD

RSA Professor of Mathematics

Head, Department of Mathematics

William Minicozzi, PhD

Singer Professor of Mathematics

Associate Head, Department of Mathematics

Martin Z. Bazant, PhD

E. G. Roos Professor

Professor of Chemical Engineering

Professor of Mathematics

Bonnie Berger, PhD

Simons Professor of Mathematics

Member, Health Sciences and Technology Faculty

Roman Bezrukavnikov, PhD

Alexei Borodin, PhD

John W. M. Bush, PhD

Hung Cheng, PhD

Tobias Colding, PhD

Cecil and Ida Green Distinguished Professor

Laurent Demanet, PhD

Professor of Earth, Atmospheric and Planetary Sciences

Joern Dunkel, PhD

MathWorks Professor of Mathematics

Alan Edelman, PhD

Pavel I. Etingof, PhD

Lawrence Guth, PhD

Claude E. Shannon (1940) Professor of Mathematics

Anette E. Hosoi, PhD

Neil and Jane Pappalardo Professor

Professor of Mechanical Engineering

Member, Institute for Data, Systems, and Society

David S. Jerison, PhD

Steven G. Johnson, PhD

Professor of Physics

Victor Kac, PhD

Kenneth N. Kamrin, PhD

Jonathan Adam Kelner, PhD

Ju-Lee Kim, PhD

Frank Thomson Leighton, PhD

George Lusztig, PhD

Edward A. Abdun-Nur (1924) Professor of Mathematics

Davesh Maulik, PhD

Richard B. Melrose, PhD

Ankur Moitra, PhD

Norbert Wiener Professor of Mathematics

Associate Director, Institute for Data, Systems, and Society

Elchanan Mossel, PhD

Tomasz S. Mrowka, PhD

Pablo A. Parrilo, PhD

Joseph F. and Nancy P. Keithley Professor in Electrical Engineering

Professor of Electrical Engineering and Computer Science

Bjorn Poonen, PhD

Distinguished Professor in Science

(On leave, spring)

Alexander Postnikov, PhD

Philippe Rigollet, PhD

Rodolfo R. Rosales, PhD

Paul Seidel, PhD

Levinson Professor of Mathematics

Scott Roger Sheffield, PhD

Leighton Family Professor of Mathematics

Peter W. Shor, PhD

Henry Adams Morss and Henry Adams Morss, Jr. (1934) Professor of Mathematics

Michael Sipser, PhD

Donner Professor of Mathematics

Gigliola Staffilani, PhD

Abby Rockefeller Mauzé Professor of Mathematics

Daniel W. Stroock, PhD

Professor Post-Tenure of Mathematics

Martin J. Wainwright, PhD

Cecil H. Green Professor in Electrical Engineering

Zhiwei Yun, PhD

Wei Zhang, PhD

Associate Professors

Tristan Collins, PhD

Class of 1948 Career Development Professor

Associate Professor of Mathematics

Semyon Dyatlov, PhD

Andrew Lawrie, PhD

Andrei Negut, PhD

Nike Sun, PhD

Yufei Zhao, PhD

Assistant Professors

Daniel Alvarez-Gavela, PhD

Assistant Professor of Mathematics

Jeremy Hahn, PhD

Rockwell International Career Development Professor

(On leave, fall)

Dor Minzer, PhD

Tristan Ozuch-Meersseman, PhD

Lisa Piccirillo, PhD

Lisa Sauermann, PhD

John Urschel, PhD

Visiting Associate Professors

Leonid Rybnikov, PhD

Visiting Simons Associate Professor of Mathematics

Adjunct Professors

Henry Cohn, PhD

Adjunct Professor of Mathematics

Jonathan Bloom, PhD

Lecturer in Mathematics

Slava Gerovitch, PhD

Peter J. Kempthorne, PhD

Tanya Khovanova, PhD

CLE Moore Instructors

Qin Deng, PhD

CLE Moore Instructor of Mathematics

Marjorie Drake, PhD

Giada Franz, PhD

Yuchen Fu, PhD

Jimmy He, PhD

Felipe Hernandez, PhD

Malo Pierig Jezequel, PhD

Ruojing Jiang, PhD

Konstantinos Kavvadias, PhD

Aaron Landesman, PhD

Miguel Moreira, PhD

Changkeun Oh, PhD

Jia Shi, PhD

Minh-Tam Trinh, PhD

David Yang, PhD

Jingze Zhu, PhD

Jonathan Zung, PhD

Instructors

Karol Bacik, PhD

Instructor of Applied Mathematics

Mitali Bafna, PhD

Omri Ben-Eliezer, PhD

Elijah Bodish, PhD

Instructor of Mathematics

Pengning Chao, PhD

Ziang Chen, PhD

Nicholas Derr, PhD

Manik Dhar, PhD

Andrew James Horning, PhD

Artem Kalmykov, PhD

Anya Katsevich, PhD

David Milton Kouskoulas, PhD

Dominique Maldague, PhD

Dan Mikulincer, PhD

Keaton Naff, PhD

Alex Pieloch, PhD

Bauyrzhan Primkulov, PhD

Melissa Sherman-Bennett, PhD

Michael Simkin, PhD

Foster Tom, PhD

Kent Vashaw, PhD

Research Staff

Principal research scientists.

Andrew Victor Sutherland II, PhD

Principal Research Scientist of Mathematics

Research Scientists

Shiva Chidambaram, PhD

Research Scientist of Mathematics

Edgar Costa, PhD

David Roe, PhD

Samuel Schiavone, PhD

Raymond van Bommel, PhD

Professors Emeriti

Michael Artin, PhD

Professor Emeritus of Mathematics

Daniel Z. Freedman, PhD

Professor Emeritus of Physics

Harvey P. Greenspan, PhD

Victor W. Guillemin, PhD

Sigurdur Helgason, PhD

Steven L. Kleiman, PhD

Daniel J. Kleitman, PhD

Haynes R. Miller, PhD

James R. Munkres, PhD

Richard P. Stanley, PhD

Harold Stark, PhD

Gilbert Strang, PhD

Alar Toomre, PhD

David A. Vogan, PhD

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.1801 , 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: L. Guth. Spring: Information: W. Minicozzi

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.1801 , 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.

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 , 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: S Dyatlov. Spring: D Jerison

18.02A Calculus

Prereq: Calculus I (GIR) U (Fall, IAP, Spring; second half of term) 5-0-7 units. CALC II Credit cannot also be received for 18.02 , 18.022 , CC.1802 , 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 .

Fall, IAP: J. W. M. Bush. Spring: D. Jerison

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 , 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.

W. Minicozzi

18.03 Differential Equations

Prereq: None. Coreq: Calculus II (GIR) U (Fall, Spring) 5-0-7 units. REST Credit cannot also be received for 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: L. Demanet

18.031 System Functions and the Laplace Transform

Prereq: None. Coreq: 18.03 U (IAP) 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.

Information: H. R. Miller

18.032 Differential Equations

Prereq: None. Coreq: Calculus II (GIR) U (Spring) 5-0-7 units. REST

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.

18.04 Complex Variables with Applications

Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Fall) 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.

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.

18.06 Linear Algebra

Prereq: Calculus II (GIR) U (Fall, Spring) 4-0-8 units. REST Credit cannot also be received for 6.C06[J] , 18.700 , 18.C06[J]

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.

Fall: TBD. Spring: A. Borodin

18.C06[J] Linear Algebra and Optimization

Same subject as 6.C06[J] Prereq: Calculus II (GIR) U (Fall) 5-0-7 units. REST Credit cannot also be received for 18.06 , 18.700

Introductory course in linear algebra and optimization, assuming no prior exposure to linear algebra and starting from the basics, including vectors, matrices, eigenvalues, singular values, and least squares. Covers the basics in optimization including convex optimization, linear/quadratic programming, gradient descent, and regularization, building on insights from linear algebra. Explores a variety of applications in science and engineering, where the tools developed give powerful ways to understand complex systems and also extract structure from data.

A. Moitra, P. Parrilo

18.062[J] Mathematics for Computer Science

Same subject as 6.1200[J] Prereq: Calculus I (GIR) U (Fall, Spring) 5-0-7 units. REST

See description under subject 6.1200[J] .

Z. R. Abel, F. T. Leighton, 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.

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

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.

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

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: D. Kouskoulas. Spring: Staff

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

Fall: D. Kouskoulas. Spring: Staff

18.086 Computational Science and Engineering II

Subject meets with 18.0861 Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Spring) Not offered regularly; consult department 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 ) G (Spring) Not offered regularly; consult department 3-0-9 units

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: W. Minicozzi

18.090 Introduction to Mathematical Reasoning

Prereq: None. Coreq: Calculus II (GIR) U (Spring) 3-0-9 units. REST

Focuses on understanding and constructing mathematical arguments. Discusses foundational topics (such as infinite sets, quantifiers, and methods of proof) as well as selected concepts from algebra (permutations, vector spaces, fields) and analysis (sequences of real numbers). Particularly suitable for students desiring additional experience with proofs before going on to more advanced mathematics subjects or subjects in related areas with significant mathematical content.

S. Dyatlov, B. Poonen, P. Seidel

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 G (Fall) 2-0-2 units

See description under subject 5.95[J] .

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.

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.

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. May not be used to satisfy Mathematics major requirements.

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 18.1002 , 18.100A , 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: Q. Deng. Spring: J. Zhu

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 18.1001 , 18.100A , 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. 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: R. Melrose. Spring: G. Franz

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 18.1001 , 18.1002 , 18.100B , 18.100P , 18.100Q

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 18.1001 , 18.1002 , 18.100A , 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. More demanding than 18.100A , for students with more mathematical maturity. Places more emphasis on point-set topology and n-space.

18.100P Real Analysis

Prereq: Calculus II (GIR) U (Spring) 4-0-11 units Credit cannot also be received for 18.1001 , 18.1002 , 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.

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.

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.

M. Jezequel

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

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.

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

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.

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 .

18.104 Seminar in Analysis

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q U (Fall, Spring) 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.

Fall: T. Ozuch-Meersseman. Spring: 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.

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

18.116 Riemann Surfaces

Prereq: 18.112 Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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.

P. I. Etingof

18.117 Topics in Several Complex Variables

Prereq: 18.112 and 18.965 Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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.

18.118 Topics in Analysis

Prereq: Permission of instructor Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units Can be repeated for credit.

Topics vary from year to year.

18.125 Measure Theory and Analysis

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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.

18.137 Topics in Geometric Partial Differential Equations

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units Can be repeated for credit.

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.

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

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 .

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.

18.157 Introduction to Microlocal Analysis

Prereq: 18.155 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 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.

R. B. Melrose

18.158 Topics in Differential Equations

Prereq: 18.157 Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units Can be repeated for credit.

18.199 Graduate Analysis Seminar

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. Moitra

18.200A Principles of Discrete Applied Mathematics

Prereq: None. Coreq: 18.06 Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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.

18.204 Undergraduate Seminar in Discrete Mathematics

Prereq: (( 6.1200[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.

J. He, D. Mikulincer, M. Sherman-Bennett, A. Weigandt

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.

A. Weigandt

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.

18.218 Topics in Combinatorics

Prereq: Permission of instructor G (Spring) 3-0-9 units Can be repeated for credit.

L. Sauermann

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: Y. Zhao

18.225 Graph Theory and Additive Combinatorics

Prereq: (( 18.701 or 18.703 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q )) or permission of instructor Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

Introduction to extremal graph theory and additive combinatorics. Highlights common themes, such as the dichotomy between structure versus pseudorandomness. Topics include Turan-type problems, Szemeredi's regularity lemma and applications, pseudorandom graphs, spectral graph theory, graph limits, arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete Fourier analysis, Freiman's theorem on sumsets and structure. Discusses current research topics and open problems.

18.226 Probabilistic Methods in Combinatorics

Prereq: ( 18.211 , 18.600 , and ( 18.100A , 18.100B , 18.100P , or 18.100Q )) or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

Introduction to the probabilistic method, a fundamental and powerful technique in combinatorics and theoretical computer science. Focuses on methodology as well as combinatorial applications. Suitable for students with strong interest and background in mathematical problem solving. Topics include linearity of expectations, alteration, second moment, Lovasz local lemma, correlation inequalities, Janson inequalities, concentration inequalities, entropy method.

Continuous Applied Mathematics

18.300 principles of continuum applied mathematics.

Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Fall) 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.

B. Geshkovski

18.303 Linear Partial Differential Equations: Analysis and Numerics

Prereq: 18.06 or 18.700 U (Fall) 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.

V. Heinonen

18.305 Advanced Analytic Methods in Science and Engineering

Prereq: 18.04 , 18.075 , or 18.112 Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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.

18.306 Advanced Partial Differential Equations with Applications

Prereq: ( 18.03 or 18.032 ) and ( 18.04 , 18.075 , or 18.112 ) Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 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

18.330 introduction to numerical analysis.

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.

18.335[J] Introduction to Numerical Methods

Same subject as 6.7310[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.

A. J. Horning

18.336[J] Fast Methods for Partial Differential and Integral Equations

Same subject as 6.7340[J] Prereq: 6.7300[J] , 16.920[J] , 18.085 , 18.335[J] , or permission of instructor G (Fall, Spring) 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.

18.337[J] Parallel Computing and Scientific Machine Learning

Same subject as 6.7320[J] Prereq: 18.06 , 18.700 , or 18.701 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Introduction to scientific machine learning with an emphasis on developing scalable differentiable programs. Covers scientific computing topics (numerical differential equations, dense and sparse linear algebra, Fourier transformations, parallelization of large-scale scientific simulation) simultaneously with modern data science (machine learning, deep neural networks, automatic differentiation), focusing on the emerging techniques at the connection between these areas, such as neural differential equations and physics-informed deep learning. Provides direct experience with the modern realities of optimizing code performance for supercomputers, GPUs, and multicores in a high-level language.

18.338 Eigenvalues of Random Matrices

Prereq: 18.701 or permission of instructor G (Fall) 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 Julia helpful, but not required.

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 (Fall) Not offered regularly; consult department 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] .

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.

B. Primkulov

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

18.355 Fluid Mechanics

Prereq: 2.25 , 12.800 , or 18.354[J] Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 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.

18.357 Interfacial Phenomena

Prereq: 2.25 , 12.800 , 18.354[J] , 18.355 , or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 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.

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 2023-2024: Not offered Acad Year 2024-2025: 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 2023-2024: Not offered Acad Year 2024-2025: 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.

18.369[J] Mathematical Methods in Nanophotonics

Same subject as 8.315[J] Prereq: 8.07 , 18.303 , or permission of instructor Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 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 G (Spring) 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 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 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.

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.

18.385[J] Nonlinear Dynamics and Chaos

Same subject as 2.036[J] Prereq: 18.03 or 18.032 Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: 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.

18.397 Mathematical Methods in Physics

Prereq: 18.745 or some familiarity with Lie theory G (Fall) Not offered regularly; consult department 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.

Theoretical Computer Science

18.400[j] computability and complexity theory.

Same subject as 6.1400[J] Prereq: ( 6.1200[J] and 6.1210 ) or permission of instructor U (Spring) 4-0-8 units

See description under subject 6.1400[J] .

R. Williams, R. Rubinfeld

18.404 Theory of Computation

Subject meets with 6.5400[J] , 18.4041[J] Prereq: 6.1200[J] or 18.200 U (Fall) 4-0-8 units

A more extensive and theoretical treatment of the material in 6.1400[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.

18.4041[J] Theory of Computation

Same subject as 6.5400[J] Subject meets with 18.404 Prereq: 6.1200[J] or 18.200 G (Fall) 4-0-8 units

18.405[J] Advanced Complexity Theory

Same subject as 6.5410[J] Prereq: 18.404 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 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 2023-2024: Not offered Acad Year 2024-2025: G (Fall, 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.

Fall: D. Minzer. Spring: A. Moitra

18.410[J] Design and Analysis of Algorithms

Same subject as 6.1220[J] Prereq: 6.1200[J] and 6.1210 U (Fall, Spring) 4-0-8 units

See description under subject 6.1220[J] .

E. Demaine, M. Goemans

18.413 Introduction to Computational Molecular Biology

Subject meets with 18.417 Prereq: 6.1210 or permission of instructor U (Spring) Not offered regularly; consult department 3-0-9 units

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

18.415[J] Advanced Algorithms

Same subject as 6.5210[J] Prereq: 6.1220[J] and ( 6.1200[J] , 6.3700 , or 18.600 ) Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 5-0-7 units

See description under subject 6.5210[J] .

A. Moitra, D. R. Karger

18.416[J] Randomized Algorithms

Same subject as 6.5220[J] Prereq: ( 6.1200[J] or 6.3700 ) and ( 6.1220[J] or 6.5210[J] ) Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 5-0-7 units

See description under subject 6.5220[J] .

D. R. Karger

18.417 Introduction to Computational Molecular Biology

Subject meets with 18.413 Prereq: 6.1210 or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

18.418[J] Topics in Computational Molecular Biology

Same subject as HST.504[J] Prereq: 6.8701 , 18.417 , or permission of instructor G (Fall) 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.

18.424 Seminar in Information Theory

Prereq: ( 6.3700 , 18.05 , or 18.600 ) and ( 18.06 , 18.700 , or 18.701 ) U (Fall) 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.

18.425[J] Cryptography and Cryptanalysis

Same subject as 6.5620[J] Prereq: 6.1220[J] G (Fall) 3-0-9 units

See description under subject 6.5620[J] .

S. Goldwasser, S. Micali, V. Vaikuntanathan

18.434 Seminar in Theoretical Computer Science

Prereq: 6.1220[J] U (Fall, 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.

Fall: E. Mossel. Spring: D. Minzer

18.435[J] Quantum Computation

Same subject as 2.111[J] , 6.6410[J] , 8.370[J] Prereq: 8.05 , 18.06 , 18.700 , 18.701 , or 18.C06[J] 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, P. Shor

18.436[J] Quantum Information Science

Same subject as 6.6420[J] , 8.371[J] Prereq: 18.435[J] 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 6.5250[J] Prereq: 6.1220[J] Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

See description under subject 6.5250[J] .

M. Ghaffari, N. A. Lynch

18.453 Combinatorial Optimization

Subject meets with 18.4531 Prereq: 18.06 , 18.700 , or 18.701 Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Spring) 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.

Information: M. X. Goemans

18.4531 Combinatorial Optimization

Subject meets with 18.453 Prereq: 18.06 , 18.700 , or 18.701 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

18.455 Advanced Combinatorial Optimization

Prereq: 18.453 or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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

Same subject as 6.7230[J] Prereq: 6.7210[J] or 15.093[J] Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

See description under subject 6.7230[J] .

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 2023-2024: Not offered Acad Year 2024-2025: U (Fall) 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.

18.510 Introduction to Mathematical Logic and Set Theory

Prereq: None Acad Year 2023-2024: U (Fall) Acad Year 2024-2025: Not offered 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.

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 6.3700 , 6.3702

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.

Fall: S. Sheffield. Spring: J. Kelner

18.615 Introduction to Stochastic Processes

Prereq: 6.3700 or 18.600 G (Fall) 3-0-9 units

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

18.619[J] Discrete Probability and Stochastic Processes (New)

Same subject as 6.7720[J] , 15.070[J] Prereq: 6.3702 , 6.7700[J] , 18.100A , 18.100B , or 18.100Q G (Spring) 3-0-9 units

See description under subject 15.070[J] .

G. Bresler, D. Gamarnik, E. Mossel, Y. Polyanskiy

18.642 Topics in Mathematics with Applications in Finance

Prereq: 18.03 , 18.06 , and ( 18.05 or 18.600 ) U (Fall) 3-0-9 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.

P. Kempthorne, V. Strela, J. Xia

18.650[J] Fundamentals of Statistics

Same subject as IDS.014[J] Subject meets with 18.6501 Prereq: 6.3700 or 18.600 U (Fall, Spring) 4-0-8 units

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

Fall: P. Rigollet. Spring: A. Katsevich

18.6501 Fundamentals of Statistics

Subject meets with 18.650[J] , IDS.014[J] Prereq: 6.3700 or 18.600 G (Fall, Spring) 4-0-8 units

18.655 Mathematical Statistics

Prereq: ( 18.650[J] and ( 18.100A , 18.100A , 18.100P , or 18.100Q )) or 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. Prior exposure to both probability and statistics at the university level is assumed.

P. Kempthorne

18.656[J] Mathematical Statistics: a Non-Asymptotic Approach

Same subject as 9.521[J] , IDS.160[J] Prereq: ( 6.7700[J] , 18.06 , and 18.6501 ) or permission of instructor G (Spring) 3-0-9 units

See description under subject 9.521[J] .

S. Rakhlin, P. Rigollet

18.657 Topics in Statistics

Topics vary from term to term.

P. Rigollet

18.675 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.

Y. Shenfeld

18.676 Stochastic Calculus

Prereq: 18.675 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.

18.677 Topics in Stochastic Processes

Prereq: 18.675 G (Spring) 3-0-9 units Can be repeated for credit.

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 6.C06[J] , 18.06 , 18.C06[J]

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 .

18.701 Algebra I

Prereq: 18.100A , 18.100B , 18.100P , 18.100Q , 18.090 , 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.

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.

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.

18.704 Seminar in Algebra

Prereq: 18.701 , ( 18.06 and 18.703 ), or ( 18.700 and 18.703 ) U (Fall, 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. Some experience with proofs required. Enrollment limited.

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.

18.706 Noncommutative Algebra

Prereq: 18.702 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 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 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units Can be repeated for credit.

18.715 Introduction to Representation Theory

Prereq: 18.702 or 18.703 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 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.

18.721 Introduction to Algebraic Geometry

Prereq: 18.702 and 18.901 U (Spring) 3-0-9 units

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

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.

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.

18.727 Topics in Algebraic Geometry

Prereq: 18.725 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

18.737 Algebraic Groups

Prereq: 18.705 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 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.

18.745 Lie Groups and Lie Algebras I

Prereq: ( 18.701 or 18.703 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Fall) 3-0-9 units

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

18.747 Infinite-dimensional Lie Algebras

Prereq: 18.745 Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

18.748 Topics in Lie Theory

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

18.755 Lie Groups and Lie Algebras II

Prereq: 18.745 or permission of instructor G (Spring) 3-0-9 units

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

18.757 Representations of Lie Groups

Prereq: 18.745 or 18.755 Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 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.

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.

M.-T. Trinh

18.782 Introduction to Arithmetic Geometry

Prereq: 18.702 Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Spring) 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.

S. Chidambaram

18.783 Elliptic Curves

Subject meets with 18.7831 Prereq: 18.702 , 18.703 , or permission of instructor Acad Year 2023-2024: U (Fall) Acad Year 2024-2025: 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: 18.702 , 18.703 , or permission of instructor Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

18.784 Seminar in Number Theory

Prereq: 18.701 or ( 18.703 and ( 18.06 or 18.700 )) U (Spring) 3-0-9 units

A. Landesman

18.785 Number Theory I

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

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.

18.787 Topics in Number Theory

Mathematics laboratory, 18.821 project laboratory in mathematics.

Prereq: Two mathematics subjects numbered 18.100 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: A. Negut. Spring: L. Piccirillo

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

Same subject as 5.961[J] , 8.396[J] , 9.980[J] , 12.396[J] Prereq: None G (Spring; second half of term) 2-0-1 units

See description under subject 8.396[J] . Limited to 80.

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

Same subject as 5.962[J] , 8.397[J] , 9.981[J] , 12.397[J] Prereq: None G (Spring; first half of term) 2-0-1 units

See description under subject 8.397[J] . Limited to 80.

Topology and Geometry

18.900 geometry and topology in the plane.

Prereq: 18.03 or 18.06 U (Spring) 3-0-9 units

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

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: A. Pieloch. Spring: R. Jiang

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

18.904 Seminar in Topology

Prereq: 18.901 U (Fall) 3-0-9 units

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.

D. Alvarez-Gavela

18.906 Algebraic Topology II

Prereq: 18.905 and ( 18.101 or 18.965 ) 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. S. Mrowka

18.917 Topics in Algebraic Topology

Prereq: 18.906 Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 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.

Information: T. Schlank

18.919 Graduate Topology Seminar

Prereq: 18.906 G (Spring) 3-0-9 units

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

18.937 Topics in Geometric Topology

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

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.

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

18.952 Theory of Differential Forms

Prereq: 18.101 and ( 18.700 or 18.701 ) U (Spring) Not offered regularly; consult department 3-0-9 units

Multilinear algebra: tensors and exterior forms. Differential forms on R n : 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 R n . 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.

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.

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.

18.968 Topics in Geometry

Prereq: 18.965 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

18.979 Graduate Geometry Seminar

Prereq: Permission of instructor G (Spring) Not offered regularly; consult department 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.

18.994 Seminar in Geometry

Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Spring) 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.

18.999 Research in Mathematics

Prereq: Permission of instructor G (Fall, IAP, 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.

18.C20[J] Introduction to Computational Science and Engineering

Same subject as 9.C20[J] , 16.C20[J] , CSE.C20[J] Prereq: 6.100A ; Coreq: 8.01 and 18.01 U (Fall, Spring; second half of term) 3-0-3 units Credit cannot also be received for 6.100B

See description under subject 16.C20[J] .

D. L. Darmofal, N. Seethapathi

18.C25[J] Real World Computation with Julia (New)

Same subject as 1.C25[J] , 6.C25[J] , 12.C25[J] , 16.C25[J] , 22.C25[J] Prereq: 6.100A , 18.03 , and 18.06 U (Fall) 3-0-9 units

Focuses on algorithms and techniques for writing and using modern technical software in a job, lab, or research group environment that may consist of interdisciplinary teams, where performance may be critical, and where the software needs to be flexible and adaptable. Topics include automatic differentiation, matrix calculus, scientific machine learning, parallel and GPU computing, and performance optimization with introductory applications to climate science, economics, agent-based modeling, and other areas. Labs and projects focus on performant, readable, composable algorithms, and software. Programming will be in Julia. Expects students to have some familiarity with Python, Matlab, or R. No Julia experience necessary.

A. Edelman, R. Ferrari, B. Forget, C. Leiseron,Y. Marzouk, J. Williams

18.UR Undergraduate Research

Undergraduate research opportunities in mathematics. Permission required in advance to register for this subject. For further information, consult the departmental coordinator.

18.THG Graduate Thesis

Program of research leading to the writing of a Ph.D. thesis; to be arranged by the student and an appropriate MIT faculty member.

18.S096 Special Subject in Mathematics

Prereq: Permission of instructor U (IAP, Spring) 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 Special Subject in Mathematics

Prereq: Permission of instructor U (IAP) Units arranged [P/D/F] Can be repeated for credit.

18.S190 Special Subject in Mathematics

Prereq: Permission of instructor U (IAP) Units arranged Can be repeated for credit.

18.S191 Special Subject in Mathematics

Prereq: Permission of instructor U (Fall) Not offered regularly; consult department Units arranged Can be repeated for credit.

18.S995 Special Subject in Mathematics

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: 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.

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.

18.S997 Special Subject in Mathematics

18.s998 special subject in mathematics.

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Doctoral Programs in Computational Science and Engineering

Application & admission information.

The Center for Computational Science and Engineering (CCSE) offers two doctoral programs in computational science and engineering (CSE) – one leading to a standalone PhD degree in CSE offered entirely by CCSE ( CSE PhD ) and the other leading to an interdisciplinary PhD degree offered jointly with participating departments in the School of Engineering and the School of Science ( Dept-CSE PhD ).

While both programs enable students to specialize at the doctoral level in a computation-related field via focused coursework and a thesis, they differ in essential ways. The standalone CSE PhD program is intended for students who plan to pursue research in cross-cutting methodological aspects of computational science. The resulting doctoral degree in Computational Science and Engineering is awarded by CCSE via the the Schwarzman College of Computing. In contrast, the interdisciplinary Dept-CSE PhD program is intended for students who are interested in computation in the context of a specific engineering or science discipline. For this reason, this degree is offered jointly with participating departments across the Institute; the interdisciplinary degree is awarded in a specially crafted thesis field that recognizes the student’s specialization in computation within the chosen engineering or science discipline.

Applicants to the standalone CSE PhD program are expected to have an undergraduate degree in CSE, applied mathematics, or another field that prepares them for an advanced degree in CSE. Applicants to the Dept-CSE PhD program should have an undergraduate degree in a related core disciplinary area as well as a strong foundation in applied mathematics, physics, or related fields. When completing the MIT CSE graduate application , students are expected to declare which of the two programs they are interested in. Admissions decisions will take into account these declared interests, along with each applicant’s academic background, preparation, and fit to the program they have selected. All applicants are asked to specify MIT CCSE-affiliated faculty that best match their research interests; applicants to the Dept-CSE PhD program also select the home department(s) that best match. At the discretion of the admissions committee, Dept-CSE PhD applications might also be shared with a home department beyond those designated in the application. CSE PhD admissions decisions are at the sole discretion of CCSE; Dept-CSE PhD admission decisions are conducted jointly between CCSE and the home departments.

Please note: These are both doctoral programs in Computational Science and Engineering; applicants interested in Computer Science must apply to the Department of Electrical Engineering and Computer Science .

Important Dates

September 15: Application Opens December 1: Deadline to apply for admission* December – March: Application review period January – March: Decisions released on rolling basis

*All supplemental materials (e.g., transcripts, test scores, letters of recommendation) must also be received by December 1. Application review begins on that date, and incomplete applications may not be reviewed. Please be sure that your recommenders are aware of this hard deadline, as we do not make exceptions. We also do not allow students to upload/submit material beyond what is required, such as degree certificates, extra recommendations, publications, etc.

A complete electronic CSE application includes the following:

Three letters of recommendation ;
Students admitted to the program will be required to supply official transcripts. Discrepancies between unofficial and official transcripts may result in the revocation of the admission offer.
Statement of objectives (limited to approximately one page) and responses to department-specific prompts for Dept-CSE PhD applicants;
Official GRE General Test score report , sent to MIT by ETS via institute code 3514 GRE REQUIREMENT WAIVED FOR FALL 2025 ;
Official IELTS score report sent to MIT by IELTS† (international applicants from non-English speaking countries only; see below for more information)
Resume or CV , uploaded in PDF format;
MIT graduate application fee of $75‡.

‡Application Fee

The MIT graduate application fee of $75.00 is a mandatory requirement set by the Institute payable by credit card. Please visit the MIT Graduate Admission Application Fee Waiver page for information about fee waiver eligibility and instructions.

Please note: CCSE cannot issue fee waivers; email requests for fee waivers sent to [email protected] will not receive a response.

Admissions Contact Information

Email: [email protected]

► Current MIT CSE SM Students: Please see the page for Current MIT Graduate Students .

GRE Requirement

GRE REQUIREMENT WAIVED FOR FALL 2025 All applicants are required to take the Graduate Record Examination (GRE) General Aptitude Test. The MIT code for submitting GRE score reports is 3514 (you do not need to list a department code). GRE scores must current; ETS considers scores valid for five years after the testing year in which you tested.

†English Language Proficiency Requirement

The CSE PhD program requires international applicants from non-English speaking countries to take the academic version of the International English Language Testing System (IELTS). The IELTS exam measures one’s ability to communicate in English in four major skill areas: listening, reading, writing, and speaking. A minimum IELTS score of 7 is required for admission. For more information about the IELTS, and to find out where and how to take the exam, please visit the IELTS web site .

While we will also accept the TOEFL iBT (Test of English as a Foreign Language), we strongly prefer the IELTS. The minimum TOEFL iBT score is 100.

This requirement is waived for those who can demonstrate that one or more of the following are true:

English is/was the language of instruction in your four-year undergraduate program,
English is the language of your employer/workplace for at least the last four years,
English was your language of instruction in both primary and secondary schools.

Degree Requirements for Admission

To be admitted as a regular graduate student, an applicant must have earned a bachelor’s degree or its equivalent from a college, university, or technical school of acceptable standing. Students in their final year of undergraduate study may be admitted on the condition that their bachelor’s degree is awarded before they enroll at MIT.

Applicants without an SM degree may apply to the CSE PhD program, however, the Departments of Aeronautics and Astronautics and Mechanical Engineering nominally require the completion of an SM degree before a student is considered a doctoral candidate. As a result, applicants to those departments holding only a bachelor’s degree are asked in the application to indicate whether they prefer to complete the CSE SM program or an SM through the home department.

Nondiscrimination Policy

The Massachusetts Institute of Technology is committed to the principle of equal opportunity in education and employment. To read MIT’s most up-to-date nondiscrimination policy, please visit the Reference Publication Office’s nondiscrimination statement page .

Additional Information

For more details, as well as answers to most commonly asked questions regarding the admissions process to individual participating Dept-CSE PhD departments including details on financial support, applicants are referred to the website of the participating department of interest.

Overview of the Application Procedure

Welcome to the MIT Mathematics Graduate Admissions page. This page explains the application process in general. For complete details, go to the on-line application which is available mid-September to December. These instructions are repeated there.

MIT admits students starting in the Fall term of each year only. Admission is to the PhD program only; there is no Masters program. There is no separate application for financial support; all admitted students are offered support.

Submitting GRE scores is entirely optional: We will accept scores if submitted (and are most interested in the Math Subject test result, if any) but it will not hurt your application if not included.

To apply, follow these steps:

Fill out the on-line application by 23:59, EST, December 15.

You will be submitting:

Field(s) of interest
Personal information/addresses
International student data
Three or more names and e-mail addresses of letter writers
Educational and work history, including IELTS/TOEFL scores (preferably from this year), and honors
Grades in math/science/engineering courses and overall
Statement of objectives
Outside financial support and potential outside support
Credit/debit card payment of $75
The Math department requires applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.

Arrange for submission of (official reports only)

Letters of recommendation
For international students, IELTS (or TOEFL iBT)

We recommend that before November 15 you notify your letter writers that you will be needing evaluations from them, so that they have time to prepare them and submit them by December 15. Once you have submitted your on-line application, instructions to your letter writers will be generated for you. You are responsible for making sure that your letter writers have copies of these instructions.

You self-reported your grades in step 1, but we require an official transcript for all admitted students. If/when we request this, arrange for an official copy of your college transcript to be sent to:

Academic Services, Room 2-110 Dept of Mathematics, MIT 77 Massachusetts Ave. Cambridge MA 02139-4307 USA

TOEFL reporting codes Institution code: MIT = 3514 Mathematics Department code: 72

International Students

IELTS is the English language proficiency test we prefer, but we also accept the TOEFL iBT . (On the other hand, we generally do not accept the TOEFL PBT.) To have IELTS results reported, indicate "Mathematics Department, Massachusetts Institute of Technology (MIT)" on your IELTS test application; no code or address is needed. To have TOEFL iBT results reported, use the codes above (3514 for MIT, and 72 for Mathematics).

If you are an international student, you should take the IELTS (or TOEFL iBT) by December 31. If you will receive an undergraduate degree from an English-language university in an English-speaking country after attending it for at least three years, then the Math Department will waive the English language proficiency test requirement.

Paper Forms

If for some reason, you are unable to use the on-line system, you may use paper forms. But note that on-line documents allow us to consider your application more quickly and conveniently. Your letter writers may also use paper forms, if necessary.

Please address questions about the application process to [email protected] . You can find more information about MIT graduate admissions in general at the MIT Graduate Admissions site .

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 https://oge.mit.edu/programs/computational-science-and-engineering-phd/ .

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MIT Homepage

Below is a list of the MIT Schwarzman College of Computing’s graduate degree programs. The Doctor of Philosophy (PhD) degree is awarded interchangeably with the Doctor of Science (ScD).

Prospective students apply to the department or program under which they want to register. Application instructions can be found on each program’s website as well as on the MIT Graduate Admissions website.

Center for Computational Science and Engineering

The Center for Computational Science and Engineering (CCSE) brings together faculty, students, and other researchers across MIT involved in computational science research and education. The center focuses on advancing computational approaches to science and engineering problems, and offers SM and PhD programs in computational science and engineering (CSE).

Computational Science and Engineering, SM and PhD . Interdisciplinary master’s program emphasizing advanced computational methods and applications. The CSE SM program prepares students with a common core of computational methods that serve all science and engineering disciplines, and an elective component that focuses on particular applications. Doctoral program enables students to specialize in methodological aspects of computational science via focused coursework and a thesis which involves the development and analysis of broadly applicable computational approaches that advance the state of the art.
Computational Science and Engineering, Interdisciplinary PhD. Doctoral program offered jointly with eight participating departments, focusing on the development of new computational methods relevant to science and engineering disciplines. Students specialize in a computation-related field of their choice through coursework and a doctoral thesis. The specialization in computational science and engineering is highlighted by specially crafted thesis fields.

Department of Electrical Engineering and Computer Science

The largest academic department at MIT, the Department of Electrical Engineering and Computer Science (EECS) prepares hundreds of students for leadership roles in academia, industry, government and research. Its world-class faculty have built their careers on pioneering contributions to the field of electrical engineering and computer science — a field which has transformed the world and invented the future within a single lifetime. MIT EECS consistently tops the U.S. News & World Report and other college rankings and is widely recognized for its rigorous and innovative curriculum. A joint venture between the Schwarzman College of Computing and the School of Engineering, EECS (also known as Course 6) is now composed of three overlapping sub-units in electrical engineering (EE), computer science (CS), and artificial intelligence and decision-making (AI+D).

Computation and Cognition, MEng*. Course 6-9P builds on the Bachelor of Science in Computation and Cognition to provide additional depth in the subject areas through advanced coursework and a substantial thesis.
Computer Science, PhD
Computer Science and Engineering, PhD
Computer Science, Economics, and Data Science, MEng*. New in Fall 2022, Course 6-14P builds on the Bachelor of Science in Computer Science, Economics, and Data Science to provide additional depth in economics and EECS through advanced coursework and a substantial thesis.
Computer Science and Molecular Biology, MEng*. Course 6-7P builds on the Bachelor of Science in Computer Science and Molecular Biology to provide additional depth in computational biology through coursework and a substantial thesis.
Electrical Engineering, PhD
Electrical Engineering and Computer Science, MEng* , SM* , and PhD . Master of Engineering program (Course 6-P) provides the depth of knowledge and the skills needed for advanced graduate study and for professional work, as well as the breadth and perspective essential for engineering leadership. Master of Science program emphasizes one or more of the theoretical or experimental aspects of electrical engineering or computer science as students progress toward their PhD.
Electrical Engineer / Engineer in Computer Science.** For PhD students who seek more extensive training and research experiences than are possible within the master’s program.
Thesis Program with Industry, MEng.* Combines the Master of Engineering academic program with periods of industrial practice at affiliated companies.

* Available only to qualified EECS undergraduates. ** Available only to students in the EECS PhD program who have not already earned a Master’s and to Leaders for Global Operations students.

Institute for Data, Systems, and Society

The Institute for Data, Systems, and Society advances education and research in analytical methods in statistics and data science, and applies these tools along with domain expertise and social science methods to address complex societal challenges in a diverse set of areas such as finance, energy systems, urbanization, social networks, and health.

Social and Engineering Systems, PhD. Interdisciplinary PhD program focused on addressing societal challenges by combining the analytical tools of statistics and data science with engineering and social science methods.
Technology and Policy, SM . Master’s program addresses societal challenges through research and education at the intersection of technology and policy.
Interdisciplinary Doctoral Program in Statistics . For students currently enrolled in a participating MIT doctoral program who wish to develop their understanding of 21st-century statistics and apply these concepts within their chosen field of study. Participating departments and programs: Aeronautics and Astronautics, Brain and Cognitive Sciences, Economics, Mathematics, Mechanical Engineering, Physics, Political Science, and Social and Engineering Systems.

Operations Research Center

The Operations Research Center (ORC) offers multidisciplinary graduate programs in operations research and analytics. ORC’s community of scholars and researchers work collaboratively to connect data to decisions in order to solve problems effectively — and impact the world positively.

In conjunction with the MIT Sloan School of Management, ORC offers the following degrees:

Operations Research, SM and PhD . Master’s program teaches important OR techniques — with an emphasis on practical, real-world applications — through a combination of challenging coursework and hands-on research. Doctoral program provides a thorough understanding of the theory of operations research while teaching students to how to develop and apply operations research methods in practice.
Business Analytics, MBAn. Specialized advanced master’s degree designed to prepare students for careers in data science and business analytics.

Smart. Open. Grounded. Inventive. Read our Ideas Made to Matter.

Which program is right for you?

Through intellectual rigor and experiential learning, this full-time, two-year MBA program develops leaders who make a difference in the world.

A rigorous, hands-on program that prepares adaptive problem solvers for premier finance careers.

A 12-month program focused on applying the tools of modern data science, optimization and machine learning to solve real-world business problems.

Earn your MBA and SM in engineering with this transformative two-year program.

Combine an international MBA with a deep dive into management science. A special opportunity for partner and affiliate schools only.

A doctoral program that produces outstanding scholars who are leading in their fields of research.

Bring a business perspective to your technical and quantitative expertise with a bachelor’s degree in management, business analytics, or finance.

A joint program for mid-career professionals that integrates engineering and systems thinking. Earn your master’s degree in engineering and management.

An interdisciplinary program that combines engineering, management, and design, leading to a master’s degree in engineering and management.

Executive Programs

A full-time MBA program for mid-career leaders eager to dedicate one year of discovery for a lifetime of impact.

This 20-month MBA program equips experienced executives to enhance their impact on their organizations and the world.

Non-degree programs for senior executives and high-potential managers.

A non-degree, customizable program for mid-career professionals.

Admissions Requirements

The following are general requirements you should meet to apply to the MIT Sloan PhD Program. Complete instructions concerning application requirements are available in the online application.

General Requirements

Bachelor's degree or equivalent
A strong quantitative background (the Accounting group requires calculus)
Exposure to microeconomics and macroeconomics (the Accounting group requires microeconomics)

A Guide to Business PhD Applications by Abhishek Nagaraj (PhD 2016) may be of interest.

Application Components

Statement of purpose.

Your written statement is your chance to convince the admissions committee that you will do excellent doctoral work and that you have the promise to have a successful career as an academic researcher.

GMAT/GRE Scores

We require either a valid GMAT or valid GRE score. At-home testing is allowed. Your unofficial score report from the testing institution is sufficient for application. If you are admitted to the program, you will be required to submit your official test score for verification.

We do not have a minimum score requirement. We do not offer test waivers. Registration information for the GMAT (code X5X-QS-21) and GRE (code 3510) may be obtained at www.mba.com and www.ets.org respectively.

TOEFL/IELTS Scores

We require either a valid TOEFL (minimum score 577 PBT/90 IBT ) or valid IELTS (minimum score 7) for all non-native English speakers. Your unofficial score report from the testing institution is sufficient for application. If you are admitted to the program, you will be required to submit your official test score for verification. Registration information for TOEFL (code 3510) and IELTS may be obtained at www.toefl.org and www.ielts.org respectively.

The TOEFL/IELTS test requirement is waived only if you meet one of the following criteria:

You are a native English speaker.
You attended all years of an undergraduate program conducted solely in English, and are a graduate of that program.

Please do not contact the PhD Program regarding waivers, as none will be discussed. If, upon review, the faculty are interested in your application with a missing required TOEFL or IELTS score, we may contact you at that time to request a score.

Transcripts

We require unofficial copies of transcripts for each college or university you have attended, even if no degree was awarded. If these transcripts are in a language other than English, we also require a copy of a certified translation. In addition, you will be asked to list the five most relevant courses you have taken.

Letters of Recommendation

We require three letters of recommendation. Academic letters are preferred, especially those providing evidence of research potential. We allow for an optional fourth recommendation, but no more than four recommendations are allowed.

Your resume should be no more than two pages. You may chose to include teaching, professional experience, research experience, publications, and other accomplishments in outside activities.

Writing Sample(s)

Applicants are encouraged to submit a writing sample. For applicants to the Finance group, a writing sample is required. There are no specific guidelines for your writing sample. Possible options include (but are not limited to) essays, masters’ theses, capstone projects, or research papers.

Video Essay

A video essay is required for the Accounting research group and optional for the Marketing and System Dynamics research groups. The essay is a short and informal video answering why you selected this research group and a time where you creatively solved a problem. The video can be recorded with your phone or computer, and should range from 2 to 5 minutes in length. There is no attention — zero emphasis! — on the production value of your video.

Nondiscrimination Policy: The Massachusetts Institute of Technology is committed to the principle of equal opportunity in education and employment. For complete text of MIT’s Nondiscrimination Statement, please click here .

Core Members
Affiliate Members

Interdisciplinary Doctoral Program in Statistics

Minor in Statistics and Data Science
MicroMasters program in Statistics and Data Science
Data Science and Machine Learning: Making Data-Driven Decisions
Norbert Wiener Fellowship
Stochastics and Statistics Seminar
IDSS Distinguished Seminars
IDSS Special Seminar
SDSC Special Events
Online events
IDS.190 Topics in Bayesian Modeling and Computation
Past Events
LIDS & Stats Tea

The Interdisciplinary PhD in Statistics (IDPS) is designed for students currently enrolled in a participating MIT doctoral program who wish to develop their understanding of 21st century statistics, using concepts of computation and data analysis as well as elements of classical statistics and probability within their chosen field of study.

Participating programs: Aeronautics & Astronautics Brain and Cognitive Sciences Economics Mathematics Mechanical Engineering Physics Political Science Social and Engineering Systems

How to join IDPS:

Doctoral students in participating programs may submit a selection form between the end of their second semester and penultimate semester in their doctoral program. Selection forms are due by the current semester add date, and students will be notified of a decision by the drop date.

Required documents include a CV, unofficial transcript, anticipated course plan and thesis proposal or statement of interest in statistics. For access to the selection form or for further information, please contact the IDSS Academic Office at [email protected]

Graduate Departments:

MIT Statistics + Data Science Center Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 617-253-1764

Accessibility
Interdisciplinary PhD in Aero/Astro and Statistics
Interdisciplinary PhD in Brain and Cognitive Sciences and Statistics
Interdisciplinary PhD in Economics and Statistics
Interdisciplinary PhD in Mathematics and Statistics
Interdisciplinary PhD in Mechanical Engineering and Statistics
Interdisciplinary PhD in Physics and Statistics
Interdisciplinary PhD in Political Science and Statistics
Interdisciplinary PhD in Social & Engineering Systems and Statistics
LIDS & Stats Tea
Spring 2023
Spring 2022
Spring 2021
Fall – Spring 2020
Fall 2019 – IDS.190 – Topics in Bayesian Modeling and Computation
Fall 2019 – Spring 2019
Fall 2018 and earlier

Earth, Atmospheric, and Planetary Sciences

77 Massachusetts Avenue Building 54-912 Cambridge MA, 02139

617-253-3381 [email protected]

Website: Earth, Atmospheric, and Planetary Sciences

Application Opens: September 1

Deadline: December 1 at 11:59 PM Eastern Time

Fee: $75.00

Terms of Enrollment

Interdisciplinary programs, areas of research.

Atmospheric Science
Climate Science
Geochemistry
Planetary Sciences

Financial Support

All students admitted to our doctoral programs are provided with full funding that includes a stipend, tuition, and health insurance. This may be in the form of a fellowship or research assistantship. At some time in your graduate career, you will be asked to serve as a teaching assistant so that you gain that experience. Research assistantships are the primary support for students beyond the first year.

Funding may vary by program, and newly admitted students are encouraged to apply to outside government and private agencies for fellowship support. Please see the EAPS website for more information.

Standardized Tests

Graduate Record Examination (GRE)

General test not required for Fall 2024 admissions cycle
Subject test in Chemistry or Physics not required for Fall 2024 admissions cycle*

International English Language Testing System (IELTS)

Minimum score required: 7
Electronic scores send to: MIT Graduate Admissions

Test of English as a Foreign Language (TOEFL)

600 (paper-based)
100 (internet-based)

IELTS is preferred, though both IELTS and TOEFL will be accepted. A waiver of the TOEFL/IELTS requirement may be available for those who have completed a four-year bachelor’s program taught exclusively in English.

Application Requirements

Online application
Statement of objectives
Three letters of recommendation
Transcripts
English proficiency exam scores

Special Instructions

Deadline for September admission is December 1st.

Do not try to convert your university grading scale or GPA to MIT’s scale. Enter the grades/GPA as granted by your school.

An original copy of your transcript from each college or university, translated into English, should be uploaded as an attachment in PDF format to your application. No other attachments will be accepted. Hard copies sent via post by an applicant will not be accepted. Only those applicants who are accepted for admission will be required to submit a hard copy of their transcripts. Any discrepancy between the scanned transcripts and official transcripts may result in a rejection or withdrawal of our admission offer.

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Life Sciences | Graduate | Careers | Undergraduate

Aeronautics and astronautics, biological engineering, biophysics graduate certificate program, brain and cognitive sciences, chemical engineering, civil and environmental engineering (environmental microbiology), computational and systems biology, earth, atmospheric, and planetary sciences, hst - health, sciences and technology, materials science and engineering, mechanical engineering, microbiology program, nuclear science and engineering, sloan mba with a health care focus, sts- science, technology, and society, whoi joint program, writing and humanistic studies, life sciences at mit.

Many areas of research today have a Life Sciences focus. This is primarily due to the powerful tools of molecular biology, which form a common language and allow exciting and important interdisciplinary approaches. Experience in Life Sciences-based research opens multiple career paths.

This site collates the broad array of MIT graduate degree programs with a primary focus on biological questions, or that can include a Life Sciences focus. Applications for graduate study should be made through the appropriate program. Please explore this site, and our program offerings!

Find out more about:

Life at MIT and how to apply

School of Science

School of Engineering

Sloan School of Management

School of Humanities, Arts, and Social Sciences

Open Courseware

Printable List of Programs

List of Life Sciences Subjects at MIT

Graduate Life Sciences Programs at MIT

Office of Admissions

Mit open courseware.

Developed by the Dean's Office: MIT School of Science, in cooperation with MIT departments.

Copyright 2013 Massachusetts Institute of Technology. Site designed by Chrysos Designs .

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Sample Research Proposal for PhD in Mathematics

VIDEO

Applied Mathematics PhD Program: 2023-24 Virtual Information Session
A cool problem from Harvard MIT Mathematics Tournament
Fully Funded PhD Scholarship at the Institute of Science and Technology Austria (ISTA)
How to Get Your PhD in Mathematics & Statistics (pt. 2)
2016 The Harvard-MIT Mathematics Tournament (HMMT)
MS Mathematics: Ableitung, Elastizität, Kostenkehre

COMMENTS

Admission
Welcome to the MIT Mathematics Graduate Admissions page. This page explains the application process in general. For complete details, go to the on-line application which is available mid-September to December. These instructions are repeated there. MIT admits students starting in the Fall term of each year only.
Graduate
Graduate Students 2018-2019. The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests ...
Applying to Grad School
The Basics. It's best to start preparing for the application process in the spring of your junior year. Getting ready involves writing essays, taking standardized test, asking professors for letters of recomendation, and filling out a lot of forms. Keep in mind, if you love math and want to go to grad school, you should apply.
Mathematics
77 Massachusetts Avenue Building 2-110 Cambridge MA, 02139. 617-253-2416 [email protected]. Website: Mathematics. Apply here. Application Opens: September 14
Admissions < MIT
The application and additional information may be found on the Advanced Study Program website. Admission is valid only for one term; a student must seek readmission each term to continue at the Institute. Those applying for special graduate student status for the first time must pay an application fee. To be allowed to continue as a special ...
Department of Mathematics < MIT
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 Admissions
Office of Graduate Education - Apply to become a part of the Massachusetts Institute of Technology community. MIT graduate students play a central role in the Institute's wide-ranging research activities, making a vital contribution to the educational experience of students and faculty, and ultimately leading to the success of the research itself.
MIT Mathematics
The Xi Chapter of Phi Beta Kappa has elected 54 mathematics majors, among 127 electees from MIT's Class of 2024, to become members. Founded in 1776, Phi Beta Kappa is the nation's oldest academic honor society. Membership is awarded to students in recognition of excellent academic records and commitment to the objectives of a liberal education.
MIT Mathematics
The School of Science has selected Mathematics Program Coordinator André Lee Dixon as one of the recipients of the 2024 Infinite Mile Award! "I have been consistently struck by the level of initiative and passion André brings to work," says his nominator, John Urschel PhD '21. Infinite Mile Award winners are nominated by colleagues for ...
CSE PhD
The standalone CSE PhD program is intended for students who plan to pursue research in cross-cutting methodological aspects of computational science. The resulting doctoral degree in Computational Science and Engineering is awarded by CCSE via the the Schwarzman College of Computing. In contrast, the interdisciplinary Dept-CSE PhD program is ...
Admission
skip to main content. Search. Contact Site Map
Graduate Programs
Electrical Engineering and Computer Science, MEng*, SM*, and PhD. Master of Engineering program (Course 6-P) provides the depth of knowledge and the skills needed for advanced graduate study and for professional work, as well as the breadth and perspective essential for engineering leadership. Master of Science program emphasizes one or more of ...
Admissions Requirements
Admissions Requirements. The following are general requirements you should meet to apply to the MIT Sloan PhD Program. Complete instructions concerning application requirements are available in the online application. General Requirements. Bachelor's degree or equivalent. A strong quantitative background (the Accounting group requires calculus)
Degree programs
MIT Sloan Master of Finance. January 4. MIT Sloan Master of Science in Management Studies. February 15. MIT Sloan MBA Program. September 29, January 18, April 11. MIT Sloan PhD Program. December 1. MIT-WHOI Joint Program in Oceanography / Applied Ocean Science and Engineering.
Undergraduate
The Department of Mathematics offers a Bachelor of Science in Mathematics in the following concentrations: applied mathematics , pure mathematics , general mathematics . Additionally, the Mathematics with Computer Science degree is offered to students wishing to pursue their interests in mathematics and theoretical computer science within a ...
Interdisciplinary PhD in Mathematics and Statistics
Interdisciplinary PhD in Mathematics and Statistics. Requirements: Students must complete their primary program's degree requirements along with the IDPS requirements. Statistics requirements must not unreasonably impact performance or progress in a student's primary degree program. PhD Earned on Completion: Mathematics and Statistics.
Interdisciplinary Doctoral Program in Statistics
Interdisciplinary Doctoral Program in Statistics. The Interdisciplinary PhD in Statistics (IDPS) is designed for students currently enrolled in a participating MIT doctoral program who wish to develop their understanding of 21st century statistics, using concepts of computation and data analysis as well as elements of classical statistics and probability within their chosen field of study.
Earth, Atmospheric, and Planetary Sciences
77 Massachusetts Avenue Building 54-912 Cambridge MA, 02139. 617-253-3381 [email protected]. Website: Earth, Atmospheric, and Planetary Sciences
Life Sciences
The Graduate PhD Program in Microbiology is an interdepartmental and interdisciplinary program at MIT. MIT has a long-standing tradition of excellence in microbiological research, and there are over 50 faculty from approximately 10 different departments and divisions who study or use microbes in significant ways in their research.