# Department of Mathematics

The Department of Mathematics offers training at the undergraduate, graduate, and postgraduate levels. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, and theoretical physics.

Course 18 includes two undergraduate degrees: a Bachelor of Science in Mathematics and a Bachelor of Science in Mathematics with Computer Science. Undergraduate students may choose one of three options leading to the Bachelor of Science in Mathematics: applied mathematics, pure mathematics, or general mathematics. The general mathematics option provides a great deal of flexibility and allows students to design their own programs in conjunction with their advisors. The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program.

At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics.

The CLE Moore instructorships and Applied Mathematics instructorships bring mathematicians at the postdoctoral level to MIT and provide them with training in research and teaching.

Bachelor of Science in Mathematics (Course 18)

Bachelor of Science in Mathematics with Computer Science (Course 18-C)

## Undergraduate Study

An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such fields as finance, business, or consulting. Students' programs are arranged through consultation with their faculty advisors.

Undergraduates in mathematics are encouraged to elect an undergraduate seminar during their junior or senior year. The experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students planning to pursue graduate work as well as for those going on to other careers. These seminars also provide training in the verbal and written communication of mathematics and may be used to fulfill the Communication Requirement.

Many mathematics majors take 18.821 Project Laboratory in Mathematics, which fulfills the Institute's Laboratory Requirement and counts toward the Communication Requirement.

### Bachelor of Science in Mathematics (Course 18)

#### General Mathematics Option

In addition to the General Institute Requirements, the requirements consist of Differential Equations, plus eight additional 12-unit subjects in Course 18 of essentially different content, including at least six advanced subjects (first decimal digit one or higher). One of these eight subjects must be Linear Algebra. This leaves available 84 units of unrestricted electives. The requirements are flexible in order to accommodate students who pursue programs that combine mathematics with a related field (such as physics, economics, or management) as well as students who are interested in both pure and applied mathematics. More details can be found on the degree chart.

#### Applied Mathematics Option

Applied mathematics focuses on the mathematical concepts and techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically enroll in 18.200 Principles of Discrete Applied Mathematics and 18.300 Principles of Continuum Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of applied mathematics and may be taken concurrently with 18.03 Differential Equations. Subject 18.300, offered in the spring term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

More detail on the Applied Mathematics option can be found on the degree chart.

#### Pure Mathematics Option

Pure (or "theoretical") mathematics is the study of the basic concepts and structure of mathematics. Its goal is to arrive at a deeper understanding and an expanded knowledge of mathematics itself.

Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.

The subjects 18.701 Algebra I and 18.901 Introduction to Topology are more advanced and should not be elected until a student has had experience with proofs, as in Real Analysis (18.100A, 18.100B, 18.100P or 18.100Q) or 18.700 Linear Algebra.

For more details, see the degree chart.

### Bachelor of Science in Mathematics with Computer Science (Course 18-C)

Mathematics and computer science are closely related fields. Problems in computer science are often formalized and solved with mathematical methods. It is likely that many important problems currently facing computer scientists will be solved by researchers skilled in algebra, analysis, combinatorics, logic and/or probability theory, as well as computer science.

The purpose of this program is to allow students to study a combination of these mathematical areas and potential areas of application in computer science. Required subjects include linear algebra (18.06 or 18.700) because it is so broadly used, and discrete mathematics (18.062[J] or 18.200) to give experience with proofs and the necessary tools for analyzing algorithms. The required subjects covering complexity (18.404 Theory of Computation or 18.400[J] Automata, Computability, and Complexity) and algorithms (18.410[J] Design and Analysis of Algorithms) provide an introduction to the most theoretical aspects of computer science. We also require exposure to other areas of computer science (6.031, 6.033, 6.034, or 6.036) where mathematical issues may also arise. More details can be found on the degree chart.

Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject 18.701 Algebra I for 18.06 Linear Algebra, and, if they already have strong theorem-proving skills, may substitute 18.211 Combinatorial Analysis or 18.212 Algebraic Combinatorics for 18.062[J] Mathematics for Computer Science or 18.200 Principles of Discrete Applied Mathematics.

### Minor in Mathematics

The requirements for a Minor in Mathematics are as follows: six 12-unit subjects in mathematics, beyond the Institute's Mathematics Requirement, of essentially different content, including at least three advanced subjects (first decimal digit one or higher).

See the Undergraduate Section for a general description of the minor program.

### Inquiries

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

## Graduate Study

The Mathematics Department offers programs covering a broad range of topics leading to the Doctor of Philosophy or Doctor of Science degree. Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120-130 doctoral students, about two thirds are in Pure Mathematics, one third in Applied Mathematics.

The programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, theoretical physics, and the theory of computation. In addition, many mathematically oriented subjects are offered by other departments. Students in Applied Mathematics are especially encouraged to take subjects in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

### Doctor of Philosophy or Doctor of Science

The requirements for these degrees are described on the department's website. In outline, they consist of a language requirement, an oral qualifying examination, a thesis proposal, completion of a minimum of 132 units (11 graduate subjects), and a thesis containing original research in mathematics.

### Interdisciplinary Programs

#### Computational Science and Engineering

Students with primary interest in computational science may also consider applying to the interdisciplinary Computational Science and Engineering (CSE) program, with which the Mathematics Department is affiliated. For more information, see the CSE website.

### Financial Support

Financial support is guaranteed for up to five years to students making satisfactory academic progress. Financial aid after the first year is usually in the form of a teaching or research assistantship.

### Inquiries

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

## Faculty and Teaching Staff

Michel X. Goemans, PhD

Professor of Applied Mathematics

Interim Head, Department of Mathematics

John W. M. Bush, PhD

Professor of Mathematics

Associate Head, Department of Mathematics

### Professors

Michael Artin, PhD

Professor of Mathematics

Martin Z. Bazant, PhD

Professor of Chemical Engineering

Professor of Mathematics

Executive Officer, Department of Chemical Engineering

Bonnie Berger, PhD

Simons Professor of Mathematics

Professor of Computer Science

Member, Health Sciences and Technology Faculty

(On leave, fall)

Roman Bezrukavnikov, PhD

Professor of Mathematics

(On leave, spring)

Alexei Borodin, PhD

Professor of Mathematics

Hung Cheng, PhD

Professor of Mathematics

Tobias Colding, PhD

Cecil and Ida Green Distinguished Professor

Professor of Mathematics

(On leave, spring)

Alan Edelman, PhD

Professor of Mathematics

Pavel I. Etingof, PhD

Professor of Mathematics

Victor W. Guillemin, PhD

Professor of Mathematics

Lawrence Guth, PhD

Professor of Mathematics

Anette E. Hosoi, PhD

Neil and Jane Pappalardo Professor

Professor of Mechanical Engineering

Professor of Mathematics

Associate Dean, School of Engineering

David S. Jerison, PhD

Professor of Mathematics

(On leave, spring)

Steven G. Johnson, PhD

Professor of Mathematics

Professor of Physics

Victor Kac, PhD

Professor of Mathematics

Ju-Lee Kim, PhD

Professor of Mathematics

(On leave)

Frank Thomson Leighton, PhD

Professor of Mathematics

George Lusztig, PhD

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

Arthur P. Mattuck, PhD

Professor of Mathematics

Davesh Maulik, PhD

Professor of Mathematics

Richard B. Melrose, PhD

Simons Professor of Mathematics

Haynes R. Miller, PhD

Professor of Mathematics

William Minicozzi, PhD

Singer Professor of Mathematics

Elchanan Mossel, PhD

Professor of Mathematics

Member, Institute for Data, Systems, and Society

Tomasz S. Mrowka, PhD

Professor of Mathematics

(On leave)

Bjorn Poonen, PhD

Claude E. Shannon (1940) Professor of Mathematics

Alexander Postnikov, PhD

Professor of Mathematics

Rodolfo R. Rosales, PhD

Professor of Mathematics

Paul Seidel, PhD

Levinson Professor in Mathematics

(On leave)

Scott Roger Sheffield, PhD

Leighton Family Professor of Mathematics

Member, Institute for Data, Systems, and Society

Peter W. Shor, PhD

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

Professor of Mathematics

Michael Sipser, PhD

Donner Professor of Mathematics

Dean, School of Science

Gigliola Staffilani, PhD

Abby Rockefeller Mauzé Professor of Mathematics

(On leave)

Richard P. Stanley, PhD

Professor of Mathematics

(On leave, spring)

Gilbert Strang, PhD

MathWorks Professor of Mathematics

Daniel W. Stroock, PhD

Professor of Mathematics

David A. Vogan, PhD

Norbert Wiener Professor of Mathematics

Chenyang Xu, PhD

Professor of Mathematics

Wei Zhang, PhD

Professor of Mathematics

### Associate Professors

Laurent Demanet, PhD

Associate Professor of Mathematics

Jonathan Adam Kelner, PhD

Mark Hyman, Jr. Career Development Professor

Associate Professor of Mathematics

Philippe Rigollet, PhD

Associate Professor of Mathematics

Member, Institute for Data, Systems, and Society

Jared R. Speck, PhD

Cecil and Ida Green Career Development Professor

Associate Professor of Mathematics

Gonçalo Jorge Trigo Neri Tabuada, PhD

Associate Professor of Mathematics

### Assistant Professors

Joern Dunkel, PhD

Assistant Professor of Mathematics

Semyon Dyatlov, PhD

Assistant Professor of Mathematics

(On leave)

Vadim Gorin, PhD

Assistant Professor of Mathematics

Andrew Lawrie, PhD

Assistant Professor of Mathematics

Ankur Moitra, PhD

Rockwell International Career Development Professor

Assistant Professor of Mathematics

Andrei Negut, PhD

Assistant Professor of Mathematics

(On leave, spring)

Aaron Pixton, PhD

Assistant Professor of Mathematics

(On leave, spring)

Giulia Sacca, PhD

Assistant Professor of Mathematics

Yufei Zhao, PhD

Assistant Professor of Mathematics

### Visiting Professors

Ivan Loseu, PhD

Visiting Professor of Mathematics

(Fall)

Glenn Stevens, PhD

Visiting Professor of Mathematics

### Visiting Associate Professors

Pierre Albin, PhD

Visiting Associate Professor of Mathematics

(Fall)

Tomoyuki Arakawa, PhD

Visiting Associate Professor of Mathematics

Kay Kirkpatrick, PhD

Visiting Associate Professor of Mathematics

(Fall)

### Visiting Assistant Professors

Leonid Petrov, PhD

Visiting Assistant Professor of Mathematics

### Adjunct Professors

Henry Cohn, PhD

Adjunct Professor of Mathematics

### Lecturers

Joel Benjamin Geiger, PhD

Lecturer in Mathematics

Slava Gerovitch, PhD, PhD

Lecturer in Mathematics

Peter J. Kempthorne, PhD

Lecturer in Mathematics

Tanya Khovanova, PhD

Lecturer in Mathematics

### CLE Moore Instructors

Thomas David Beck, PhD

CLE Moore Instructor of Mathematics

Stéphane Benoist, PhD

CLE Moore Instructor of Mathematics

Alexey Bufetov, PhD

CLE Moore Instructor of Mathematics

Kyeongsu Choi, PhD

CLE Moore Instructor of Mathematics

Max D. Engelstein, PhD

CLE Moore Instructor of Mathematics

Jianfeng Lin, PhD

CLE Moore Instructor of Mathematics

Heather R. Macbeth, PhD

CLE Moore Instructor of Mathematics

Christos Mantoulidis, PhD

CLE Moore Instructor of Mathematics

Cris Negron, PhD

CLE Moore Instructor of Mathematics

Georg Oberdieck, PhD

CLE Moore Instructor of Mathematics

(On sabbatical)

Yumeng Ou, PhD

CLE Moore Instructor of Mathematics

Yu Pan, PhD

CLE Moore Instructor of Mathematics

Druv Ranganathan, PhD

CLE Moore Instructor of Mathematics

Ananth Shankar, PhD

CLE Moore Instructor of Mathematics

Luca Spolaor, PhD

CLE Moore Instructor of Mathematics

(On leave)

Bobby L. E. Wilson, PhD

CLE Moore Instructor of Mathematics

Zhouli Xu, PhD

CLE Moore Instructor of Mathematics

### Instructors of Applied Mathematics

Victor-Emmanuel Brunel, PhD

Instructor of Applied Mathematics

Yash Kiran Deshpande, PhD

Instructor of Applied Mathematics

Asaf Ferber, PhD

Instructor of Applied Mathematics

Luiz Maltez Faria, PhD

Instructor of Applied Mathematics

Thomas McConville, PhD

Instructor of Applied Mathematics

Philip Pearce, PhD

Instructor of Applied Mathematics

Carlos Pérez-Arancibia, PhD

Instructor of Applied Mathematics

Mustazee Rahman, PhD

Instructor of Applied Mathematics

Andrej Risteski, PhD

Instructor of Applied Mathematics

Elina Miaylova Robeva, PhD

Instructor of Applied Mathematics

Henrik Ronellenfitsch, PhD

Instructor of Applied Mathematics

Pedro Sáenz Hervías, PhD

Instructor of Applied Mathematics

Wonseok Shin, PhD

Instructor of Applied Mathematics

Stuart Thomson, PhD

Instructor of Applied Mathematics

### Instructors of Pure Mathematics

Nicholas Edelen, PhD

Instructor of Pure Mathematics

Yevgeny Liokumovich, PhD

Instructor of Pure Mathematics

Kyler B. Siegel, PhD

Instructor of Pure Mathematics

(On leave)

## Research Staff

### Principal Research Scientists

Andrew Victor Sutherland II, PhD

Principal Research Scientist of Mathematics

### Research Scientists

David I. Spivak, PhD

Research Scientist of Mathematics

## Professors Emeriti

Richard Dudley, PhD

Professor Emeritus of Mathematics

Daniel Z. Freedman, PhD

Professor Emeritus of Mathematics

Professor Emeritus of Physics

Harvey P. Greenspan, PhD

Professor Emeritus of Mathematics

Sigurdur Helgason, PhD

Professor Emeritus of Mathematics

Steven L. Kleiman, PhD

Professor Emeritus of Mathematics

Daniel J. Kleitman, PhD

Professor Emeritus of Mathematics

James R. Munkres, PhD

Professor Emeritus of Mathematics

Gerald E. Sacks, PhD

Professor Emeritus of Mathematics

Isadore Manuel Singer, PhD

Institute Professor Emeritus

Professor Emeritus of Mathematics

Harold Stark, PhD

Professor Emeritus of Mathematics

Alar Toomre, PhD

Professor Emeritus of Mathematics

## General Mathematics

**18.01 Calculus**

Prereq: None

U (Fall, Spring)

5-0-7 units. CALC I

Credit cannot also be received for 18.01A, CC.181A, ES.1801, ES.181A

Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions.

*Fall: J. Speck
**Spring: Information: J. W. Bush
*

**18.01A Calculus**

Prereq: Knowledge of differentiation and elementary integration

U (Fall; first half of term)

5-0-7 units. CALC I

Credit cannot also be received for 18.01, CC.181A, ES.1801, ES.181A

Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Prerequisites: one year of high-school calculus or the equivalent, with a score of 4 or 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the 18.01 advanced standing exam.

*D. Maulik
*

**18.02 Calculus**

Prereq: Calculus I (GIR)

U (Fall, Spring)

5-0-7 units. CALC II

Credit cannot also be received for 18.022, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A

Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.

*Fall: J. W. Bush
**Spring: L. Guth
*

**18.02A Calculus**

Prereq: Calculus I (GIR)

U (Fall, IAP, Spring)

5-0-7 units. CALC II

Credit cannot also be received for 18.02, 18.022, CC.1802, CC.182A, ES.1802, ES.182A

First half is taught during the last six weeks of the Fall term; covers material in the first half of 18.02 (through double integrals). Second half of 18.02A can be taken either during IAP (daily lectures) or during the second half of the Spring term; it covers the remaining material in 18.02.

*T. Beck
*

**18.022 Calculus**

Prereq: Calculus I (GIR)

U (Fall)

5-0-7 units. CALC II

Credit cannot also be received for 18.02, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A

Calculus of several variables. Topics as in 18.02 but with more focus on mathematical concepts. Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions.

*P. I. Etingof
*

**18.03 Differential Equations**

Prereq: None. *Coreq: Calculus II (GIR)*

U (Fall, Spring)

5-0-7 units. REST

Credit cannot also be received for 18.032, CC.1803, ES.1803

Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams.

*Fall: A. Negut
**Spring: B. Poonen
*

**18.031 System Functions and the Laplace Transform**

Prereq: None. *Coreq: 18.03*

U (Fall, Spring; second half of term)

1-0-2 units

Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering.

*P. Pearce
*

**18.032 Differential Equations (18.034)**

Prereq: None. *Coreq: Calculus II (GIR)*

U (Spring)

5-0-7 units. REST

Credit cannot also be received for 18.03, CC.1803, ES.1803

Covers much of the same material as 18.03 with more emphasis on theory. The point of view is rigorous and results are proven. Local existence and uniqueness of solutions.

*Information: J. W. Bush
*

**18.04 Complex Variables with Applications**

Prereq: Calculus II (GIR); 18.03 or 18.032

U (Spring)

4-0-8 units

Credit cannot also be received for 18.075, 18.0751

Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations.

*Information: J. W. Bush
*

**18.05 Introduction to Probability and Statistics**

Prereq: Calculus II (GIR)

U (Spring)

4-0-8 units. REST

Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.

*Information: J. W. Bush
*

**18.06 Linear Algebra**

Prereq: Calculus II (GIR)

U (Fall, Spring)

4-0-8 units. REST

Credit cannot also be received for 18.700

Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB. Compared with 18.700, more emphasis on matrix algorithms and many applications.

*Fall: S. G. Johnson
**Spring: A. Edelman
*

**18.062[J] Mathematics for Computer Science**

Same subject as 6.042[J]

Prereq: Calculus I (GIR)

U (Fall, Spring)

5-0-7 units. REST

See description under subject 6.042[J].

*F. T. Leighton, A. R. Meyer, A. Moitra
*

**18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning**

Subject meets with 18.0651

Prereq: 18.06

U (Spring)

3-0-9 units

Reviews linear algebra with applications to life sciences, finance, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and hidden Markov models.

*G. Strang
*

**18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning**

Subject meets with 18.065

Prereq: 18.06

G (Spring)

3-0-9 units

Reviews linear algebra with applications to life sciences, finance, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and hidden Markov models. Students in Course 18 must register for the undergraduate version, 18.065.

*G. Strang
*

**18.075 Methods for Scientists and Engineers**

Subject meets with 18.0751

Prereq: Calculus II (GIR); 18.03

U (Spring)

3-0-9 units

Credit cannot also be received for 18.04

Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations.

*H. Cheng
*

**18.0751 Methods for Scientists and Engineers**

Subject meets with 18.075

Prereq: Calculus II (GIR); 18.03

G (Spring)

3-0-9 units

Credit cannot also be received for 18.04

Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations. Students in Courses 6, 8, 12, 18, and 22 must register for undergraduate version, 18.075.

*H. Cheng
*

**18.085 Computational Science and Engineering I**

Subject meets with 18.0851

Prereq: Calculus II (GIR); 18.03 or 18.032

U (Fall, Spring, Summer)

3-0-9 units

Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.

*Fall: G. Strang
**Spring: P. Saenz
*

**18.0851 Computational Science and Engineering I**

Subject meets with 18.085

Prereq: Calculus II (GIR); 18.03 or 18.032

G (Fall, Spring, Summer)

3-0-9 units

Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications. Students in Course 18 must register for the undergraduate version, 18.085.

*Fall: G. Strang
**Spring: P. Saenz
*

**18.086 Computational Science and Engineering II**

Subject meets with 18.0861

Prereq: Calculus II (GIR); 18.03 or 18.032

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: U (Spring)

3-0-9 units

Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods.

*Information: G. Strang
*

**18.0861 Computational Science and Engineering II**

Subject meets with 18.086

Prereq: Calculus II (GIR); 18.03 or 18.032

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods. Students in Course 18 must register for the undergraduate version, 18.086.

*Information: G. Strang
*

**18.089 Review of Mathematics**

Prereq: Permission of instructor

G (Summer)

5-0-7 units

One-week review of one-variable calculus (18.01), followed by concentrated study covering multivariable calculus (18.02), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances.

*Information: J. W. Bush
*

**18.094[J] Teaching College-Level Science and Engineering**

Same subject as 1.95[J], 5.95[J], 7.59[J], 8.395[J]

Subject meets with 2.978

Prereq: None

G (Fall)

2-0-2 units

See description under subject 5.95[J].

*J. Rankin
*

**18.095 Mathematics Lecture Series**

Prereq: Calculus I (GIR)

U (IAP)

2-0-4 units

Can be repeated for credit.

Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session.

*Information: J. W. Bush
*

**18.098 Internship in Mathematics**

Prereq: Permission of instructor

U (Fall, IAP, Spring, Summer)

Units arranged [P/D/F]

Can be repeated for credit.

Provides academic credit for students pursuing internships to gain practical experience in the applications of mathematical concepts and methods.

*Information: J. W. Bush
*

**18.099 Independent Study**

Prereq: Permission of instructor

U (Fall, IAP, Spring, Summer)

Units arranged

Can be repeated for credit.

Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval.

*Information: J. W. Bush
*

## Analysis

**18.1001 Real Analysis**

Subject meets with 18.100A

Prereq: Calculus II (GIR)

G (Fall, Spring)

3-0-9 units

Credit cannot also be received for 18.100B, 18.100P, 18.100Q

Four options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. Option A: Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. Option B: More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Option P: 15-unit (4-0-11) variant of Option A, with further instruction and practice in written communication. Option Q: 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication. Students in Course 18 must register for one of the undergraduate versions of this subject: 18.100A, 18.100B, 18.100P, or 18.100Q.

*Fall: A. P. Mattuck
**Spring: H. Macbeth
*

**18.1002 Real Analysis**

Subject meets with 18.100B

Prereq: Calculus II (GIR)

G (Fall, Spring)

3-0-9 units

Credit cannot also be received for 18.100A, 18.100Q

Four options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. Option A: Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. Option B: More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Option P: 15-unit (4-0-11) variant of Option A, with further instruction and practice in written communication. Option Q: 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication. Students in Course 18 must register for one of the undergraduate versions of this subject: 18.100A, 18.100B, 18.100P, or 18.100Q.

*Fall: D. Jerison
**Spring: Information: R. B. Melrose
*

**18.100A Real Analysis**

Subject meets with 18.1001

Prereq: Calculus II (GIR)

U (Fall, Spring)

3-0-9 units

Credit cannot also be received for 18.100B, 18.100P, 18.100Q

Four options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. *Option A:* Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. *Option B:* More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. *Option P:* 15-unit (4-0-11) variant of Option A, with further instruction and practice in written communication. *Option Q:* 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication.

*Fall: A. P. Mattuck (18.100A), D. Jerison (18.100B), V. Gorin (18.100Q)
**Spring: H. Macbeth (18.100A), J. Speck (18.100B), information: R.B. Melrose (18.100P)
*

**18.100B Real Analysis**

Subject meets with 18.1002

Prereq: Calculus II (GIR)

U (Fall, Spring)

3-0-9 units

Credit cannot also be received for 18.100A, 18.100Q

Three options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchangeFour options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. *Option A:* Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. *Option B:* More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. *Option P:* 15-unit (4-0-11) variant of Option A, with further instruction and practice in written communication. *Option Q:* 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication.

*Fall: A. P. Mattuck (18.100A), D. Jerison (18.100B), V. Gorin (18.100Q)
**Spring: H. Macbeth (18.100A), information: R.B. Melrose (18.100B), information: R.B. Melrose (18.100P)
*

**18.100P Real Analysis**

Prereq: Calculus II (GIR)

U (Spring)

4-0-11 units

Credit cannot also be received for 18.1001, 18.100A, 18.100B, 18.100Q

Four options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. *Option A:* Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. *Option B:* More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. *Option P:* 15-unit (4-0-11) variant of Option A, with further instruction and practice in written communication. *Option Q:* 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication.
Enrollment limited in Options P and Q.

*Information: R. B. Melrose
*

**18.100Q Real Analysis**

Prereq: Calculus II (GIR)

U (Fall)

4-0-11 units

Credit cannot also be received for 18.1001, 18.1002, 18.100A, 18.100B, 18.100P

Four options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. *Option A:* Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. *Option B:* More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. *Option P:* 15-unit (4-0-11) variant of Option A, with further instruction and practice in written communication. *Option Q:* 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication.
Enrollment limited in Options P and Q.

*Fall: 18.100A: A. P. Mattuck
18.100B: D. Jerison
18.100Q: V. Gorin
*

*Spring: 18.100A: H. Macbeth*

18.100B: Information: R.B. Melrose

18.100P: Information: R.B. Melrose

18.100B: Information: R.B. Melrose

18.100P: Information: R.B. Melrose

**18.101 Analysis and Manifolds**

Subject meets with 18.1011

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

U (Fall)

3-0-9 units

Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.901 helpful but not required.

*V. W. Guillemin
*

**18.1011 Analysis and Manifolds**

Subject meets with 18.101

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

G (Fall)

3-0-9 units

Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.9011 helpful but not required. Students in Course 18 must register for the undergraduate version, 18.101.

*V. W. Guillemin
*

**18.102 Introduction to Functional Analysis**

Subject meets with 18.1021

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

U (Spring)

3-0-9 units

Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem.

*R. B. Melrose
*

**18.1021 Introduction to Functional Analysis**

Subject meets with 18.102

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

G (Spring)

3-0-9 units

Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem. Students in Course 18 must register for the undergraduate version, 18.102.

*R. B. Melrose
*

**18.103 Fourier Analysis: Theory and Applications**

Subject meets with 18.1031

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

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
*

**18.1031 Fourier Analysis: Theory and Applications**

Subject meets with 18.103

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

G (Fall)

3-0-9 units

Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. Students in Course 18 must register for the undergraduate version, 18.103.

*A. Lawrie
*

**18.104 Seminar in Analysis**

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q

U (Spring)

3-0-9 units

Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

*Y. Ou
*

**18.112 Functions of a Complex Variable**

Subject meets with 18.1121

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

U (Fall)

3-0-9 units

Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.

*A. Borodin
*

**18.1121 Functions of a Complex Variable**

Subject meets with 18.112

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

G (Fall)

3-0-9 units

Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions. Students in Course 18 must register for the undergraduate version, 18.112.

*A. Borodin
*

**18.116 Riemann Surfaces**

Prereq: 18.112

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory.

*Information: R. B. Melrose
*

**18.117 Topics in Several Complex Variables**

Prereq: 18.112, 18.965

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Can be repeated for credit.

Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds.

*B. Poonen
*

**18.118 Topics in Analysis (New)**

Prereq: Permission of instructor.

Acad Year 2017-2018: G (Fall)

Acad Year 2018-2019: Not offered

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*L. Guth
*

**18.125 Measure Theory and Analysis**

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q

G (Spring)

3-0-9 units

Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.

*D. W. Stroock
*

**18.137 Topics in Geometric Partial Differential Equations**

Prereq: Permission of Instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*Information: R. B. Melrose
*

**18.152 Introduction to Partial Differential Equations**

Subject meets with 18.1521

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

U (Spring)

3-0-9 units

Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems.

*J. Speck
*

**18.1521 Introduction to Partial Differential Equations**

Subject meets with 18.152

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

G (Spring)

3-0-9 units

Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems. Students in Course 18 must register for the undergraduate version, 18.152.

*J. Speck
*

**18.155 Differential Analysis I**

Prereq: 18.102 or 18.103

G (Fall)

3-0-9 units

First part of a two-subject sequence. Review of Lebesgue integration. Lp spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Recommended prerequisite: 18.112.

*R. B. Melrose
*

**18.156 Differential Analysis II**

Prereq: 18.155

G (Spring)

3-0-9 units

Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations.

*A. Lawrie
*

**18.157 Introduction to Microlocal Analysis**

Prereq: 18.155

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

The semi-classical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semi-classical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject.

*V. W. Guillemin
*

**18.158 Topics in Differential Equations**

Prereq: 18.157

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*G. Staffilani
*

**18.175 Theory of Probability**

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q

G (Spring)

3-0-9 units

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

*V. Gorin
*

**18.176 Stochastic Calculus**

Prereq: 18.175

G (Spring)

3-0-9 units

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

*S. Benoist
*

**18.177 Topics in Stochastic Processes**

Prereq: 18.175

G (Fall, Spring)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*Fall: S. Sheffield
**Spring: A. Borodin
*

**18.199 Graduate Analysis Seminar**

Prereq: Permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Can be repeated for credit.

Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance.

*V. W. Guillemin
*

## Discrete Applied Mathematics

**18.200 Principles of Discrete Applied Mathematics**

Prereq: None. *Coreq: 18.06*

U (Spring)

4-0-11 units

Credit cannot also be received for 18.200A

Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.

*M. X. Goemans, A. Moitra
*

**18.200A Principles of Discrete Applied Mathematics**

Prereq: None. *Coreq: 18.06*

U (Fall)

3-0-9 units

Credit cannot also be received for 18.200

Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming.

*A. Risteski
*

**18.204 Undergraduate Seminar in Discrete Mathematics**

Prereq: 18.200 or 18.062[J]; 18.06, 18.700, or 18.701; or permission of instructor

U (Fall, Spring)

3-0-9 units

Seminar in combinatorics, graph theory, and discrete mathematics in general. Participants read and present papers from recent mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited.

*Fall: A. Ferber
**Spring: M. Rahman
*

**18.211 Combinatorial Analysis**

Prereq: Calculus II (GIR); 18.06, 18.700, or 18.701

U (Fall)

3-0-9 units

Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful.

*M. Rahman
*

**18.212 Algebraic Combinatorics**

Prereq: 18.701 or 18.703

U (Spring)

3-0-9 units

Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.

*Information: R. P. Stanley
*

**18.217 Combinatorial Theory**

Prereq: Permission of instructor

G (Fall)

3-0-9 units

Can be repeated for credit.

Content varies from year to year.

*R. P. Stanley
*

**18.218 Topics in Combinatorics**

Prereq: Permission of instructor

G (Spring)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*A. Postnikov
*

**18.219 Seminar in Combinatorics**

Prereq: Permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Can be repeated for credit.

Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class.

*Information: R. P. Stanley
*

## Continuous Applied Mathematics

**18.300 Principles of Continuum Applied Mathematics**

Prereq: Calculus II (GIR); 18.03 or 18.032

U (Spring)

3-0-9 units

Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment.

*L. Faria
*

**18.303 Linear Partial Differential Equations: Analysis and Numerics**

Prereq: 18.06 or 18.700

U (Spring)

3-0-9 units

Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science and engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. Some programming required for homework and final project.

*C. Perez-Arancibia
*

**18.305 Advanced Analytic Methods in Science and Engineering**

Prereq: 18.04, 18.075, or 18.112

G (Fall)

3-0-9 units

Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques; WKB method on partial differential equations.

*H. Cheng
*

**18.306 Advanced Partial Differential Equations with Applications**

Prereq: 18.03 or 18.032; 18.04, 18.075, or 18.112

G (Spring)

3-0-9 units

Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc.

*R. R. Rosales
*

**18.327 Topics in Applied Mathematics**

Prereq: Permission of instructor

Acad Year 2017-2018: G (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*L. Demanet
*

**18.330 Introduction to Numerical Analysis**

Prereq: Calculus II (GIR); 18.03 or 18.032

U (Fall)

3-0-9 units

Basic techniques for the efficient numerical solution of problems in science and engineering. Root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra. Knowledge of programming in Fortran, C, or MATLAB helpful.

*M. Taus
*

**18.335[J] Introduction to Numerical Methods**

Same subject as 6.337[J]

Prereq: 18.06, 18.700, or 18.701

G (Spring)

3-0-9 units

Advanced introduction to numerical analysis. Surveys major topics that arise at various levels of solving classic numerical problems, such as systems of linear equations, eigenvalue equations, and least squares problems. Specific topics include matrix factorizations (QR, SVD, LU, Cholesky); direct and iterative methods to solve linear systems (Gaussian elimination, Krylov subspace methods); numerical algorithms to solve eigenvalue equations (Rayleigh quotient iteration, inverse iteration, QR algorithm); conditioning of problems and stability of algorithms; and floating point arithmetic.

*W. Shin
*

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

Same subject as 6.335[J]

Prereq: 6.336[J], 16.920[J], 18.085, 18.335[J], or permission of instructor

G (Fall)

3-0-9 units

Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.

*C. Perez-Arancibia
*

**18.337[J] Numerical Computing and Interactive Software**

Same subject as 6.338[J]

Prereq: 18.06, 18.700, or 18.701

G (Fall)

3-0-9 units

Interdisciplinary introduction to computing with Julia. Covers scientific computing and data analysis problems. Combines knowledge from computer science and computational science illustrating Julia's approach to scientific computing. Sample scientific computing topics include dense and sparse linear algebra, Fourier transforms, data handling, machine learning, and N-body problems. Provides direct experience with the modern realities of programming supercomputers, GPUs, and multicores in a high-level language.

*A. Edelman
*

**18.338 Eigenvalues of Random Matrices**

Prereq: 18.701 or permission of instructor

Acad Year 2017-2018: G (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*A. Edelman
*

**18.352[J] Nonlinear Dynamics: The Natural Environment**

Same subject as 12.009[J]

Prereq: Physics I (GIR), Calculus II (GIR); *Coreq: 18.03*

U (Spring)

3-0-9 units

See description under subject 12.009[J].

*D. H. Rothman
*

**18.353[J] Nonlinear Dynamics: Chaos**

Same subject as 2.050[J], 12.006[J]

Prereq: 18.03 or 18.032; Physics II (GIR)

U (Fall)

3-0-9 units

See description under subject 12.006[J].

*P-T. Brun
*

**18.354[J] Nonlinear Dynamics: Continuum Systems**

Same subject as 1.062[J], 12.207[J]

Subject meets with 18.3541

Prereq: 18.03 or 18.032; Physics II (GIR)

U (Spring)

3-0-9 units

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

*L. Bourouiba
*

**18.3541 Nonlinear Dynamics: Continuum Systems**

Subject meets with 1.062[J], 12.207[J], 18.354[J]

Prereq: 18.03 or 18.032; Physics II (GIR)

G (Spring)

3-0-9 units

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

*L. Bourouiba
*

**18.355 Fluid Mechanics**

Prereq: 18.354[J], 2.25, or 12.800

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

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

*J. W. Bush
*

**18.357 Interfacial Phenomena**

Prereq: 18.354[J], 18.355, 12.800, 2.25, or permission of instructor

Acad Year 2017-2018: G (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*J. W. Bush
*

**18.358[J] Nonlinear Dynamics and Turbulence**

Same subject as 1.686[J], 2.033[J]

Subject meets with 1.068

Prereq: Permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

See description under subject 1.686[J].

*L. Bourouiba
*

**18.367 Waves and Imaging (New)**

Prereq: Permission of instructor

Acad Year 2017-2018: G (Fall)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*L. Demanet
*

**18.369[J] Mathematical Methods in Nanophotonics**

Same subject as 8.315[J]

Prereq: 18.305 or permission of instructor

Acad Year 2017-2018: G (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*S. G. Johnson
*

**18.376[J] Wave Propagation**

Same subject as 1.138[J], 2.062[J]

Prereq: 2.003[J], 18.075

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

See description under subject 2.062[J].

*T. R. Akylas, R. R. Rosales
*

**18.377[J] Nonlinear Dynamics and Waves**

Same subject as 1.685[J], 2.034[J]

Prereq: Permission of instructor

Acad Year 2017-2018: G (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*T. R. Akylas
*

**18.384 Undergraduate Seminar in Physical Mathematics**

Prereq: 18.300, 18.353[J], 18.354[J], or permission of instructor

U (Fall)

3-0-9 units

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

*S. Thomson
*

**18.385[J] Nonlinear Dynamics and Chaos**

Same subject as 2.036[J]

Prereq: 18.03 or 18.032

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

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

*R. R. Rosales
*

**18.397 Mathematical Methods in Physics (18.276)**

Prereq: 18.745 or some familiarity with Lie theory

Acad Year 2017-2018: G (Fall)

Acad Year 2018-2019: Not offered

3-0-9 units

Can be repeated for credit.

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

*V. G. Kac
*

## Theoretical Computer Science

**18.400[J] Automata, Computability, and Complexity**

Same subject as 6.045[J]

Prereq: 6.042[J]

U (Spring)

4-0-8 units

See description under subject 6.045[J].

*R. Williams
*

**18.404 Theory of Computation**

Subject meets with 6.840[J], 18.4041[J]

Prereq: 18.200 or 18.062[J]

U (Fall)

4-0-8 units

A more extensive and theoretical treatment of the material in 6.045[J]/18.400[J], emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.

*M. Sipser
*

**18.4041[J] Theory of Computation**

Same subject as 6.840[J]

Subject meets with 18.404

Prereq: 18.200 or 18.062[J]

G (Fall)

4-0-8 units

A more extensive and theoretical treatment of the material in 6.045[J]/18.400[J], emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Students in Course 18 must register for the undergraduate version, 18.404.

*M. Sipser
*

**18.405[J] Advanced Complexity Theory**

Same subject as 6.841[J]

Prereq: 18.404

Acad Year 2017-2018: G (Fall)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*R. Williams
*

**18.408 Topics in Theoretical Computer Science**

Prereq: Permission of instructor

G (Fall, Spring)

3-0-9 units

Can be repeated for credit.

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

*Fall: A. Moitra
**Spring: J. A. Kelner
*

**18.410[J] Design and Analysis of Algorithms**

Same subject as 6.046[J]

Prereq: 6.006

U (Fall, Spring)

4-0-8 units

See description under subject 6.046[J].

*E. Demaine, M. Goemans
*

**18.415[J] Advanced Algorithms**

Same subject as 6.854[J]

Prereq: 6.041B, 6.042[J], or 18.600; 6.046[J]

G (Fall)

5-0-7 units

See description under subject 6.854[J].

*A. Moitra, D. R. Karger
*

**18.416[J] Randomized Algorithms**

Same subject as 6.856[J]

Prereq: 6.854[J], 6.041B or 6.042[J]

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

5-0-7 units

See description under subject 6.856[J].

*D. R. Karger
*

**18.417 Introduction to Computational Molecular Biology**

Prereq: 6.01, 6.006, or permission of instructor

G (Fall)

Not offered regularly; consult department

3-0-9 units

Introduces the basic computational methods used to model and predict the structure of biomolecules (proteins, DNA, RNA). Covers classical techniques in the field (molecular dynamics, Monte Carlo, dynamic programming) to more recent advances in analyzing and predicting RNA and protein structure, ranging from Hidden Markov Models and 3-D lattice models to attribute Grammars and tree Grammars.

*Information: B. Berger
*

**18.418 Topics in Computational Molecular Biology**

Prereq: 18.417, 6.047, or permission of instructor

G (Spring)

3-0-9 units

Can be repeated for credit.

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

*B. Berger
*

**18.424 Seminar in Information Theory**

Prereq: 18.05, 18.600, or 6.041B; 18.06, 18.700, or 18.701

U (Spring)

3-0-9 units

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

*P. W. Shor
*

**18.425[J] Cryptography and Cryptanalysis**

Same subject as 6.875[J]

Prereq: 6.046[J]

G (Spring)

3-0-9 units

See description under subject 6.875[J].

*S. Goldwasser, S. Micali
*

**18.434 Seminar in Theoretical Computer Science**

Prereq: 18.410[J]

U (Fall)

3-0-9 units

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

*Y. K. Deshpande
*

**18.435[J] Quantum Computation**

Same subject as 2.111[J], 8.370[J]

Prereq: Permission of instructor

G (Fall)

3-0-9 units

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

*I. Chuang, E. Farhi, S. Lloyd, P. Shor
*

**18.436[J] Quantum Information Science**

Same subject as 6.443[J], 8.371[J]

Prereq: 18.435[J]

G (Spring)

3-0-9 units

See description under subject 8.371[J].

*I. Chuang
*

**18.437[J] Distributed Algorithms**

Same subject as 6.852[J]

Prereq: 6.046[J]

G (Spring)

3-0-9 units

See description under subject 6.852[J].

*N. A. Lynch
*

**18.453 Combinatorial Optimization**

Subject meets with 18.4531

Prereq: 18.06, 18.700, or 18.701

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: U (Spring)

3-0-9 units

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

*M. X. Goemans
*

**18.4531 Combinatorial Optimization**

Subject meets with 18.453

Prereq: 18.06, 18.700, or 18.701

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

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

*M. X. Goemans
*

**18.455 Advanced Combinatorial Optimization**

Prereq: 18.453 or permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

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

*M. X. Goemans
*

## Logic

**18.504 Seminar in Logic**

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.510, 18.700, or 18.701

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: U (Spring)

3-0-9 units

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

*H. Cohn
*

**18.510 Introduction to Mathematical Logic and Set Theory**

Prereq: None

Acad Year 2017-2018: U (Fall)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*H. Cohn
*

**18.515 Mathematical Logic**

Prereq: Permission of instructor

G (Spring)

Not offered regularly; consult department

3-0-9 units

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

*Information: B. Poonen
*

## Probability and Statistics

**18.600 Probability and Random Variables**

Prereq: Calculus II (GIR)

U (Fall, Spring)

4-0-8 units. REST

Credit cannot also be received for 15.079, 15.0791

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

*Fall: J. A. Kelner
**Spring: S. Sheffield
*

**18.615 Introduction to Stochastic Processes**

Prereq: 18.600 or 6.041B

G (Spring)

3-0-9 units

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

*A. Bufetov
*

**18.642 Topics in Mathematics with Applications in Finance**

Prereq: 18.03; 18.06; 18.05 or 18.600

U (Fall)

4-0-11 units

Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required. Instruction and practice in written communication provided. Limited to 30.

*P. Kempthorne, V. Strela, J. Xia
*

**18.650[J] Fundamentals of Statistics**

Same subject as IDS.014[J]

Subject meets with 18.6501

Prereq: 18.600 or 6.041B

U (Fall, Spring)

4-0-8 units

Credit cannot also be received for 15.075[J], IDS.013[J]

A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.

*Fall: P. Rigollet
**Spring: V.-E. Brunel
*

**18.6501 Fundamentals of Statistics**

Subject meets with 18.650[J], IDS.014[J]

Prereq: 18.600 or 6.041B

G (Fall, Spring)

4-0-8 units

Credit cannot also be received for 15.075[J], IDS.013[J]

A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics. Students in Course 18 must register for the undergraduate version, 18.650[J].

*Fall: P. Rigollet
**Spring: V.-E. Brunel
*

**18.655 Mathematical Statistics**

Prereq: Permission of instructor

G (Spring)

3-0-9 units

Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis.

*P. Kempthorne
*

**18.657 Topics in Statistics**

Prereq: Permission of instructor

G (Spring)

3-0-9 units

Can be repeated for credit.

Topics vary from term to term.

*P. Rigollet
*

## Algebra and Number Theory

**18.700 Linear Algebra**

Prereq: Calculus II (GIR)

U (Fall)

3-0-9 units. REST

Credit cannot also be received for 18.06

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

*G. Oberdieck
*

**18.701 Algebra I**

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

U (Fall)

3-0-9 units

More extensive and theoretical than the 18.700-18.703 sequence. Experience with proofs necessary. First term: group theory, geometry, and linear algebra. Second term: group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, Galois theory.

*M. Artin
*

**18.702 Algebra II**

Prereq: 18.701

U (Spring)

3-0-9 units

More extensive and theoretical than the 18.700-18.703 sequence. Experience with proofs necessary. First term: group theory, geometry, and linear algebra. Second term: group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, Galois theory.

*Information: M. Artin
*

**18.703 Modern Algebra**

Prereq: Calculus II (GIR)

U (Spring)

3-0-9 units

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

*C. Negron
*

**18.704 Seminar in Algebra**

Prereq: 18.701; or 18.06, 18.703; or 18.700, 18.703

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. Some experience with proofs required. Enrollment limited.

*V. G. Kac
*

**18.705 Commutative Algebra**

Prereq: 18.702

G (Fall)

3-0-9 units

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

*A. Pixton
*

**18.706 Noncommutative Algebra**

Prereq: 18.702

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Topics may include one or more of the following: Coxeter groups, Hecke algebras, their canonical bases and their representations.

*G. Lusztig
*

**18.708 Topics in Algebra**

Prereq: 18.705

Acad Year 2017-2018: G (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*Information: P. I. Etingof
*

**18.715 Introduction to Representation Theory**

Prereq: 18.702 or 18.703

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

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

*B. Poonen
*

**18.721 Introduction to Algebraic Geometry**

Prereq: 18.702, 18.901

Acad Year 2017-2018: U (Spring)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*M. Artin
*

**18.725 Algebraic Geometry I**

Prereq: None. *Coreq: 18.705*

G (Fall)

3-0-9 units

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

*G. Sacca
*

**18.726 Algebraic Geometry II**

Prereq: 18.725

G (Spring)

3-0-9 units

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

*G. Sacca
*

**18.727 Topics in Algebraic Geometry**

Prereq: 18.725

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*Information: R. Bezrukavnikov
*

**18.737 Algebraic Groups**

Prereq: 18.705

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

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

*G. Lusztig
*

**18.745 Introduction to Lie Algebras**

Prereq: 18.701 or 18.703

G (Spring)

3-0-9 units

Topics may include structure of finite-dimensional Lie algebras; theorems of Engel and Lie; Cartan subalgebras and regular elements; trace form and Cartan's criterion; Chevalley's conjugacy theorem; classification and construction of semisimple Lie algebras; Weyl group; universal enveloping algebra and the Casimir operator; Weyl's complete reducibility theorem, Levi and Maltsev theorems; Verma modules; classification of irreducible finite-dimensional representations of semisimple Lie algebras; Weyl's character and dimension formulas.

*G. Lusztig
*

**18.747 Infinite-dimensional Lie Algebras**

Prereq: 18.745

G (Fall)

3-0-9 units

Topics vary from year to year.

*R. Bezrukavnikov
*

**18.748 Topics in Lie Theory**

Prereq: Permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*P. I. Etingof
*

**18.755 Introduction to Lie Groups**

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.700 or 18.701

G (Fall)

3-0-9 units

A general introduction to manifolds and Lie groups. The role of Lie groups in mathematics and physics. Exponential mapping. Correspondence with Lie algebras. Homogeneous spaces and transformation groups. Adjoint representation. Covering groups. Automorphism groups. Invariant differential forms and cohomology of Lie groups and homogeneous spaces. 18.101 recommended but not required.

*D. Vogan
*

**18.757 Representations of Lie Groups**

Prereq: 18.745 or 18.755

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

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

*Information: R. Bezrukavnikov
*

**18.781 Theory of Numbers**

Prereq: None

U (Spring)

3-0-9 units

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

*A. Shankar
*

**18.782 Introduction to Arithmetic Geometry**

Prereq: 18.702

Acad Year 2017-2018: U (Fall)

Acad Year 2018-2019: Not offered

3-0-9 units

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

*D. Ranganathan
*

**18.783 Elliptic Curves**

Subject meets with 18.7831

Prereq: 18.703, or *Coreq: 18.702*, or permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: U (Spring)

3-0-9 units

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

*A. Sutherland
*

**18.7831 Elliptic Curves**

Subject meets with 18.783

Prereq: 18.703, *Coreq: 18.702*, or permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

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

*A. Sutherland
*

**18.784 Seminar in Number Theory**

Prereq: 18.06; 18.100A, 18.100B, 18.100P, or 18.100Q

U (Fall)

3-0-9 units

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

*A. Shankar
*

**18.785 Number Theory I**

Prereq: None. *Coreq: 18.705*

G (Fall)

3-0-9 units

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

*A. Sutherland
*

**18.786 Number Theory II**

Prereq: 18.785

G (Spring)

3-0-9 units

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

*A. Sutherland
*

**18.787 Topics in Number Theory**

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Can be repeated for credit.

Topics vary from year to year.

*B. Poonen
*

## Mathematics Laboratory

**18.821 Project Laboratory in Mathematics**

Prereq: Two mathematics subjects numbered 18.100 or above

U (Fall, Spring)

3-6-3 units. Institute LAB

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

*Fall: R. Bezrukavnikov
**Spring: H. R. Miller
*

## Topology and Geometry

**18.901 Introduction to Topology**

Subject meets with 18.9011

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

U (Fall, Spring)

3-0-9 units

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

*Fall: G. Lusztig
**Spring: J. Lin
*

**18.9011 Introduction to Topology**

Subject meets with 18.901

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

G (Fall, Spring)

3-0-9 units

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

*Fall: G. Lusztig
**Spring: J. Lin
*

**18.904 Seminar in Topology**

Prereq: 18.901

U (Fall)

3-0-9 units

Topics vary from year to year and include the fundamental group and covering spaces. Time permitting, also covers the relationship between these objects and the theory of knots. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

*Z. Xu
*

**18.905 Algebraic Topology I**

Prereq: 18.701 or 18.703; 18.901

G (Fall)

3-0-9 units

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

*G. Tabuada
*

**18.906 Algebraic Topology II**

Prereq: 18.905

G (Spring)

3-0-9 units

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

*Z. Xu
*

**18.917 Topics in Algebraic Topology**

Prereq: 18.906

G (Spring)

3-0-9 units

Can be repeated for credit.

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

*G. Tabuada
*

**18.919 Graduate Topology Seminar**

Prereq: 18.906

G (Fall)

3-0-9 units

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

*H. R. Miller
*

**18.937 Topics in Geometric Topology**

Prereq: Permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Can be repeated for credit.

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

*E. Murphy
*

**18.950 Differential Geometry**

Subject meets with 18.9501

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

U (Spring)

3-0-9 units

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

*B. Wilson
*

**18.9501 Differential Geometry**

Subject meets with 18.950

Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q; 18.06, 18.700, or 18.701

G (Spring)

3-0-9 units

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

*B. Wilson
*

**18.952 Theory of Differential Forms**

Prereq: 18.101; 18.700 or 18.701

U (Spring)

3-0-9 units

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

*V. W. Guillemin
*

**18.965 Geometry of Manifolds I**

Prereq: 18.101, 18.950 or 18.952

G (Fall)

3-0-9 units

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

*W. Minicozzi
*

**18.966 Geometry of Manifolds II**

Prereq: 18.965

G (Spring)

3-0-9 units

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

*Fall: W. Minicozzi
**Spring: T. Colding
*

**18.968 Topics in Geometry**

Prereq: 18.965

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Fall)

3-0-9 units

Can be repeated for credit.

Content varies from year to year.

*T. Colding
*

**18.979 Graduate Geometry Seminar**

Prereq: Permission of instructor

Acad Year 2017-2018: Not offered

Acad Year 2018-2019: G (Spring)

3-0-9 units

Can be repeated for credit.

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

*T. Mrowka
*

**18.994 Seminar in Geometry**

Prereq: 18.101, 18.102, 18.103, or 18.112

U (Spring)

3-0-9 units

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

*W. Minicozzi
*

**18.999 Research in Mathematics**

Prereq: Permission of instructor

G (Fall, Spring, Summer)

Units arranged

Can be repeated for credit.

Opportunity for study of graduate-level topics in mathematics under the supervision of a member of the department. For graduate students desiring advanced work not provided in regular subjects.

*Information: W. Minicozzi
*

**18.UR Undergraduate Research**

Prereq: Permission of instructor

U (Fall, IAP, Spring, Summer)

Units arranged [P/D/F]

Can be repeated for credit.

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

*Information: J. W. Bush
*

**18.THG Graduate Thesis**

Prereq: Permission of instructor

G (Fall, IAP, Spring, Summer)

Units arranged

Can be repeated for credit.

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

*Information: W. Minicozzi
*

**18.S096 Special Subject in Mathematics**

Prereq: Permission of instructor

U (Fall, IAP, Spring)

Units arranged

Can be repeated for credit.

Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. 18.S097 is graded P/D/F.

*Fall: E. Mossel
**Spring: P. Kempthorne
*

**18.S097 Special Subject in Mathematics**

Prereq: Permission of instructor

U (IAP, Spring)

Units arranged [P/D/F]

Can be repeated for credit.

Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. 18.S097 is graded P/D/F.

*Information: J. W. Bush
*

**18.S995 Special Subject in Mathematics**

Prereq: Permission of instructor

G (Fall)

Units arranged

Can be repeated for credit.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval.

*J. Dunkel
*

**18.S996 Special Subject in Mathematics**

Prereq: Permission of instructor

G (Spring)

Units arranged

Can be repeated for credit.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.

*E. Mossel
*

**18.S997 Special Subject in Mathematics**

Prereq: Permission of instructor

G (Fall)

Units arranged

Can be repeated for credit.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.

*Y. Zhao
*

**18.S998 Special Subject in Mathematics**

Prereq: Permission of instructor

G (IAP, Spring)

Not offered regularly; consult department

Units arranged

Can be repeated for credit.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.

*Information: J. W. Bush
*