DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS
Bachelor of Science in Mathematics (Course 18)

The Department of Mathematics (http://math.mit.edu) oers

General Mathematics Option
training at the undergraduate, graduate, and postgraduate levels.
In addition to the General Institute Requirements, the requirements
Its expertise covers a broad spectrum of elds ranging from the
consist of Dierential Equations, plus eight additional 12-unit
traditional areas of "pure" mathematics, such as analysis, algebra,
subjects in Course 18 of essentially dierent content, including at
geometry, and topology, to applied mathematics areas such as
least six advanced subjects (rst decimal digit one or higher) that are
combinatorics, computational biology, fluid dynamics, theoretical
distributed over at least three distinct areas (at least three distinct
computer science, and theoretical physics.
rst decimal digits). One of these eight subjects must be Linear
Course 18 includes two undergraduate degrees: a Bachelor of Algebra. This leaves available 84 units of unrestricted electives. The
Science in Mathematics and a Bachelor of Science in Mathematics requirements are flexible in order to accommodate students who
with Computer Science. Undergraduate students may choose one pursue programs that combine mathematics with a related eld
of three options leading to the Bachelor of Science in Mathematics: (such as physics, economics, or management) as well as students
applied mathematics, pure mathematics, or general mathematics. who are interested in both pure and applied mathematics. More
The general mathematics option provides a great deal of flexibility details can be found on the degree chart (http://catalog.mit.edu/
and allows students to design their own programs in conjunction degree-charts/mathematics-course-18/#generalmathematicstext).
with their advisors. The Mathematics with Computer Science degree
is oered for students who want to pursue interests in mathematics Applied Mathematics Option
and theoretical computer science within a single undergraduate Applied mathematics focuses on the mathematical concepts and
program. techniques applied in science, engineering, and computer science.
Particular attention is given to the following principles and their
At the graduate level, the Mathematics Department oers the PhD in mathematical formulations: propagation, equilibrium, stability,
Mathematics, which culminates in the exposition of original research optimization, computation, statistics, and random processes.
in a dissertation. Graduate students also receive training and gain
experience in the teaching of mathematics. Sophomores interested in applied mathematics typically enroll
in 18.200 Principles of Discrete Applied Mathematics and 18.300
The CLE Moore instructorships and Applied Mathematics Principles of Continuum Applied Mathematics. Subject 18.200 is
instructorships bring mathematicians at the postdoctoral level to MIT devoted to the discrete aspects of applied mathematics and may be
and provide them with training in research and teaching. taken concurrently with 18.03 Dierential Equations. Subject 18.300,
oered in the spring term, is devoted to continuous aspects and
makes considerable use of dierential equations.
Undergraduate Study
The subjects in Group I of the program correspond roughly to those
An undergraduate degree in mathematics provides an excellent areas of applied mathematics that make heavy use of discrete
basis for graduate work in mathematics or computer science, or mathematics, while Group II emphasizes those subjects that
for employment in such elds as nance, business, or consulting. deal mainly with continuous processes. Some subjects, such as
Students' programs are arranged through consultation with their probability or numerical analysis, have both discrete and continuous
faculty advisors. aspects.

Undergraduates in mathematics are encouraged to elect an Students planning to go on to graduate work in applied mathematics
undergraduate seminar during their junior or senior year. The should also take some basic subjects in analysis and algebra.
experience gained from active participation in a seminar conducted
by a research mathematician has proven to be valuable for students More detail on the Applied Mathematics option can be found on the
planning to pursue graduate work as well as for those going on to degree chart (http://catalog.mit.edu/degree-charts/mathematics-
other careers. These seminars also provide training in the verbal and course-18/#appliedmathematicstext).
written communication of mathematics and may be used to fulll the
Communication Requirement. Pure Mathematics Option
Pure (or "theoretical") mathematics is the study of the basic concepts
Many mathematics majors take 18.821 Project Laboratory in and structure of mathematics. Its goal is to arrive at a deeper
Mathematics, which fullls the Institute's Laboratory Requirement understanding and an expanded knowledge of mathematics itself.
and counts toward the Communication Requirement.
Traditionally, pure mathematics has been classied into three
general elds: analysis, which deals with continuous aspects of
mathematics; algebra, which deals with discrete aspects; and

Department of Mathematics   |   3
DEPARTMENT OF MATHEMATICS

geometry. The undergraduate program is designed so that students

become familiar with each of these areas. Students also may wish to Inquiries
explore other topics such as logic, number theory, complex analysis, For further information, see the department's website (http://
and subjects within applied mathematics. math.mit.edu/academics/undergrad) or contact Math Academic
Services, 617-253-2416.
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,
Graduate Study
18.100P or 18.100Q) or 18.700 Linear Algebra.
The Mathematics Department oers programs covering a broad
For more details, see the degree chart (http:// range of topics leading to the Doctor of Philosophy or Doctor of
catalog.mit.edu/degree-charts/mathematics-course-18/ Science degree. Candidates are admitted to either the Pure or
#theoreticalmathematicstext). Applied Mathematics programs but are free to pursue interests in
both groups. Of the roughly 115-125 doctoral students, about two
Bachelor of Science in Mathematics with Computer Science thirds are in Pure Mathematics, one third in Applied Mathematics.
(Course 18-C)
Mathematics and computer science are closely related elds. The programs in Pure and Applied Mathematics oer basic and
Problems in computer science are oen formalized and solved with advanced classes in analysis, algebra, geometry, Lie theory, logic,
mathematical methods. It is likely that many important problems number theory, probability, statistics, topology, astrophysics,
currently facing computer scientists will be solved by researchers combinatorics, fluid dynamics, numerical analysis, theoretical
skilled in algebra, analysis, combinatorics, logic and/or probability physics, and the theory of computation. In addition, many
theory, as well as computer science. mathematically oriented subjects are oered by other departments.
Students in Applied Mathematics are especially encouraged to
The purpose of this program is to allow students to study a take subjects in engineering and scientic subjects related to their
combination of these mathematical areas and potential areas of research.
application in computer science. Required subjects include linear
algebra (18.06 or 18.700) because it is so broadly used, and discrete All students pursue research under the supervision of the faculty
mathematics (18.062[J] or 18.200) to give experience with proofs and and are encouraged to take advantage of the many seminars and
the necessary tools for analyzing algorithms. The required subjects colloquia at MIT and in the Boston area.
covering complexity (18.404 Theory of Computation or 18.400[J]
Computability and Complexity Theory) and algorithms (18.410[J] Doctor of Philosophy or Doctor of Science
Design and Analysis of Algorithms) provide an introduction to the The requirements for these degrees are described on the
most theoretical aspects of computer science.  We also require department's website (http://math.mit.edu/academics/grad/
exposure to other areas of computer science (6.031, 6.033, 6.034, timeline). In outline, they consist of an oral qualifying examination,
or 6.036) where mathematical issues may also arise. More details a thesis proposal, completion of a minimum of 96 units (8 graduate
can be found on the degree chart (http://catalog.mit.edu/degree- subjects), experience in classroom teaching, and a thesis containing
charts/mathematics-computer-science-course-18-c). original research in mathematics.

Some flexibility is allowed in this program. In particular, students Interdisciplinary Programs

may substitute the more advanced subject 18.701 Algebra I for 18.06
Linear Algebra, and, if they already have strong theorem-proving Computational Science and Engineering
skills, may substitute 18.211 Combinatorial Analysis or 18.212 Students with primary interest in computational science may also
Algebraic Combinatorics for 18.062[J] Mathematics for Computer consider applying to the interdisciplinary Computational Science and
Science or 18.200 Principles of Discrete Applied Mathematics. Engineering (CSE) program, with which the Mathematics Department
is aliated. For more information, see the CSE website (http://
Minor in Mathematics gradadmissions.mit.edu/programs/cse).
The requirements for a Minor in Mathematics are as follows: six 12-
unit subjects in mathematics, beyond the Institute's Mathematics Mathematics and Statistics
Requirement, of essentially dierent content, including at least three The Interdisciplinary Doctoral Program in Statistics provides training
advanced subjects (rst decimal digit one or higher). in statistics, including classical statistics and probability as well as
computation and data analysis, to students who wish to integrate
See the Undergraduate Section for a general description of the minor
these valuable skills into their primary academic program. The
program (http://catalog.mit.edu/mit/undergraduate-education/
program is administered jointly by the departments of Aeronautics
academic-programs/minors).
and Astronautics, Economics, Mathematics, Mechanical Engineering,
and Political Science, and the Statistics and Data Science Center

4   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

within the Institute for Data, Systems, and Society. It is open to Laurent Demanet, PhD
current doctoral students in participating departments. For more Professor of Mathematics
information, including department-specic requirements, see the Professor of Earth, Atmospheric and Planetary Sciences
full program description (http://catalog.mit.edu/interdisciplinary/
graduate-programs/phd-statistics) under Interdisciplinary Graduate Alan Edelman, PhD
Programs. Professor of Mathematics
(On leave, spring)
Financial Support Pavel I. Etingof, PhD
Financial support is guaranteed for up to ve years to students Professor of Mathematics
making satisfactory academic progress. Financial aid aer the rst
year is usually in the form of a teaching or research assistantship. Victor W. Guillemin, PhD
Professor of Mathematics
Inquiries
For further information, see the department's website (http:// Lawrence Guth, PhD
math.mit.edu/academics/grad) or contact Math Academic Services, Claude E. Shannon (1940) Professor of Mathematics
617-253-2416. Anette E. Hosoi, PhD
Neil and Jane Pappalardo Professor
Professor of Mechanical Engineering
Faculty and Teaching Sta Professor of Mathematics
Michel X. Goemans, PhD Associate Dean, School of Engineering
Professor of Mathematics Member, Institute for Data, Systems, and Society
Head, Department of Mathematics
David S. Jerison, PhD
William Minicozzi, PhD Professor of Mathematics
Singer Professor of Mathematics
Steven G. Johnson, PhD
Associate Head, Department of Mathematics
Professor of Mathematics
Professor of Physics
Professors
Michael Artin, PhD Victor Kac, PhD
Professor Post-Tenure of Mathematics Professor of Mathematics

Martin Z. Bazant, PhD Jonathan Adam Kelner, PhD

E. G. Roos Professor Professor of Mathematics
Professor of Chemical Engineering
Professor of Mathematics Ju-Lee Kim, PhD
Professor of Mathematics
Bonnie Berger, PhD
Simons Professor of Mathematics Frank Thomson Leighton, PhD
Member, Health Sciences and Technology Faculty Professor of Mathematics
(On leave, spring)
Roman Bezrukavnikov, PhD
Professor of Mathematics George Lusztig, PhD
Edward A. Abdun-Nur (1924) Professor of Mathematics
Alexei Borodin, PhD
Professor of Mathematics Davesh Maulik, PhD
Professor of Mathematics
John W. M. Bush, PhD
Professor of Mathematics Richard B. Melrose, PhD
Professor of Mathematics
Hung Cheng, PhD
Professor of Mathematics Haynes R. Miller, PhD
Professor of Mathematics
Tobias Colding, PhD
Cecil and Ida Green Distinguished Professor
Professor of Mathematics

Department of Mathematics   |   5
DEPARTMENT OF MATHEMATICS

Elchanan Mossel, PhD Wei Zhang, PhD

Professor of Mathematics Professor of Mathematics
Member, Institute for Data, Systems, and Society
Associate Professors
Tomasz S. Mrowka, PhD Joern Dunkel, PhD
Professor of Mathematics Associate Professor of Mathematics
Pablo A. Parrilo, PhD (On leave, spring)
Joseph F. and Nancy P. Keithley Professor Semyon Dyatlov, PhD
Professor of Electrical Engineering and Computer Science Associate Professor of Mathematics
Professor of Mathematics
Core Faculty, Institute for Data, Systems, and Society Ankur Moitra, PhD
Associate Professor of Mathematics
Bjorn Poonen, PhD
Distinguished Professor in Science Andrei Negut, PhD
Professor of Mathematics Class of 1947 Career Development Chair
Associate Professor of Mathematics
Alexander Postnikov, PhD
Professor of Mathematics Nike Sun, PhD
Associate Professor of Mathematics
Philippe Rigollet, PhD
Professor of Mathematics Assistant Professors
Member, Institute for Data, Systems, and Society Tristan Collins, PhD
Rodolfo R. Rosales, PhD Assistant Professor of Mathematics
Professor of Mathematics Peter Hintz, PhD
Paul Seidel, PhD Assistant Professor of Mathematics
Levinson Professor of Mathematics (On leave, fall)

Scott Roger Sheeld, PhD Andrew Lawrie, PhD

Leighton Family Professor of Mathematics Assistant Professor of Mathematics
Member, Institute for Data, Systems, and Society Dor Minzer, PhD
Peter W. Shor, PhD Assistant Professor of Mathematics
Henry Adams Morss and Henry Adams Morss, Jr. (1934) Professor Lisa Piccirillo, PhD
Professor of Mathematics Assistant Professor of Mathematics
Michael Sipser, PhD Yufei Zhao, PhD
Donner Professor of Mathematics Class of 1956 Career Development Chair
Gigliola Stalani, PhD Assistant Professor of Mathematics
Abby Rockefeller Mauzé Professor of Mathematics
Visiting Professors
Gilbert Strang, PhD Thomas Lam, PhD
MathWorks Professor of Mathematics Visiting Professor of Mathematics
Member, Institute for Data, Systems, and Society
David Sanders, PhD
Daniel W. Stroock, PhD Visiting Professor of Mathematics
Professor Post-Tenure of Mathematics
Visiting Associate Professors
Chenyang Xu, PhD Thibaut Le Gouic, PhD
Professor of Mathematics Visiting Associate Professor of Mathematics
(On leave)

Zhiwei Yun, PhD Adjunct Professors

Professor of Mathematics Henry Cohn, PhD
Adjunct Professor of Mathematics

6   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

Yilin Wang, PhD

Lecturers CLE Moore Instructor of Mathematics
Jennifer French, PhD
Lecturer in Digital Learning Abigail Ward, PhD
CLE Moore Instructor of Mathematics
Slava Gerovitch, PhD
Lecturer in Mathematics Junho Whang, PhD
CLE Moore Instructor of Mathematics
Peter J. Kempthorne, PhD
Lecturer in Mathematics Xueying Yu, PhD
CLE Moore Instructor of Mathematics
Tanya Khovanova, PhD
Lecturer in Mathematics Yiming Zhao, PhD
CLE Moore Instructor of Mathematics
CLE Moore Instructors Ziquan Zhuang, PhD
Charlotte Chan, PhD CLE Moore Instructor of Mathematics
CLE Moore Instructor of Mathematics

Tony Feng, PhD Instructors of Applied Mathematics

CLE Moore Instructor of Mathematics Peter Baddoo, PhD
Instructor of Applied Mathematics
Promit Ghosal, PhD
CLE Moore Instructor of Mathematics Keaton Burns, PhD
Instructor of Applied Mathematics
Jeremy Hahn, PhD
CLE Moore Instructor of Mathematics Gary Choi, PhD
Instructor of Applied Mathematics
Pei-Ken Hung, PhD
CLE Moore Instructor of Mathematics Diego Cifuentes, PhD
Instructor of Applied Mathematics
Benjamin Landon, PhD
CLE Moore Instructor of Mathematics Souvik Dhara, PhD
Schramm Fellow Instructor
Yang Li, PhD
CLE Moore Instructor of Mathematics Matthew Durey, PhD
Instructor of Applied Mathematics
Tristan Ozuch-Meersseman, PhD
CLE Moore Instructor of Mathematics William Cole Franks, PhD
Instructor of Applied Mathematics
Casey Rodriguez, PhD
CLE Moore Instructor of Mathematics Nir Gadish, PhD
Instructor of Applied Mathematics
Matthew Rosenzweig, PhD
CLE Moore Instructor of Mathematics Julia Gaudio, PhD
Instructor of Applied Mathematics
Junliang Shen, PhD
CLE Moore Instructor of Mathematics Felix Gotti, PhD
Instructor of Applied Mathematics
Yair Shenfeld, PhD
CLE Moore Instructor of Mathematics Vili Heinonen, PhD
Instructor of Applied Mathematics
Alexander Smith, PhD
CLE Moore Instructor of Mathematics Zilin Jiang, PhD
Instructor of Applied Mathematics
Minh-Tam Trinh, PhD
CLE Moore Instructor of Mathematics Ousmane Kodio, PhD
Instructor of Applied Mathematics
Chen Wan, PhD
CLE Moore Instructor of Mathematics

Department of Mathematics   |   7
DEPARTMENT OF MATHEMATICS

Lu Lu, PhD David Roe, PhD

Instructor of Applied Mathematics Research Scientist of Mathematics

Tyler Maunu, PhD Samuel Schiavone, PhD

Instructor of Applied Mathematics Research Scientist of Mathematics

Christopher Rackauckas, PhD David I. Spivak, PhD

Instructor of Applied Mathematics Research Scientist of Mathematics

Instructors of Pure Mathematics Raymond van Bommel, PhD

Daniel Alvarez-Gavela, PhD Research Scientist of Mathematics
Instructor of Pure Mathematics
Professors Emeriti
Li Chen, PhD
Instructor of Pure Mathematics Daniel Z. Freedman, PhD
Professor Emeritus of Mathematics
Anthony Conway, PhD Professor Emeritus of Physics
Instructor of Pure Mathematics
Harvey P. Greenspan, PhD
Irving Dai, PhD Professor Emeritus of Mathematics
Instructor of Pure Mathematics
Sigurdur Helgason, PhD
Daniel Kriz, PhD Professor Emeritus of Mathematics
Instructor of Pure Mathematics
Steven L. Kleiman, PhD
Dominique Maldague, PhD Professor Emeritus of Mathematics
Instructor of Pure Mathematics
Daniel J. Kleitman, PhD
Jonathan Wang, PhD Professor Emeritus of Mathematics
Instructor of Pure Mathematics
Arthur P. Mattuck, PhD
Nicholas Wilkins, PhD Professor Emeritus of Mathematics
Instructor of Pure Mathematics
James R. Munkres, PhD
Professor Emeritus of Mathematics
Research Sta
Isadore Manuel Singer, PhD
Principal Research Scientists Institute Professor Emeritus
Andrew Victor Sutherland II, PhD Professor Emeritus of Mathematics
Principal Research Scientist of Mathematics
Richard P. Stanley, PhD
Research Scientists Professor Emeritus of Mathematics
Pawan Bharadwaj Pisupati, PhD
Harold Stark, PhD
Research Scientist of Mathematics
Professor Emeritus of Mathematics
Edgar Costa, PhD
Alar Toomre, PhD
Research Scientist of Mathematics
Professor Emeritus of Mathematics
Francesc Fité, PhD
David A. Vogan, PhD
Research Scientist of Mathematics
Norbert Wiener Professor Emeritus of Mathematics
Wanlin Li, PhD
Research Scientist of Mathematics

Philippe Ricoux, PhD

Research Scientist of Mathematics

8   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

General Mathematics 18.02A Calculus

Prereq: Calculus I (GIR)
18.01 Calculus U (Fall, IAP, Spring; second half of term)
Prereq: None 5-0-7 units. CALC II
U (Fall, Spring) First half is taught during the last six weeks of the Fall term; covers
5-0-7 units. CALC I material in the rst half of 18.02 (through double integrals). Second
Credit cannot also be received for 18.01A, ES.1801, ES.181A half of 18.02A can be taken either during IAP (daily lectures) or
Dierentiation and integration of functions of one variable, during the second half of the Spring term; it covers the remaining
with applications. Informal treatment of limits and continuity. material in 18.02. In person not required.
Dierentiation: denition, rules, application to graphing, rates, Fall, IAP: J. W. M. Bush. Spring: W. Minicozzi
approximations, and extremum problems. Indenite integration;
separable rst-order dierential equations. Denite integral; 18.022 Calculus
fundamental theorem of calculus. Applications of integration Prereq: Calculus I (GIR)
to geometry and science. Elementary functions. Techniques of U (Fall)
integration. Polar coordinates. L'Hopital's rule. Improper integrals. 5-0-7 units. CALC II
Innite series: geometric, p-harmonic, simple comparison tests, Credit cannot also be received for 18.02, CC.1802, ES.1802, ES.182A
power series for some elementary functions. In person not required. Calculus of several variables. Topics as in 18.02 but with more
Fall: L. Guth. Spring: Information: W. Minicozzi focus on mathematical concepts. Vector algebra, dot product,
matrices, determinant. Functions of several variables, continuity,
18.01A Calculus dierentiability, derivative. Parametrized curves, arc length,
Prereq: Knowledge of dierentiation and elementary integration curvature, torsion. Vector elds, gradient, curl, divergence. Multiple
U (Fall; rst half of term) integrals, change of variables, line integrals, surface integrals.
5-0-7 units. CALC I Stokes' theorem in one, two, and three dimensions.
Credit cannot also be received for 18.01, ES.1801, ES.181A G. Stalani
Six-week review of one-variable calculus, emphasizing material
not on the high-school AB syllabus: integration techniques and 18.03 Dierential Equations
applications, improper integrals, innite series, applications to Prereq: None. Coreq: Calculus II (GIR)
other topics, such as probability and statistics, as time permits. U (Fall, Spring)
Prerequisites: one year of high-school calculus or the equivalent, 5-0-7 units. REST
with a score of 5 on the AB Calculus test (or the AB portion of the BC Credit cannot also be received for 18.032, CC.1803, ES.1803
test, or an equivalent score on a standard international exam), or Study of dierential equations, including modeling physical
equivalent college transfer credit, or a passing grade on the rst half systems. Solution of rst-order ODEs by analytical, graphical,
of the 18.01 advanced standing exam. and numerical methods. Linear ODEs with constant coecients.
D. Jerison Complex numbers and exponentials. Inhomogeneous equations:
polynomial, sinusoidal, and exponential inputs. Oscillations,
18.02 Calculus damping, resonance. Fourier series. Matrices, eigenvalues,
Prereq: Calculus I (GIR) eigenvectors, diagonalization. First order linear systems: normal
U (Fall, Spring) modes, matrix exponentials, variation of parameters. Heat equation,
5-0-7 units. CALC II wave equation. Nonlinear autonomous systems: critical point
Credit cannot also be received for 18.022, CC.1802, ES.1802, ES.182A analysis, phase plane diagrams.
Calculus of several variables. Vector algebra in 3-space, Fall: T. Collins. Spring: A. Lawrie
determinants, matrices. Vector-valued functions of one variable,
space motion. Scalar functions of several variables: partial
dierentiation, gradient, optimization techniques. Double integrals
and line integrals in the plane; exact dierentials and conservative
elds; Green's theorem and applications, triple integrals, line and
surface integrals in space, Divergence theorem, Stokes' theorem;
applications. In person not required.
Fall: S. Dyatlov. Spring: W. Minicozzi

Department of Mathematics   |   9
DEPARTMENT OF MATHEMATICS

18.031 System Functions and the Laplace Transform 18.06 Linear Algebra
Prereq: None. Coreq: 18.03 Prereq: Calculus II (GIR)
U (IAP) U (Fall, Spring)
1-0-2 units 4-0-8 units. REST
Credit cannot also be received for 18.700
Studies basic continuous control theory as well as representation
of functions in the complex frequency domain. Covers generalized Basic subject on matrix theory and linear algebra, emphasizing
functions, unit impulse response, and convolution; and Laplace topics useful in other disciplines, including systems of equations,
transform, system (or transfer) function, and the pole diagram. vector spaces, determinants, eigenvalues, singular value
Includes examples from mechanical and electrical engineering. decomposition, and positive denite matrices. Applications to
Information: H. R. Miller least-squares approximations, stability of dierential equations,
networks, Fourier transforms, and Markov processes. Uses linear
18.032 Dierential Equations algebra soware. Compared with 18.700, more emphasis on matrix
Prereq: None. Coreq: Calculus II (GIR) algorithms and many applications.
U (Spring) Fall: E. Mossel.  Spring: A. Negut
5-0-7 units. REST
Credit cannot also be received for 18.03, CC.1803, ES.1803 18.062[J] Mathematics for Computer Science
Same subject as 6.042[J]
Covers much of the same material as 18.03 with more emphasis on Prereq: Calculus I (GIR)
theory. The point of view is rigorous and results are proven. Local U (Fall, Spring)
existence and uniqueness of solutions. In person not required. 5-0-7 units. REST
T. Ozuch-Meersseman
See description under subject 6.042[J].
18.04 Complex Variables with Applications Z. R. Abel, F. T. Leighton, A. Moitra
Prereq: Calculus II (GIR) and (18.03 or 18.032)
U (Spring) 18.065 Matrix Methods in Data Analysis, Signal Processing, and
4-0-8 units Machine Learning
Credit cannot also be received for 18.075, 18.0751 Subject meets with 18.0651
Prereq: 18.06
Complex algebra and functions; analyticity; contour integration, U (Spring)
Cauchy's theorem; singularities, Taylor and Laurent series; residues, 3-0-9 units
evaluation of integrals; multivalued functions, potential theory in
two dimensions; Fourier analysis, Laplace transforms, and partial Reviews linear algebra with applications to life sciences, nance,
dierential equations. In person not required. engineering, and big data. Covers singular value decomposition,
Y. Wang weighted least squares, signal and image processing, principal
component analysis, covariance and correlation matrices, directed
18.05 Introduction to Probability and Statistics and undirected graphs, matrix factorizations, neural nets, machine
Prereq: Calculus II (GIR) learning, and computations with large matrices. In person not
U (Spring) required.
4-0-8 units. REST G. Strang

Elementary introduction with applications. Basic probability

models. Combinatorics. Random variables. Discrete and continuous
probability distributions. Statistical estimation and testing.
Condence intervals. Introduction to linear regression.
J. Orlo

10   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.0651 Matrix Methods in Data Analysis, Signal Processing, 18.085 Computational Science and Engineering I
and Machine Learning Subject meets with 18.0851
Subject meets with 18.065 Prereq: Calculus II (GIR) and (18.03 or 18.032)
Prereq: 18.06 U (Fall, Spring, Summer)
G (Spring) 3-0-9 units
3-0-9 units
Review of linear algebra, applications to networks, structures, and
Reviews linear algebra with applications to life sciences, nance, estimation, nite dierence and nite element solution of dierential
engineering, and big data. Covers singular value decomposition, equations, Laplace's equation and potential flow, boundary-value
weighted least squares, signal and image processing, principal problems, Fourier series, discrete Fourier transform, convolution.
component analysis, covariance and correlation matrices, directed Frequent use of MATLAB in a wide range of scientic and engineering
and undirected graphs, matrix factorizations, neural nets, machine applications.
learning, and computations with large matrices. Students in Course Fall: L. Demanet. Spring: L. Lu
18 must register for the undergraduate version, 18.065. In person not
required. 18.0851 Computational Science and Engineering I
G. Strang Subject meets with 18.085
Prereq: Calculus II (GIR) and (18.03 or 18.032)
18.075 Methods for Scientists and Engineers G (Fall, Spring, Summer)
Subject meets with 18.0751 3-0-9 units
Prereq: Calculus II (GIR) and 18.03
U (Spring) Review of linear algebra, applications to networks, structures,
3-0-9 units and estimation, nite dierence and nite element solution of
Credit cannot also be received for 18.04 dierential equations, Laplace's equation and potential flow,
boundary-value problems, Fourier series, discrete Fourier transform,
Covers functions of a complex variable; calculus of residues. convolution. Frequent use of MATLAB in a wide range of scientic and
Includes ordinary dierential equations; Bessel and Legendre engineering applications. Students in Course 18 must register for the
functions; Sturm-Liouville theory; partial dierential equations; heat undergraduate version, 18.085.
equation; and wave equations. Fall: L. Demanet. Spring: L. Lu
H. Cheng
18.086 Computational Science and Engineering II
18.0751 Methods for Scientists and Engineers Subject meets with 18.0861
Subject meets with 18.075 Prereq: Calculus II (GIR) and (18.03 or 18.032)
Prereq: Calculus II (GIR) and 18.03 Acad Year 2020-2021: Not oered
G (Spring) Acad Year 2021-2022: U (Spring)
3-0-9 units 3-0-9 units
Credit cannot also be received for 18.04
Initial value problems: nite dierence methods, accuracy and
Covers functions of a complex variable; calculus of residues. stability, heat equation, wave equations, conservation laws and
Includes ordinary dierential equations; Bessel and Legendre shocks, level sets, Navier-Stokes. Solving large systems: elimination
functions; Sturm-Liouville theory; partial dierential equations; heat with reordering, iterative methods, preconditioning, multigrid,
equation; and wave equations. Students in Courses 6, 8, 12, 18, and Krylov subspaces, conjugate gradients. Optimization and minimum
22 must register for undergraduate version, 18.075. principles: weighted least squares, constraints, inverse problems,
H. Cheng calculus of variations, saddle point problems, linear programming,
duality, adjoint methods.
Information: W. G. Strang

Department of Mathematics   |   11
DEPARTMENT OF MATHEMATICS

18.0861 Computational Science and Engineering II 18.098 Internship in Mathematics

Subject meets with 18.086 Prereq: Permission of instructor
Prereq: Calculus II (GIR) and (18.03 or 18.032) U (Fall, IAP, Spring, Summer)
Acad Year 2020-2021: Not oered Units arranged [P/D/F]
Acad Year 2021-2022: G (Spring) Can be repeated for credit.
3-0-9 units
Provides academic credit for students pursuing internships to gain
Initial value problems: nite dierence methods, accuracy and practical experience in the applications of mathematical concepts
stability, heat equation, wave equations, conservation laws and and methods.
shocks, level sets, Navier-Stokes. Solving large systems: elimination Information: W. Minicozzi
with reordering, iterative methods, preconditioning, multigrid,
Krylov subspaces, conjugate gradients. Optimization and minimum 18.099 Independent Study
principles: weighted least squares, constraints, inverse problems, Prereq: Permission of instructor
calculus of variations, saddle point problems, linear programming, U (Fall, IAP, Spring, Summer)
duality, adjoint methods. Students in Course 18 must register for the Units arranged
undergraduate version, 18.086. Can be repeated for credit.
Information: W. G. Strang
Studies (during IAP) or special individual reading (during regular
18.089 Review of Mathematics terms). Arranged in consultation with individual faculty members
Prereq: Permission of instructor and subject to departmental approval.  May not be used to satisfy
G (Summer) Mathematics major requirements.
5-0-7 units Information: W. Minicozzi

One-week review of one-variable calculus (18.01), followed by

Analysis
concentrated study covering multivariable calculus (18.02), two
hours per day for ve weeks. Primarily for graduate students in 18.1001 Real Analysis
Course 2N. Degree credit allowed only in special circumstances. Prereq: Calculus II (GIR)
Information: W. Minicozzi G (Fall, Spring)
3-0-9 units
18.094[J] Teaching College-Level Science and Engineering Credit cannot also be received for 18.1002, 18.100A, 18.100B,
Same subject as 1.95[J], 5.95[J], 7.59[J], 8.395[J] 18.100P, 18.100Q
Subject meets with 2.978
Prereq: None Covers fundamentals of mathematical analysis: convergence of
Acad Year 2020-2021: Not oered sequences and series, continuity, dierentiability, Riemann integral,
Acad Year 2021-2022: G (Fall) sequences and series of functions, uniformity, interchange of limit
2-0-2 units operations. Shows the utility of abstract concepts and teaches
understanding and construction of proofs. Proofs and denitions are
See description under subject 5.95[J]. less abstract than in 18.100B. Gives applications where possible.
J. Rankin Concerned primarily with the real line. Students in Course 18 must
register for undergraduate version 18.100A.
18.095 Mathematics Lecture Series Fall: C. Rodriguez. Spring: X. Yu
Prereq: Calculus I (GIR)
U (IAP)
2-0-4 units
Can be repeated for credit.

Ten lectures by mathematics faculty members on interesting

topics from both classical and modern mathematics. All lectures
accessible to students with calculus background and an interest in
mathematics. At each lecture, reading and exercises are assigned.
Students prepare these for discussion in a weekly problem session.
Information: W. Minicozzi

12   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.1002 Real Analysis 18.100P Real Analysis

Subject meets with 18.100A, 18.100B Prereq: Calculus II (GIR)
Prereq: Calculus II (GIR) U (Spring)
G (Fall, Spring) 4-0-11 units
3-0-9 units Credit cannot also be received for 18.1001, 18.1002, 18.100A,
Credit cannot also be received for 18.1001, 18.100P, 18.100Q 18.100B, 18.100Q

Covers fundamentals of mathematical analysis: convergence of Covers fundamentals of mathematical analysis: convergence of
sequences and series, continuity, dierentiability, Riemann integral, sequences and series, continuity, dierentiability, Riemann integral,
sequences and series of functions, uniformity, interchange of limit sequences and series of functions, uniformity, interchange of limit
operations. Shows the utility of abstract concepts and teaches operations. Shows the utility of abstract concepts and teaches
understanding and construction of proofs. More demanding than understanding and construction of proofs. Proofs and denitions are
18.100A, for students with more mathematical maturity. Places more less abstract than in 18.100B. Gives applications where possible.
emphasis on point-set topology and n-space. Students in Course Concerned primarily with the real line. Includes instruction and
18 must register for undergraduate version 18.100B.  In person not practice in written communication. Enrollment limited.
required. B. Landon
Fall: T. Colding. Spring: P-K Hung
18.100Q Real Analysis
18.100A Real Analysis Prereq: Calculus II (GIR)
Subject meets with 18.1002, 18.100B U (Fall)
Prereq: Calculus II (GIR) 4-0-11 units
U (Fall, Spring) Credit cannot also be received for 18.1001, 18.1002, 18.100A,
3-0-9 units 18.100B, 18.100P
Credit cannot also be received for 18.1001, 18.100P, 18.100Q
Covers fundamentals of mathematical analysis: convergence of
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, dierentiability, Riemann integral,
sequences and series, continuity, dierentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit
sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches
operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than
understanding and construction of proofs. Proofs and denitions are 18.100A, for students with more mathematical maturity. Places more
less abstract than in 18.100B. Gives applications where possible. emphasis on point-set topology and n-space. Includes instruction
Concerned primarily with the real line. and practice in written communication. Enrollment limited.
Fall: C. Rodriguez. Spring: X. Yu Y. Zhao

18.100B Real Analysis 18.101 Analysis and Manifolds

Subject meets with 18.1002, 18.100A Subject meets with 18.1011
Prereq: Calculus II (GIR) Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
U (Fall, Spring) 18.100Q)
3-0-9 units U (Fall)
Credit cannot also be received for 18.1001, 18.100P, 18.100Q 3-0-9 units

Covers fundamentals of mathematical analysis: convergence of Introduction to the theory of manifolds: vector elds and densities
sequences and series, continuity, dierentiability, Riemann integral, on manifolds, integral calculus in the manifold setting and the
sequences and series of functions, uniformity, interchange of limit manifold version of the divergence theorem. 18.901 helpful but not
operations. Shows the utility of abstract concepts and teaches required.
understanding and construction of proofs. More demanding than R. B. Melrose
18.100A, for students with more mathematical maturity. Places more
emphasis on point-set topology and n-space. In person not required.
Fall: T. Colding. Spring: P-K Hung

Department of Mathematics   |   13
DEPARTMENT OF MATHEMATICS

18.1011 Analysis and Manifolds 18.1031 Fourier Analysis: Theory and Applications
Subject meets with 18.101 Subject meets with 18.103
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
18.100Q) 18.100Q)
G (Fall) G (Fall)
3-0-9 units 3-0-9 units

Introduction to the theory of manifolds: vector elds and densities Roughly half the subject devoted to the theory of the Lebesgue
on manifolds, integral calculus in the manifold setting and the integral with applications to probability, and half to Fourier series
manifold version of the divergence theorem. 18.9011 helpful but not and Fourier integrals. Students in Course 18 must register for the
required. Students in Course 18 must register for the undergraduate undergraduate version, 18.103.
version, 18.101. A. Lawrie
R. B. Melrose
18.104 Seminar in Analysis
18.102 Introduction to Functional Analysis Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Subject meets with 18.1021 U (Spring)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 3-0-9 units
18.100Q)
U (Spring) Students present and discuss material from books or journals. Topics
3-0-9 units vary from year to year. Instruction and practice in written and oral
communication provided. In person not required. Enrollment limited.
Normed spaces, completeness, functionals, Hahn-Banach theorem, G. Stalani
duality, operators. Lebesgue measure, measurable functions,
integrability, completeness of L-p spaces. Hilbert space. Compact, 18.112 Functions of a Complex Variable
Hilbert-Schmidt and trace class operators. Spectral theorem. Subject meets with 18.1121
C. Rodriguez Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
18.100Q)
18.1021 Introduction to Functional Analysis U (Fall)
Subject meets with 18.102 3-0-9 units
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
18.100Q) Studies the basic properties of analytic functions of one complex
G (Spring) variable. Conformal mappings and the Poincare model of non-
3-0-9 units Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral
formula. Taylor and Laurent decompositions. Singularities,
Normed spaces, completeness, functionals, Hahn-Banach theorem, residues and computation of integrals. Harmonic functions and
duality, operators. Lebesgue measure, measurable functions, Dirichlet's problem for the Laplace equation. The partial fractions
integrability, completeness of L-p spaces. Hilbert space. Compact, decomposition. Innite series and innite product expansions. The
Hilbert-Schmidt and trace class operators. Spectral theorem. Gamma function. The Riemann mapping theorem. Elliptic functions.
Students in Course 18 must register for the undergraduate version, A. Borodin
18.102.
C. Rodriguez

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.
A. Lawrie

14   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.1121 Functions of a Complex Variable 18.125 Measure Theory and Analysis

Subject meets with 18.112 Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or G (Spring)
18.100Q) 3-0-9 units
G (Fall)
3-0-9 units Provides a rigorous introduction to Lebesgue's theory of measure
and integration. Covers material that is essential in analysis,
Studies the basic properties of analytic functions of one complex probability theory, and dierential geometry.
variable. Conformal mappings and the Poincare model of non- D. W. Stroock
Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral
formula. Taylor and Laurent decompositions. Singularities, 18.137 Topics in Geometric Partial Dierential Equations
residues and computation of integrals. Harmonic functions and Prereq: Permission of instructor
Dirichlet's problem for the Laplace equation. The partial fractions Acad Year 2020-2021: Not oered
decomposition. Innite series and innite product expansions. The Acad Year 2021-2022: G (Fall)
Gamma function. The Riemann mapping theorem. Elliptic functions. 3-0-9 units
Students in Course 18 must register for the undergraduate version, Can be repeated for credit.
18.112.
A. Borodin Topics vary from year to year.
T. Colding
18.116 Riemann Surfaces
Prereq: 18.112 18.152 Introduction to Partial Dierential Equations
Acad Year 2020-2021: Not oered Subject meets with 18.1521
Acad Year 2021-2022: G (Spring) Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
3-0-9 units 18.100Q)
U (Spring)
Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of 3-0-9 units
elliptic functions and modular forms. Some applications, such as to
number theory. Introduces three main types of partial dierential equations:
T. S. Mrowka diusion, elliptic, and hyperbolic. Includes mathematical tools,
real-world examples and applications, such as the Black-Scholes
18.117 Topics in Several Complex Variables equation, the European options problem, water waves, scalar
Prereq: 18.112 and 18.965 conservation laws, rst order equations and trac problems.
Acad Year 2020-2021: Not oered D. Jerison
Acad Year 2021-2022: G (Spring)
3-0-9 units 18.1521 Introduction to Partial Dierential Equations
Can be repeated for credit. Subject meets with 18.152
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
Harmonic theory on complex manifolds, Hodge decomposition 18.100Q)
theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of G (Spring)
Stein manifolds. As time permits students also study holomorphic 3-0-9 units
vector bundles on Kahler manifolds.
B. Poonen Introduces three main types of partial dierential equations:
diusion, elliptic, and hyperbolic. Includes mathematical tools,
18.118 Topics in Analysis real-world examples and applications, such as the Black-Scholes
Prereq: Permission of instructor equation, the European options problem, water waves, scalar
G (Fall) conservation laws, rst order equations and trac problems.
Not oered regularly; consult department Students in Course 18 must register for the undergraduate version,
3-0-9 units 18.152.
Can be repeated for credit. D. Jerison

Topics vary from year to year.

L. Guth

Department of Mathematics   |   15
DEPARTMENT OF MATHEMATICS

18.155 Dierential Analysis I 18.199 Graduate Analysis Seminar

Prereq: 18.102 or 18.103 Prereq: Permission of instructor
G (Fall) G (Fall)
3-0-9 units Not oered regularly; consult department
3-0-9 units
First part of a two-subject sequence. Review of Lebesgue integration. Can be repeated for credit.
Lp spaces. Distributions. Fourier transform. Sobolev spaces.
Spectral theorem, discrete and continuous spectrum. Homogeneous Studies original papers in dierential analysis and dierential
distributions. Fundamental solutions for elliptic, hyperbolic and equations. Intended for rst- and second-year graduate students.
parabolic dierential operators. Recommended prerequisite: 18.112. Permission must be secured in advance.
In person not required. V. W. Guillemin
T. S. Mrowka
Discrete Applied Mathematics
18.156 Dierential Analysis II
Prereq: 18.155 18.200 Principles of Discrete Applied Mathematics
G (Spring) Prereq: None. Coreq: 18.06
3-0-9 units U (Spring)
Second part of a two-subject sequence. Covers variable coecient 4-0-11 units
elliptic, parabolic and hyperbolic partial dierential equations. Credit cannot also be received for 18.200A
L. Guth Study of illustrative topics in discrete applied mathematics,
including probability theory, information theory, coding theory,
18.157 Introduction to Microlocal Analysis secret codes, generating functions, and linear programming.
Prereq: 18.155 Instruction and practice in written communication provided.
Acad Year 2020-2021: G (Spring) Enrollment limited.
Acad Year 2021-2022: Not oered M. X. Goemans, D. Cifuentes
3-0-9 units

The semi-classical theory of partial dierential equations. Discussion 18.200A Principles of Discrete Applied Mathematics
of Pseudodierential operators, Fourier integral operators, Prereq: None. Coreq: 18.06
asymptotic solutions of partial dierential equations, and the U (Fall)
spectral theory of Schroedinger operators from the semi-classical 3-0-9 units
perspective. Heavy emphasis placed on the symplectic geometric Credit cannot also be received for 18.200
underpinnings of this subject. Study of illustrative topics in discrete applied mathematics,
P. Hintz including probability theory, information theory, coding theory,
secret codes, generating functions, and linear programming.
18.158 Topics in Dierential Equations D. Cifuentes
Prereq: 18.157
Acad Year 2020-2021: G (Spring) 18.204 Undergraduate Seminar in Discrete Mathematics
Acad Year 2021-2022: Not oered Prereq: ((6.042[J] or 18.200) and (18.06, 18.700, or 18.701)) or
3-0-9 units permission of instructor
Can be repeated for credit. U (Fall, Spring)
Topics vary from year to year. In person not required. 3-0-9 units
R. B. Melrose 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.
S. Dhara, N. Gadish, J. Gaudio

16   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.211 Combinatorial Analysis 18.225 Graph Theory and Additive Combinatorics (New)
Prereq: Calculus II (GIR) and (18.06, 18.700, or 18.701) Prereq: ((18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or
U (Fall) 18.100Q)) or permission of instructor
3-0-9 units Acad Year 2020-2021: Not oered
Acad Year 2021-2022: G (Fall)
Combinatorial problems and methods for their solution. 3-0-9 units
Enumeration, generating functions, recurrence relations,
construction of bijections. Introduction to graph theory. Prior Introduction to extremal graph theory and additive combinatorics.
experience with abstraction and proofs is helpful. Highlights common themes, such as the dichotomy between
Z. Jiang structure versus pseudorandomness. Topics include Turan-
type problems, Szemeredi's regularity lemma and applications,
18.212 Algebraic Combinatorics pseudorandom graphs, spectral graph theory, graph limits,
Prereq: 18.701 or 18.703 arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete
U (Spring) Fourier analysis, Freiman's theorem on sumsets and structure.
3-0-9 units Discusses current research topics and open problems.
Y. Zhao
Applications of algebra to combinatorics. Topics include walks
in graphs, the Radon transform, groups acting on posets, Young 18.226 Probabilistic Methods in Combinatorics (New)
tableaux, electrical networks. Prereq: (18.211, 18.600, and (18.100A, 18.100B, 18.100P, or 18.100Q))
A. Postnikov or permission of instructor
Acad Year 2020-2021: G (Fall)
18.217 Combinatorial Theory Acad Year 2021-2022: Not oered
Prereq: Permission of instructor 3-0-9 units
G (Fall)
3-0-9 units Introduction to the probabilistic method, a fundamental and
Can be repeated for credit. powerful technique in combinatorics and theoretical computer
science. Focuses on methodology as well as combinatorial
Content varies from year to year. In person not required. applications. Suitable for students with strong interest and
A. Postnikov background in mathematical problem solving. Topics include
linearity of expectations, alteration, second moment, Lovasz local
18.218 Topics in Combinatorics lemma, correlation inequalities, Janson inequalities, concentration
Prereq: Permission of instructor inequalities, entropy method.
G (Spring) Y. Zhao
3-0-9 units
Can be repeated for credit.
Continuous Applied Mathematics
Topics vary from year to year.
D. Minzer 18.300 Principles of Continuum Applied Mathematics
Prereq: Calculus II (GIR) and (18.03 or 18.032)
18.219 Seminar in Combinatorics U (Spring)
Prereq: Permission of instructor 3-0-9 units
G (Fall)
Not oered regularly; consult department Covers fundamental concepts in continuous applied mathematics.
3-0-9 units Applications from trac flow, fluids, elasticity, granular flows, etc.
Can be repeated for credit. Also covers continuum limit; conservation laws, quasi-equilibrium;
kinematic waves; characteristics, simple waves, shocks; diusion
Content varies from year to year. Readings from current research (linear and nonlinear); numerical solution of wave equations;
papers in combinatorics. Topics to be chosen and presented by the nite dierences, consistency, stability; discrete and fast Fourier
class. transforms; spectral methods; transforms and series (Fourier,
Information: Y. Zhao Laplace). Additional topics may include sonic booms, Mach cone,
caustics, lattices, dispersion and group velocity. Uses MATLAB
computing environment.
M. Durey

Department of Mathematics   |   17
DEPARTMENT OF MATHEMATICS

18.303 Linear Partial Dierential Equations: Analysis and 18.327 Topics in Applied Mathematics
Numerics Prereq: Permission of instructor
Prereq: 18.06 or 18.700 Acad Year 2020-2021: Not oered
U (Spring) Acad Year 2021-2022: G (Spring)
3-0-9 units 3-0-9 units
Can be repeated for credit.
Provides students with the basic analytical and computational
tools of linear partial dierential equations (PDEs) for practical Topics vary from year to year.
applications in science and engineering, including heat/diusion, L. Demanet
wave, and Poisson equations. Analytics emphasize the viewpoint of
linear algebra and the analogy with nite matrix problems. Studies 18.330 Introduction to Numerical Analysis
operator adjoints and eigenproblems, series solutions, Green's Prereq: Calculus II (GIR) and (18.03 or 18.032)
functions, and separation of variables. Numerics focus on nite- U (Spring)
dierence and nite-element techniques to reduce PDEs to matrix 3-0-9 units
problems, including stability and convergence analysis and implicit/
explicit timestepping. Some programming required for homework Basic techniques for the ecient numerical solution of problems in
and nal project. science and engineering. Root nding, interpolation, approximation
V. Heinonen of functions, integration, dierential equations, direct and iterative
methods in linear algebra. Knowledge of programming in a language
18.305 Advanced Analytic Methods in Science and Engineering such as MATLAB, Python, or Julia is helpful. In person not required.
Prereq: 18.04, 18.075, or 18.112 D. Sanders
G (Fall)
3-0-9 units 18.335[J] Introduction to Numerical Methods
Same subject as 6.337[J]
Covers expansion around singular points: the WKB method on Prereq: 18.06, 18.700, or 18.701
ordinary and partial dierential equations; the method of stationary G (Spring)
phase and the saddle point method; the two-scale method and the 3-0-9 units
method of renormalized perturbation; singular perturbation and
boundary-layer techniques; WKB method on partial dierential Advanced introduction to numerical analysis: accuracy and eciency
equations. In person not required. of numerical algorithms. In-depth coverage of sparse-matrix/iterative
H. Cheng and dense-matrix algorithms in numerical linear algebra (for linear
systems and eigenproblems). Floating-point arithmetic, backwards
18.306 Advanced Partial Dierential Equations with Applications error analysis, conditioning, and stability. Other computational
Prereq: (18.03 or 18.032) and (18.04, 18.075, or 18.112) topics (e.g., numerical integration or nonlinear optimization) may
G (Spring) also be surveyed.  Final project involves some programming.
3-0-9 units S. Johnson

Concepts and techniques for partial dierential equations, especially

nonlinear. Diusion, 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, trac flow, etc. In person not required.
R. R. Rosales

18   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.336[J] Fast Methods for Partial Dierential and Integral 18.352[J] Nonlinear Dynamics: The Natural Environment
Equations Same subject as 12.009[J]
Same subject as 6.335[J] Prereq: Calculus II (GIR) and Physics I (GIR); Coreq: 18.03
Prereq: 6.336[J], 16.920[J], 18.085, 18.335[J], or permission of Acad Year 2020-2021: Not oered
instructor Acad Year 2021-2022: U (Spring)
G (Fall) 3-0-9 units
3-0-9 units
See description under subject 12.009[J].
Unied introduction to the theory and practice of modern, near D. H. Rothman
linear-time, numerical methods for large-scale partial-dierential
and integral equations. Topics include preconditioned iterative 18.353[J] Nonlinear Dynamics: Chaos
methods; generalized Fast Fourier Transform and other butterfly- Same subject as 2.050[J], 12.006[J]
based methods; multiresolution approaches, such as multigrid Prereq: Physics II (GIR) and (18.03 or 18.032)
algorithms and hierarchical low-rank matrix decompositions; U (Fall)
and low and high frequency Fast Multipole Methods. Example 3-0-9 units
applications include aircra design, cardiovascular system
modeling, electronic structure computation, and tomographic See description under subject 12.006[J].
imaging. M. Durey
K. Burns
18.354[J] Nonlinear Dynamics: Continuum Systems
18.337[J] Parallel Computing and Scientic Machine Learning Same subject as 1.062[J], 12.207[J]
Same subject as 6.338[J] Subject meets with 18.3541
Prereq: 18.06, 18.700, or 18.701 Prereq: Physics II (GIR) and (18.03 or 18.032)
G (Fall) U (Spring)
3-0-9 units 3-0-9 units

Introduction to scientic machine learning with an emphasis on General mathematical principles of continuum systems. From
developing scalable dierentiable programs. Covers scientic microscopic to macroscopic descriptions in the form of linear
computing topics (numerical dierential equations, dense and or nonlinear (partial) dierential equations. Exact solutions,
sparse linear algebra, Fourier transformations, parallelization dimensional analysis, calculus of variations and singular
of large-scale scientic simulation) simultaneously with modern perturbation methods. Stability, waves and pattern formation in
data science (machine learning, deep neural networks, automatic continuum systems. Subject matter illustrated using natural fluid
dierentiation), focusing on the emerging techniques at the and solid systems found, for example, in geophysics and biology.
connection between these areas, such as neural dierential O. Kodio
equations and physics-informed deep learning. Provides direct
experience with the modern realities of optimizing code performance 18.3541 Nonlinear Dynamics: Continuum Systems
for supercomputers, GPUs, and multicores in a high-level language. Subject meets with 1.062[J], 12.207[J], 18.354[J]
C. Rackauckas Prereq: Physics II (GIR) and (18.03 or 18.032)
G (Spring)
18.338 Eigenvalues of Random Matrices 3-0-9 units
Prereq: 18.701 or permission of instructor General mathematical principles of continuum systems. From
Acad Year 2020-2021: G (Fall) microscopic to macroscopic descriptions in the form of linear
Acad Year 2021-2022: Not oered or nonlinear (partial) dierential equations. Exact solutions,
3-0-9 units dimensional analysis, calculus of variations and singular
Covers the modern main results of random matrix theory as it perturbation methods. Stability, waves and pattern formation in
is currently applied in engineering and science. Topics include continuum systems. Subject matter illustrated using natural fluid
matrix calculus for nite and innite matrices (e.g., Wigner's semi- and solid systems found, for example, in geophysics and biology.
circle and Marcenko-Pastur laws), free probability, random graphs, Students in Courses 1, 12, and 18 must register for undergraduate
combinatorial methods, matrix statistics, stochastic operators, version, 18.354[J].
passage to the continuum limit, moment methods, and compressed O. Kodio
sensing. Knowledge of MATLAB hepful, but not required.
A. Edelman

Department of Mathematics   |   19
DEPARTMENT OF MATHEMATICS

18.355 Fluid Mechanics 18.367 Waves and Imaging

Prereq: 2.25, 12.800, or 18.354[J] Prereq: Permission of instructor
Acad Year 2020-2021: Not oered Acad Year 2020-2021: G (Spring)
Acad Year 2021-2022: G (Fall) Acad Year 2021-2022: Not oered
3-0-9 units 3-0-9 units

Topics include the development of Navier-Stokes equations, inviscid The mathematics of inverse problems involving waves, with
flows, boundary layers, lubrication theory, Stokes flows, and surface examples taken from reflection seismology, synthetic aperture
tension. Fundamental concepts illustrated through problems drawn radar, and computerized tomography. Suitable for graduate students
from a variety of areas, including geophysics, biology, and the from all departments who have anities with applied mathematics.
dynamics of sport. Particular emphasis on the interplay between Topics include acoustic, elastic, electromagnetic wave equations;
dimensional analysis, scaling arguments, and theory. Includes geometrical optics; scattering series and inversion; migration and
classroom and laboratory demonstrations. backprojection; adjoint-state methods; Radon and curvilinear
J. W. Bush Radon transforms; microlocal analysis of imaging; optimization,
regularization, and sparse regression.
18.357 Interfacial Phenomena L. Demanet
Prereq: 2.25, 12.800, 18.354[J], 18.355, or permission of instructor
Acad Year 2020-2021: G (Spring) 18.369[J] Mathematical Methods in Nanophotonics
Acad Year 2021-2022: Not oered Same subject as 8.315[J]
3-0-9 units Prereq: 8.07, 18.303, or permission of instructor
Acad Year 2020-2021: Not oered
Fluid systems dominated by the influence of interfacial tension. Acad Year 2021-2022: G (Spring)
Elucidates the roles of curvature pressure and Marangoni stress in 3-0-9 units
a variety of hydrodynamic settings. Particular attention to drops
and bubbles, soap lms and minimal surfaces, wetting phenomena, High-level approaches to understanding complex optical media,
water-repellency, surfactants, Marangoni flows, capillary origami structured on the scale of the wavelength, that are not generally
and contact line dynamics. Theoretical developments are analytically soluable. The basis for understanding optical
accompanied by classroom demonstrations. Highlights the role of phenomena such as photonic crystals and band gaps, anomalous
surface tension in biology. diraction, mechanisms for optical connement, optical bers (new
J. W. Bush and old), nonlinearities, and integrated optical devices. Methods
covered include linear algebra and eigensystems for Maxwell's
18.358[J] Nonlinear Dynamics and Turbulence equations, symmetry groups and representation theory, Bloch's
Same subject as 1.686[J], 2.033[J] theorem, numerical eigensolver methods, time and frequency-
Subject meets with 1.068 domain computation, perturbation theory, and coupled-mode
Prereq: 1.060A theories.
Acad Year 2020-2021: Not oered S. G. Johnson
Acad Year 2021-2022: G (Spring)
3-2-7 units 18.376[J] Wave Propagation
Same subject as 1.138[J], 2.062[J]
See description under subject 1.686[J]. Prereq: 2.003[J] and 18.075
L. Bourouiba Acad Year 2020-2021: G (Spring)
Acad Year 2021-2022: Not oered
3-0-9 units

See description under subject 2.062[J].

T. R. Akylas, R. R. Rosales

20   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.377[J] Nonlinear Dynamics and Waves 18.397 Mathematical Methods in Physics

Same subject as 1.685[J], 2.034[J] Prereq: 18.745 or some familiarity with Lie theory
Prereq: Permission of instructor G (Fall)
Acad Year 2020-2021: Not oered Not oered regularly; consult department
Acad Year 2021-2022: G (Spring) 3-0-9 units
3-0-9 units Can be repeated for credit.

A unied treatment of nonlinear oscillations and wave phenomena Content varies from year to year. Recent developments in quantum
with applications to mechanical, optical, geophysical, fluid, eld theory require mathematical techniques not usually covered in
electrical and flow-structure interaction problems. Nonlinear free and standard graduate subjects.
forced vibrations; nonlinear resonances; self-excited oscillations; V. G. Kac
lock-in phenomena. Nonlinear dispersive and nondispersive waves;
resonant wave interactions; propagation of wave pulses and
Theoretical Computer Science
nonlinear Schrodinger equation. Nonlinear long waves and breaking;
theory of characteristics; the Korteweg-de Vries equation; solitons 18.400[J] Computability and Complexity Theory
and solitary wave interactions. Stability of shear flows. Some topics Same subject as 6.045[J]
and applications may vary from year to year. Prereq: 6.042[J]
R. R. Rosales U (Spring)
4-0-8 units
18.384 Undergraduate Seminar in Physical Mathematics
Prereq: 12.006[J], 18.300, 18.354[J], or permission of instructor See description under subject 6.045[J].
U (Fall) R. Williams, R. Rubinfeld
3-0-9 units
18.404 Theory of Computation
Covers the mathematical modeling of physical systems, with Subject meets with 6.840[J], 18.4041[J]
emphasis on the reading and presentation of papers. Addresses Prereq: 6.042[J] or 18.200
a broad range of topics, with particular focus on macroscopic U (Fall)
physics and continuum systems: fluid dynamics, solid mechanics, 4-0-8 units
and biophysics. Instruction and practice in written and oral
communication provided. Enrollment limited. A more extensive and theoretical treatment of the material in
O. Kodio 6.045[J]/18.400[J], emphasizing computability and computational
complexity theory. Regular and context-free languages. Decidable
18.385[J] Nonlinear Dynamics and Chaos and undecidable problems, reducibility, recursive function theory.
Same subject as 2.036[J] Time and space measures on computation, completeness, hierarchy
Prereq: 18.03 or 18.032 theorems, inherently complex problems, oracles, probabilistic
Acad Year 2020-2021: G (Fall) computation, and interactive proof systems.
Acad Year 2021-2022: Not oered M. Sipser
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. In person not
required.
R. R. Rosales

Department of Mathematics   |   21
DEPARTMENT OF MATHEMATICS

18.4041[J] Theory of Computation 18.415[J] Advanced Algorithms

Same subject as 6.840[J] Same subject as 6.854[J]
Subject meets with 18.404 Prereq: 6.046[J] and (6.042[J], 18.600, or 6.041)
Prereq: 6.042[J] or 18.200 G (Fall)
G (Fall) 5-0-7 units
4-0-8 units
See description under subject 6.854[J].
A more extensive and theoretical treatment of the material in A. Moitra, D. R. Karger
6.045[J]/18.400[J], emphasizing computability and computational
complexity theory. Regular and context-free languages. Decidable 18.416[J] Randomized Algorithms
and undecidable problems, reducibility, recursive function theory. Same subject as 6.856[J]
Time and space measures on computation, completeness, hierarchy Prereq: (6.041 or 6.042[J]) and (6.046[J] or 6.854[J])
theorems, inherently complex problems, oracles, probabilistic Acad Year 2020-2021: G (Spring)
computation, and interactive proof systems. Students in Course 18 Acad Year 2021-2022: Not oered
must register for the undergraduate version, 18.404. 5-0-7 units
M. Sipser
See description under subject 6.856[J].
18.405[J] Advanced Complexity Theory D. R. Karger
Same subject as 6.841[J]
Prereq: 18.404 18.417 Introduction to Computational Molecular Biology
Acad Year 2020-2021: Not oered Prereq: 6.006, 6.01, or permission of instructor
Acad Year 2021-2022: G (Fall) G (Fall)
3-0-9 units Not oered regularly; consult department
3-0-9 units
Current research topics in computational complexity theory.
Nondeterministic, alternating, probabilistic, and parallel Introduces the basic computational methods used to model and
computation models. Boolean circuits. Complexity classes and predict the structure of biomolecules (proteins, DNA, RNA). Covers
complete sets. The polynomial-time hierarchy. Interactive proof classical techniques in the eld (molecular dynamics, Monte Carlo,
systems. Relativization. Denitions of randomness. Pseudo- dynamic programming) to more recent advances in analyzing and
randomness and derandomizations. Interactive proof systems and predicting RNA and protein structure, ranging from Hidden Markov
probabilistically checkable proofs. Models and 3-D lattice models to attribute Grammars and tree
R. Williams Grammars.
Information: B. Berger
18.408 Topics in Theoretical Computer Science
Prereq: Permission of instructor 18.418[J] Topics in Computational Molecular Biology
G (Spring) Same subject as HST.504[J]
3-0-9 units Prereq: 6.047, 18.417, or permission of instructor
Can be repeated for credit. G (Spring)
3-0-9 units
Study of areas of current interest in theoretical computer science. Can be repeated for credit.
Topics vary from term to term.
A. Moitra, J. A. Kelner Covers current research topics in computational molecular biology.
Recent research papers presented from leading conferences such as
18.410[J] Design and Analysis of Algorithms the International Conference on Computational Molecular Biology
Same subject as 6.046[J] (RECOMB) and the Conference on Intelligent Systems for Molecular
Prereq: 6.006 Biology (ISMB). Topics include original research (both theoretical
U (Fall, Spring) and experimental) in comparative genomics, sequence and structure
4-0-8 units analysis, molecular evolution, proteomics, gene expression,
transcriptional regulation, biological networks, drug discovery,
See description under subject 6.046[J]. and privacy. Recent research by course participants also covered.
E. Demaine, M. Goemans Participants will be expected to present individual projects to the
class.
B. Berger

22   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.424 Seminar in Information Theory 18.437[J] Distributed Algorithms

Prereq: (6.041, 18.05, or 18.600) and (18.06, 18.700, or 18.701) Same subject as 6.852[J]
U (Spring) Prereq: 6.046[J]
3-0-9 units Acad Year 2020-2021: G (Fall)
Acad Year 2021-2022: Not oered
Considers various topics in information theory, including data 3-0-9 units
compression, Shannon's Theorems, and error-correcting codes.
Students present and discuss the subject matter. Instruction and See description under subject 6.852[J].
practice in written and oral communication provided. Enrollment N. A. Lynch
limited.
P. W. Shor 18.453 Combinatorial Optimization
Subject meets with 18.4531
18.425[J] Cryptography and Cryptanalysis Prereq: 18.06, 18.700, or 18.701
Same subject as 6.875[J] Acad Year 2020-2021: U (Spring)
Prereq: 6.046[J] Acad Year 2021-2022: Not oered
G (Fall) 3-0-9 units
3-0-9 units
Thorough treatment of linear programming and combinatorial
See description under subject 6.875[J]. optimization. Topics include matching theory, network flow, matroid
S. Goldwasser, S. Micali, V. Vaikuntanathan optimization, and how to deal with NP-hard optimization problems.
Prior exposure to discrete mathematics (such as 18.200) helpful.
18.434 Seminar in Theoretical Computer Science Information: W. C. Franks
Prereq: 6.046[J]
U (Fall) 18.4531 Combinatorial Optimization
3-0-9 units Subject meets with 18.453
Prereq: 18.06, 18.700, or 18.701
Topics vary from year to year. Students present and discuss Acad Year 2020-2021: G (Spring)
the subject matter. Instruction and practice in written and oral Acad Year 2021-2022: Not oered
communication provided. Enrollment limited. 3-0-9 units
J. Gaudio
Thorough treatment of linear programming and combinatorial
18.435[J] Quantum Computation optimization. Topics include matching theory, network flow, matroid
Same subject as 2.111[J], 8.370[J] optimization, and how to deal with NP-hard optimization problems.
Prereq: Permission of instructor Prior exposure to discrete mathematics (such as 18.200) helpful.
G (Fall) Students in Course 18 must register for the undergraduate version,
3-0-9 units 18.453.
W. C. Franks
Provides an introduction to the theory and practice of quantum
computation. Topics covered: physics of information processing; 18.455 Advanced Combinatorial Optimization
quantum algorithms including the factoring algorithm and Prereq: 18.453 or permission of instructor
Grover's search algorithm; quantum error correction; quantum Acad Year 2020-2021: Not oered
communication and cryptography. Knowledge of quantum mechanics Acad Year 2021-2022: G (Spring)
helpful but not required. 3-0-9 units
I. Chuang, A. Harrow, S. Lloyd, P. Shor
Advanced treatment of combinatorial optimization with an emphasis
18.436[J] Quantum Information Science on combinatorial aspects. Non-bipartite matchings, submodular
Same subject as 6.443[J], 8.371[J] functions, matroid intersection/union, matroid matching,
Prereq: 18.435[J] submodular flows, multicommodity flows, packing and connectivity
G (Spring) problems, and other recent developments.
3-0-9 units M. X. Goemans

See description under subject 8.371[J].

I. Chuang, A. Harrow

Department of Mathematics   |   23
DEPARTMENT OF MATHEMATICS

18.456[J] Algebraic Techniques and Semidenite Optimization Probability and Statistics

Same subject as 6.256[J]
Prereq: 6.251[J] or 15.093[J] 18.600 Probability and Random Variables
Acad Year 2020-2021: G (Spring) Prereq: Calculus II (GIR)
Acad Year 2021-2022: Not oered U (Fall, Spring)
3-0-9 units 4-0-8 units. REST
See description under subject 6.256[J]. Credit cannot also be received for 6.041, 6.431, 15.079, 15.0791
P. Parrilo Probability spaces, random variables, distribution functions.
Binomial, geometric, hypergeometric, Poisson distributions.
Logic Uniform, exponential, normal, gamma and beta distributions.
Conditional probability, Bayes theorem, joint distributions.
18.504 Seminar in Logic Chebyshev inequality, law of large numbers, and central limit
Prereq: (18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, theorem. Credit cannot also be received for 6.041A or 6.041B.
18.100P, or 18.100Q) J. A. Kelner, S. Sheeld
Acad Year 2020-2021: U (Fall)
Acad Year 2021-2022: Not oered 18.615 Introduction to Stochastic Processes
3-0-9 units Prereq: 6.041 or 18.600
G (Spring)
Students present and discuss the subject matter taken from current 3-0-9 units
journals or books. Topics vary from year to year. Instruction and
practice in written and oral communication provided. Enrollment Basics of stochastic processes. Markov chains, Poisson processes,
limited. random walks, birth and death processes, Brownian motion.
H. Cohn P. Kempthorne

18.510 Introduction to Mathematical Logic and Set Theory 18.642 Topics in Mathematics with Applications in Finance
Prereq: None Prereq: 18.03, 18.06, and (18.05 or 18.600)
Acad Year 2020-2021: Not oered U (Fall)
Acad Year 2021-2022: U (Fall) 3-0-9 units
3-0-9 units
Introduction to mathematical concepts and techniques used in
Propositional and predicate logic. Zermelo-Fraenkel set theory. nance. Lectures focusing on linear algebra, probability, statistics,
Ordinals and cardinals. Axiom of choice and transnite induction. stochastic processes, and numerical methods are interspersed
Elementary model theory: completeness, compactness, and with lectures by nancial sector professionals illustrating the
Lowenheim-Skolem theorems. Godel's incompleteness theorem. corresponding application in the industry. Prior knowledge of
H. Cohn economics or nance helpful but not required.
P. Kempthorne, V. Strela, J. Xia
18.515 Mathematical Logic
Prereq: Permission of instructor
G (Spring)
Not oered 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. Quantier
elimination. Recursive sets and functions. Incompleteness and
undecidability. Ordinals and cardinals. Set-theoretic formalization of
mathematics.
Information: B. Poonen

24   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

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

Same subject as IDS.014[J] (New)
Subject meets with 18.6501 Same subject as 9.521[J], IDS.160[J]
Prereq: 6.041 or 18.600 Prereq: (6.436[J], 18.06, and 18.6501) or permission of instructor
U (Fall, Spring) G (Spring)
4-0-8 units 3-0-9 units
Credit cannot also be received for 15.075[J], IDS.013[J]
See description under subject 9.521[J].
In-depth introduction to the theoretical foundations of statistical S. Rakhlin, P. Rigollet
methods that are useful in many applications. Enables students to
understand the role of mathematics in the research and development 18.657 Topics in Statistics
of ecient statistical methods. Topics include methods for Prereq: Permission of instructor
estimation (maximum likelihood estimation, method of moments, Acad Year 2020-2021: Not oered
M-estimation), hypothesis testing (Wald's test, likelihood ratio Acad Year 2021-2022: G (Spring)
test, T tests, goodness of t), Bayesian statistics, linear regression, 3-0-9 units
generalized linear models, and principal component analysis. Can be repeated for credit.
Fall: P. Rigollet. Spring: T. Maunu
Topics vary from term to term.
18.6501 Fundamentals of Statistics P. Rigollet
Subject meets with 18.650[J], IDS.014[J]
Prereq: 6.041 or 18.600 18.675 Theory of Probability
G (Fall, Spring) Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
4-0-8 units G (Fall)
Credit cannot also be received for 15.075[J], IDS.013[J] 3-0-9 units

In-depth introduction to the theoretical foundations of statistical Sums of independent random variables, central limit phenomena,
methods that are useful in many applications. Enables students to innitely divisible laws, Levy processes, Brownian motion,
understand the role of mathematics in the research and development conditioning, and martingales. Prior exposure to probability (e.g.,
of ecient statistical methods. Topics include methods for 18.600) recommended.
estimation (maximum likelihood estimation, method of moments, Y. Wang
M-estimation), hypothesis testing (Wald's test, likelihood ratio
test, T tests, goodness of t), Bayesian statistics, linear regression, 18.676 Stochastic Calculus
generalized linear models, and principal component analysis. Prereq: 18.675
Fall: P. Rigollet. Spring: T. Maunu G (Spring)
3-0-9 units
18.655 Mathematical Statistics Introduction to stochastic processes, building on the fundamental
Prereq: (18.650[J] and (18.100A, 18.100A, 18.100P, or 18.100Q)) or example of Brownian motion. Topics include Brownian motion,
permission of instructor continuous parameter martingales, Ito's theory of stochastic
G (Fall) dierential equations, Markov processes and partial dierential
3-0-9 units equations, and may also include local time and excursion theory.
Decision theory, estimation, condence intervals, hypothesis Students should have familiarity with Lebesgue integration and its
testing. Introduces large sample theory. Asymptotic eciency of application to probability. In person not required.
estimates. Exponential families. Sequential analysis. Prior exposure N. Sun
to both probability and statistics at the university level is assumed.
T. Maunu 18.677 Topics in Stochastic Processes
Prereq: 18.675
G (Spring)
3-0-9 units
Can be repeated for credit.

Topics vary from year to year.

A. Borodin

Department of Mathematics   |   25
DEPARTMENT OF MATHEMATICS

Algebra and Number Theory 18.704 Seminar in Algebra

Prereq: 18.701, (18.06 and 18.703), or (18.700 and 18.703)
18.700 Linear Algebra U (Fall, Spring)
Prereq: Calculus II (GIR) 3-0-9 units
U (Fall) Topics vary from year to year. Students present and discuss
3-0-9 units. REST the subject matter. Instruction and practice in written and oral
Credit cannot also be received for 18.06 communication provided. Some experience with proofs required.
Vector spaces, systems of linear equations, bases, linear Enrollment limited.
independence, matrices, determinants, eigenvalues, inner products, Fall: C. Chan.  Spring: J. Wang
quadratic forms, and canonical forms of matrices. More emphasis on
theory and proofs than in 18.06. 18.705 Commutative Algebra
V. G. Kac Prereq: 18.702
G (Fall)
18.701 Algebra I 3-0-9 units
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of Exactness, direct limits, tensor products, Cayley-Hamilton theorem,
instructor integral dependence, localization, Cohen-Seidenberg theory,
U (Fall) Noether normalization, Nullstellensatz, chain conditions, primary
3-0-9 units decomposition, length, Hilbert functions, dimension theory,
18.701-18.702 is more extensive and theoretical than the completion, Dedekind domains.
18.700-18.703 sequence. Experience with proofs necessary. 18.701 R. Bezrukavnikov
focuses on group theory, geometry, and linear algebra.
B. Poonen 18.706 Noncommutative Algebra
Prereq: 18.702
18.702 Algebra II Acad Year 2020-2021: G (Fall)
Prereq: 18.701 Acad Year 2021-2022: Not oered
U (Spring) 3-0-9 units
3-0-9 units Topics may include Wedderburn theory and structure of Artinian
Continuation of 18.701. Focuses on group representations, rings, rings, Morita equivalence and elements of category theory,
ideals, elds, polynomial rings, modules, factorization, integers in localization and Goldie's theorem, central simple algebras and the
quadratic number elds, eld extensions, and Galois theory. Brauer group, representations, polynomial identity rings, invariant
M. Artin theory growth of algebras, Gelfand-Kirillov dimension.
Z. Yun
18.703 Modern Algebra
Prereq: Calculus II (GIR) 18.708 Topics in Algebra
U (Spring) Prereq: 18.705
3-0-9 units Acad Year 2020-2021: Not oered
Acad Year 2021-2022: G (Spring)
Focuses on traditional algebra topics that have found greatest 3-0-9 units
application in science and engineering as well as in mathematics: Can be repeated for credit.
group theory, emphasizing nite groups; ring theory, including
ideals and unique factorization in polynomial and Euclidean rings; Topics vary from year to year.
eld theory, including properties and applications of nite elds. Z. Yun
18.700 and 18.703 together form a standard algebra sequence.
V. G. Kac

26   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.715 Introduction to Representation Theory 18.727 Topics in Algebraic Geometry

Prereq: 18.702 or 18.703 Prereq: 18.725
Acad Year 2020-2021: G (Fall) Acad Year 2020-2021: Not oered
Acad Year 2021-2022: Not oered Acad Year 2021-2022: G (Spring)
3-0-9 units 3-0-9 units
Can be repeated for credit.
Algebras, representations, Schur's lemma. Representations of SL(2).
Representations of nite groups, Maschke's theorem, characters, Topics vary from year to year.
applications. Induced representations, Burnside's theorem, Mackey A. Negut
formula, Frobenius reciprocity. Representations of quivers.
G. Lusztig 18.737 Algebraic Groups
Prereq: 18.705
18.721 Introduction to Algebraic Geometry Acad Year 2020-2021: G (Spring)
Prereq: 18.702 and 18.901 Acad Year 2021-2022: Not oered
Acad Year 2020-2021: Not oered 3-0-9 units
Acad Year 2021-2022: U (Spring)
3-0-9 units Structure of linear algebraic groups over an algebraically closed
eld, with emphasis on reductive groups. Representations of groups
Presents basic examples of complex algebraic varieties, ane and over a nite eld using methods from etale cohomology. Some
projective algebraic geometry, sheaves, cohomology. results from algebraic geometry are stated without proof.
M. Artin B. Poonen

18.725 Algebraic Geometry I 18.745 Lie Groups and Lie Algebras I

Prereq: None. Coreq: 18.705 Prereq: (18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or
G (Fall) 18.100Q)
3-0-9 units G (Fall)
3-0-9 units
Introduces the basic notions and techniques of modern algebraic
geometry. Covers fundamental notions and results about algebraic Covers fundamentals of the theory of Lie algebras and related
varieties over an algebraically closed eld; relations between groups. Topics may include theorems of Engel and Lie; enveloping
complex algebraic varieties and complex analytic varieties; algebra, Poincare-Birkho-Witt theorem; classication and
and examples with emphasis on algebraic curves and surfaces. construction of semisimple Lie algebras; the center of their
Introduction to the language of schemes and properties of enveloping algebras; elements of representation theory; compact Lie
morphisms. Knowledge of elementary algebraic topology, groups and/or nite Chevalley groups. In person not required.
elementary dierential geometry recommended, but not required. P. I. Etingof
D. Maulik
18.747 Innite-dimensional Lie Algebras
18.726 Algebraic Geometry II Prereq: 18.745
Prereq: 18.725 Acad Year 2020-2021: Not oered
G (Spring) Acad Year 2021-2022: G (Fall)
3-0-9 units 3-0-9 units

Continuation of the introduction to algebraic geometry given in Topics vary from year to year.
18.725. More advanced properties of the varieties and morphisms of P. I. Etingof
schemes, as well as sheaf cohomology.
D. Maulik 18.748 Topics in Lie Theory
Prereq: Permission of instructor
Acad Year 2020-2021: G (Fall)
Acad Year 2021-2022: Not oered
3-0-9 units
Can be repeated for credit.

Topics vary from year to year. In person not required.

J-L. Kim

Department of Mathematics   |   27
DEPARTMENT OF MATHEMATICS

18.755 Lie Groups and Lie Algebras II 18.783 Elliptic Curves

Prereq: 18.745 or permission of instructor Subject meets with 18.7831
G (Spring) Prereq: 18.702, 18.703, or permission of instructor
3-0-9 units Acad Year 2020-2021: U (Spring)
Acad Year 2021-2022: Not oered
A more in-depth treatment of Lie groups and Lie algebras. Topics 3-0-9 units
may include homogeneous spaces and groups of automorphisms;
representations of compact groups and their geometric realizations, Computationally focused introduction to elliptic curves, with
Peter-Weyl theorem; invariant dierential forms and cohomology of applications to number theory and cryptography. Topics include
Lie groups and homogeneous spaces; complex reductive Lie groups, point-counting, isogenies, pairings, and the theory of complex
classication of real reductive groups. In person not required. multiplication, with applications to integer factorization, primality
P. I. Etingof proving, and elliptic curve cryptography. Includes a brief introduction
to modular curves and the proof of Fermat's Last Theorem.
18.757 Representations of Lie Groups A. Sutherland
Prereq: 18.745 or 18.755
G (Spring) 18.7831 Elliptic Curves
Not oered regularly; consult department Subject meets with 18.783
3-0-9 units Prereq: 18.702, 18.703, or permission of instructor
Acad Year 2020-2021: G (Spring)
Covers representations of locally compact groups, with emphasis on Acad Year 2021-2022: Not oered
compact groups and abelian groups. Includes Peter-Weyl theorem 3-0-9 units
and Cartan-Weyl highest weight theory for compact Lie groups.
Information: R. Bezrukavnikov Computationally focused introduction to elliptic curves, with
applications to number theory and cryptography. Topics include
18.781 Theory of Numbers point-counting, isogenies, pairings, and the theory of complex
Prereq: None multiplication, with applications to integer factorization, primality
U (Spring) proving, and elliptic curve cryptography. Includes a brief introduction
3-0-9 units to modular curves and the proof of Fermat's Last Theorem. Students
in Course 18 must register for the undergraduate version, 18.783.
An elementary introduction to number theory with no algebraic A. Sutherland
prerequisites. Primes, congruences, quadratic reciprocity,
diophantine equations, irrational numbers, continued fractions, 18.784 Seminar in Number Theory
partitions. In person not required. Prereq: 18.701 or (18.703 and (18.06 or 18.700))
J-L Kim U (Fall)
3-0-9 units
18.782 Introduction to Arithmetic Geometry
Prereq: 18.702 Topics vary from year to year. Students present and discuss
Acad Year 2020-2021: U (Fall) the subject matter. Instruction and practice in written and oral
Acad Year 2021-2022: Not oered communication provided. In person not required. Enrollment limited.
3-0-9 units D. Kriz

Exposes students to arithmetic geometry, motivated by the problem 18.785 Number Theory I
of nding rational points on curves. Includes an introduction to p- Prereq: None. Coreq: 18.705
adic numbers and some fundamental results from number theory G (Fall)
and algebraic geometry, such as the Hasse-Minkowski theorem and 3-0-9 units
the Riemann-Roch theorem for curves. Additional topics may include
Mordell's theorem, the Weil conjectures, and Jacobian varieties. Dedekind domains, unique factorization of ideals, splitting of
D. Roe primes. Lattice methods, niteness of the class group, Dirichlet's
unit theorem. Local elds, ramication, discriminants. Zeta and
L-functions, analytic class number formula. Adeles and ideles.
Statements of class eld theory and the Chebotarev density
theorem.
W. Zhang

28   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.786 Number Theory II Topology and Geometry

Prereq: 18.785
G (Spring) 18.900 Geometry and Topology in the Plane
3-0-9 units Prereq: 18.03 or 18.06
Continuation of 18.785. More advanced topics in number theory, U (Fall)
such as Galois cohomology, proofs of class eld theory, modular 3-0-9 units
forms and automorphic forms, Galois representations, or quadratic Covers selected topics in geometry and topology, which can be
forms. visualized in the two-dimensional plane. Polygons and polygonal
W. Zhang paths. Billiards. Closed curves and immersed curves. Algebraic
curves. Triangulations and complexes. Hyperbolic geometry.
18.787 Topics in Number Theory Geodesics and curvature. Other topics may be included as time
Prereq: Permission of instructor permits.
Acad Year 2020-2021: Not oered P. Seidel
Acad Year 2021-2022: G (Fall)
3-0-9 units 18.901 Introduction to Topology
Can be repeated for credit. Subject meets with 18.9011
Topics vary from year to year. Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of
W. Zhang instructor
U (Fall, Spring)
3-0-9 units
Mathematics Laboratory
Introduces topology, covering topics fundamental to modern analysis
18.821 Project Laboratory in Mathematics and geometry. Topological spaces and continuous functions,
Prereq: Two mathematics subjects numbered 18.10 or above connectedness, compactness, separation axioms, covering spaces,
U (Fall, Spring) and the fundamental group.
3-6-3 units. Institute LAB Fall: I. Dai. Spring: G. Lusztig

Guided research in mathematics, employing the scientic 18.9011 Introduction to Topology

method. Students confront puzzling and complex mathematical Subject meets with 18.901
situations, through the acquisition of data by computer, pencil Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of
and paper, or physical experimentation, and attempt to explain instructor
them mathematically. Students choose three projects from a large G (Fall, Spring)
collection of options. Each project results in a laboratory report 3-0-9 units
subject to revision; oral presentation on one or two projects. Projects
drawn from many areas, including dynamical systems, number Introduces topology, covering topics fundamental to modern analysis
theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and geometry. Topological spaces and continuous functions,
and probability. In person not required. Enrollment limited. connectedness, compactness, separation axioms, covering spaces,
A. Negut, Z. Yun and the fundamental group. Students in Course 18 must register for
the undergraduate version, 18.901.
Fall: I. Dai. Spring: G. Lusztig

18.904 Seminar in Topology

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

Topics vary from year to year. Students present and discuss

the subject matter. Instruction and practice in written and oral
communication provided. In person not required. Enrollment limited.
L. Piccirillo

Department of Mathematics   |   29
DEPARTMENT OF MATHEMATICS

18.905 Algebraic Topology I 18.950 Dierential Geometry

Prereq: 18.901 and (18.701 or 18.703) Subject meets with 18.9501
G (Fall) Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
3-0-9 units 18.100Q)
Acad Year 2020-2021: Not oered
Singular homology, CW complexes, universal coecient and Künneth Acad Year 2021-2022: U (Spring)
theorems, cohomology, cup products, Poincaré duality. 3-0-9 units
J. Hahn
Introduction to dierential geometry, centered on notions of
18.906 Algebraic Topology II curvature. Starts with curves in the plane, and proceeds to higher
Prereq: 18.905 dimensional submanifolds. Computations in coordinate charts: rst
G (Spring) and second fundamental form, Christoel symbols. Discusses the
3-0-9 units distinction between extrinsic and intrinsic aspects, in particular
Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics.
Continues the introduction to Algebraic Topology from 18.905. Topics Examples such as hyperbolic space.
include basic homotopy theory, spectral sequences, characteristic T. Collins
classes, and cohomology operations. In person not required.
P. Seidel 18.9501 Dierential Geometry
Subject meets with 18.950
18.917 Topics in Algebraic Topology Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
Prereq: 18.906 18.100Q)
Acad Year 2020-2021: Not oered Acad Year 2020-2021: Not oered
Acad Year 2021-2022: G (Spring) Acad Year 2021-2022: G (Spring)
3-0-9 units 3-0-9 units
Can be repeated for credit.
Introduction to dierential geometry, centered on notions of
Content varies from year to year. Introduces new and signicant curvature. Starts with curves in the plane, and proceeds to higher
developments in algebraic topology with the focus on homotopy dimensional submanifolds. Computations in coordinate charts: rst
theory and related areas. and second fundamental form, Christoel symbols. Discusses the
Information: H. R. Miller distinction between extrinsic and intrinsic aspects, in particular
Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics.
18.919 Graduate Topology Seminar Examples such as hyperbolic space. Students in Course 18 must
Prereq: 18.906 register for the undergraduate version, 18.950.
G (Fall) T. Collins
3-0-9 units

Study and discussion of important original papers in the various 18.952 Theory of Dierential Forms
parts of algebraic topology. Open to all students who have taken Prereq: 18.101 and (18.700 or 18.701)
18.906 or the equivalent, not only prospective topologists. U (Spring)
H. R. Miller 3-0-9 units

Multilinear algebra: tensors and exterior forms. Dierential forms

18.937 Topics in Geometric Topology n
on R : exterior dierentiation, the pull-back operation and the
Prereq: Permission of instructor Poincaré lemma. Applications to physics: Maxwell's equations from
Acad Year 2020-2021: G (Spring) the dierential form perspective. Integration of forms on open sets
Acad Year 2021-2022: Not oered n
of R . The change of variables formula revisited. The degree of a
3-0-9 units dierentiable mapping. Dierential forms on manifolds and De Rham
Can be repeated for credit. theory. Integration of forms on manifolds and Stokes' theorem. The
Content varies from year to year. Introduces new and signicant push-forward operation for forms. Thom forms and intersection
developments in geometric topology. In person not required. theory. Applications to dierential topology.
T. S. Mrowka V. W. Guillemin

30   |   Department of Mathematics
 DEPARTMENT OF MATHEMATICS

18.965 Geometry of Manifolds I 18.994 Seminar in Geometry

Prereq: 18.101, 18.950, or 18.952 Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or
G (Fall) 18.100Q)
3-0-9 units Acad Year 2020-2021: Not oered
Acad Year 2021-2022: U (Fall)
Dierential forms, introduction to Lie groups, the DeRham theorem, 3-0-9 units
Riemannian manifolds, curvature, the Hodge theory. 18.966 is a
continuation of 18.965 and focuses more deeply on various aspects Students present and discuss subject matter taken from current
of the geometry of manifolds. Contents vary from year to year, and journals or books. Topics vary from year to year. Instruction and
can range from Riemannian geometry (curvature, holonomy) to practice in written and oral communication provided. Enrollment
symplectic geometry, complex geometry and Hodge-Kahler theory, or limited.
smooth manifold topology. Prior exposure to calculus on manifolds, Information: W. Minicozzi
as in 18.952, recommended. In person not required.
W. Minicozzi 18.999 Research in Mathematics
Prereq: Permission of instructor
18.966 Geometry of Manifolds II G (Fall, IAP, Spring, Summer)
Prereq: 18.965 Units arranged
G (Spring) Can be repeated for credit.
3-0-9 units
Opportunity for study of graduate-level topics in mathematics
Continuation of 18.965, focusing more deeply on various aspects under the supervision of a member of the department. For graduate
of the geometry of manifolds. Contents vary from year to year, and students desiring advanced work not provided in regular subjects.
can range from Riemannian geometry (curvature, holonomy) to Information: W. Minicozzi
symplectic geometry, complex geometry and Hodge-Kahler theory, or
smooth manifold topology. 18.UR Undergraduate Research
T. Colding Prereq: Permission of instructor
U (Fall, IAP, Spring, Summer)
18.968 Topics in Geometry Units arranged [P/D/F]
Prereq: 18.965 Can be repeated for credit.
Acad Year 2020-2021: Not oered
Acad Year 2021-2022: G (Spring) Undergraduate research opportunities in mathematics. Permission
3-0-9 units required in advance to register for this subject. For further
Can be repeated for credit. information, consult the departmental coordinator.
Information: W. Minicozzi
Content varies from year to year.
P. Seidel 18.THG Graduate Thesis
Prereq: Permission of instructor
18.979 Graduate Geometry Seminar G (Fall, IAP, Spring, Summer)
Prereq: Permission of instructor Units arranged
Acad Year 2020-2021: Not oered Can be repeated for credit.
Acad Year 2021-2022: G (Spring)
3-0-9 units Program of research leading to the writing of a Ph.D. thesis; to be
Can be repeated for credit. arranged by the student and an appropriate MIT faculty member.
Information: W. Minicozzi
Content varies from year to year. Study of classical papers in
geometry and in applications of analysis to geometry and topology.
T. Mrowka

Department of Mathematics   |   31
DEPARTMENT OF MATHEMATICS

18.S096 Special Subject in Mathematics 18.S995 Special Subject in Mathematics

Prereq: Permission of instructor Prereq: Permission of instructor
U (Fall, Spring) Acad Year 2020-2021: Not oered
Units arranged Acad Year 2021-2022: G (Fall)
Can be repeated for credit. Units arranged
Can be repeated for credit.
Opportunity for group study of subjects in mathematics not
otherwise included in the curriculum. Oerings are initiated by Opportunity for group study of advanced subjects in mathematics
members of the Mathematics faculty on an ad hoc basis, subject to not otherwise included in the curriculum. Oerings are initiated by
departmental approval. 18.S097 is graded P/D/F. members of the mathematics faculty on an ad hoc basis, subject to
Fall: A. Moitra, P. Parrilo.  Spring: H. Cohn departmental approval.
Sta
18.S097 Special Subject in Mathematics
Prereq: Permission of instructor 18.S996 Special Subject in Mathematics
U (IAP) Prereq: Permission of instructor
Units arranged [P/D/F] G (Fall)
Can be repeated for credit. Units arranged
Can be repeated for credit.
Opportunity for group study of subjects in mathematics not
otherwise included in the curriculum. Oerings are initiated by Opportunity for group study of advanced subjects in mathematics
members of the Mathematics faculty on an ad hoc basis, subject to not otherwise included in the curriculum. Oerings are initiated by
departmental approval. 18.S097 is graded P/D/F. members of the Mathematics faculty on an ad hoc basis, subject to
Information: W. Minicozzi Departmental approval.
Fall:  A. Harrow
18.S190 Special Subject in Mathematics
Prereq: Permission of instructor 18.S997 Special Subject in Mathematics
U (Fall, Spring; second half of term) Prereq: Permission of instructor
Units arranged Acad Year 2020-2021: Not oered
Can be repeated for credit. Acad Year 2021-2022: G (IAP)
Units arranged
Opportunity for group study of subjects in mathematics not Can be repeated for credit.
otherwise included in the curriculum. Oerings are initiated by
members of the Mathematics faculty on an ad hoc basis, subject to Opportunity for group study of advanced subjects in mathematics
departmental approval. not otherwise included in the curriculum. Oerings are initiated by
Sta members of the Mathematics faculty on an ad hoc basis, subject to
Departmental approval.
18.S191 Special Subject in Mathematics Sta
Prereq: Permission of instructor
U (Fall, Spring) 18.S998 Special Subject in Mathematics
Units arranged Prereq: Permission of instructor
Can be repeated for credit. Acad Year 2020-2021: Not oered
Acad Year 2021-2022: G (Spring)
Opportunity for group study of subjects in mathematics not Units arranged
otherwise included in the curriculum. Oerings are initiated by Can be repeated for credit.
members of the Mathematics faculty on an ad hoc basis, subject to
departmental approval. Opportunity for group study of advanced subjects in mathematics
Sta not otherwise included in the curriculum. Oerings are initiated by
members of the Mathematics faculty on an ad hoc basis, subject to
departmental approval.
Sta

32   |   Department of Mathematics