## Department of Mathematics

#### Chair

David Gabai

#### Acting Chair

Igor Y. Rodnianski (fall/spring)

#### Associate Chair

János Kollár

#### Departmental Representative

János Kollár

Jennifer M. Johnson

#### Director of Graduate Studies

Peter Ozsváth

Zsolt Patakfalvi

#### Professor

Michael Aizenman, also Physics

Manjul Bhargava

Sun-Yung Alice Chang

Maria Chudnovsky, also Applied and Computational Mathematics

Fernando Codá Marques

Peter Constantin, also Applied and Computational Mathematics

Mihalis C. Dafermos

Weinan E, also Applied and Computational Mathematics

Charles L. Fefferman

David Gabai

Robert C. Gunning

Alexandru D. Ionescu

Nicholas M. Katz

Sergiu Klainerman

János Kollár

Elliott H. Lieb, also Physics

John N. Mather

Sophie Morel

Assaf Naor

Peter S. Ozsváth

Igor Y. Rodnianski

Peter C. Sarnak

Paul D. Seymour, also Applied and Computational Mathematics

Yakov G. Sinai

Amit Singer, also Applied and Computational Mathematics

Christopher M. Skinner

Zoltán Szábo

Gang Tian

Paul C. Yang

Shou-Wu Zhang

#### Visiting Professor

Keith Devlin, Stanley Kelley, Jr., Visiting Professor for Distinguished Teaching

Sergio Fenley

#### Assistant Professor

Stefanos Aretakis

Zeev Dvir, also Computer Science

Adam S. Levine

Adam W. Marcus, also Applied and Computational Mathematics

Zsolt Patakfalvi

Fabio G. Pusateri

Sucharit Sarkar

Nicholas J. Sheridan

Vlad Vicol

#### Instructor

Antonio Ache

Nicolas A.S. Boumal

Ana Caraiani

Lucas Culler

Hansheng Diao

Tarek Elgindi

Aurel Mihai Fulger

Javier Gomez-Serrano

Micaela Ignatova

Jonathan Kommemi

Chun-hung Liu

Ana Menezes

Rafael Montezuma Pinero Cabral

Ori Parzanchevski

Steven Sivek

Florian Sprung

#### Senior Lecturer

Jonathan Fickenscher

Jennifer M. Johnson

Mark W. McConnell

#### Associated Faculty

John P. Burgess, Philosophy

René A. Carmona, Operations Research and Financial Engineering

Bernard Chazelle, Computer Science

Hans Halvorson, Philosophy

Yannis G. Kevrekidis, Chemical and Biological Engineering

William Massey, Operations Research and Financial Engineering

Frans Pretorius, Physics

Robert E. Tarjan, Computer Science

Robert J. Vanderbei, Operations Research and Financial Engineering

Sergio Verdú, Electrical Engineering

#### Information and Departmental Plan of Study

Most freshmen and sophomores interested in science, engineering, or finance take courses from the standard calculus and linear algebra sequence 103-104-201-202, which emphasizes concrete computations over more theoretical considerations. Note that 201 and 202 can be taken in either order.

Students who are not prepared to begin with 103 may take 100, a rigorous precalculus/prestatistics refresher offered only in the fall semester and intended for students whose highest math SAT score is below 650.

Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)-103-175. Note that 175 covers selected topics from 201, with biology and economics applications in mind. Prospective math-track economics/finance majors will need the standard sequence 103-104-201-202 instead of 175.

More mathematically inclined students, especially prospective physics majors, may opt to replace 201-202 with 203-204, for greater emphasis on theory and more challenging computational problems.

Prospective mathematics majors must take at least one course introducing formal mathematical argument and rigorous proofs. The recommended freshman sequence for prospective majors is 215-217. Prospective majors who already have substantial experience with university-level proof-based analysis courses may consider the accelerated sequence 216-218 instead. Other possible sequences for prospective majors include 214-204-203 and 203-204-215, although the latter two are relatively rare. Note that 203 and 204 can be taken in either order.

**Placement. **Students with little or no background in calculus are placed in 103, or in 100 if their SAT mathematics scores indicate insufficient background in precalculus topics. To qualify for placement in 104 or 175, a student should score 5 on the AB Advanced Placement Examination or a 4 on the BC Advanced Placement Examination. To qualify for placement into 201 or 202, a student should have a score of 5 on the BC Examination. Students who possess in addition a particularly strong interest in mathematics as well as a SAT mathematics score of at least 750 may opt for 203 or 214 or 215 or 216 instead. For more detailed placement information, consult the Department of Mathematics home page or placement officer.

#### Advanced Placement

One unit of advanced placement credit is granted when a student is placed in MAT 104 or 175. Two units of advanced placement credit are granted when a student is placed in MAT 201, 203, or 217.

#### Prerequisites

Generally, either 215-217 or 216-218 or 203-204-215 are strongly recommended for admission to the department. Prospective mathematics majors should consult the department early and plan a program that includes as much of the 215-217 or 216-218 sequence as possible. Most majors begin taking courses at the 300-level by the second semester of the sophomore year, in preparation for their junior independent work.

Further information for prospective majors is available on the department home page.

#### Program of Study

Students must complete **four core requirements**:

- one course in
**real analysis**(e.g. 320 or 325 or 425 or 385) - one course in
**complex analysis**(e.g. 330 or 335) - one course in
**algebra**(e.g. 340 or 345) - one course in
**geometry**or**topology**(e.g. 350 or 355 or 365 or 560)

It is recommended that students complete some of these core requirements by the end of the sophomore year. Completing these core courses early gives more options for junior and senior independent work.

Note: One course in **discrete mathematics** (e.g. 375, 377 or 378) can replace the geometry/topology core requirement, if desired.

**In addition to the four core requirements**, students must complete an additional four courses at the 300 level or higher, up to three of which may be cognate courses outside the mathematics department, with permission from the junior or senior advisers or departmental representative.

The departmental grade (the average grade of the eight departmental courses) together with grades and reports on independent work is the basis on which honors and prizes are awarded on graduation.

Students should refer to *Course Offerings* to check which courses are offered in a given term. Programs of study in various fields of pure mathematics and applied mathematics are available. Appropriate plans of study may be arranged for students interested in numerical analysis, discrete mathematics, optimization, physics, the biological sciences, probability and statistics, finance, economics, or computer science. For students interested in these areas, a coherent program containing up to three courses in a cognate field may be approved.

#### Independent Work

All departmental students engage in independent work, supervised by a member of the department chosen in consultation with a departmental adviser. The independent work of the junior year generally consists of participating actively in a junior seminar in both the fall and the spring semesters. Alternatively, a student may opt to replace **one** junior seminar with supervised reading in a special subject and then writing a paper based on that reading. The independent work in the senior year centers on writing a senior thesis. A substantial percentage of our majors work with faculty in other departments on their senior project.

#### Senior Departmental Examination

Each senior takes an oral examination based on the senior thesis and the broader subfield to which it contributes. A departmental committee conducts the examination in May.

### Courses

MAT 100 Precalculus/Prestatistics Fall QR

An intensive and rigorous treatment of algebra and trigonometry as preparation for further courses in calculus or statistics. Topics include functions and their graphs, equations involving polynomial and rational functions, exponentials, logarithms and trigonometry.
*
J. Johnson*

MAT 102 Survey of Calculus Not offered this year QR

One semester survey of the major concepts and computational techniques of calculus including limits, derivatives and integrals. Emphasis on basic examples and applications of calculus including approximation, differential equations, rates of change and error estimation for students who will take no further calculus. Prerequisites: MAT100 or equivalent. Restrictions: Cannot receive course credit for both MAT103 and MAT102. Provides adequate preparation for MAT175. Three classes.
* Staff*

MAT 103 Calculus I Fall, Spring QR

First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. The fall offering will emphasize applications to physics and engineering in preparation for MAT104; the spring offering will emphasize applications to economics and life sciences, in preparation for MAT175. Prerequisite: MAT100 or equivalent. Three classes.
* Staff*

MAT 104 Calculus II Fall, Spring QR

Continuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers. Prerequisite: MAT103 or equivalent. Three classes.
* Staff*

MAT 151 Problem Solving in Mathematics (see APC 151)

MAT 175 Mathematics for Economics/Life Sciences Fall, Spring QR

Survey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers.
Students preparing for math track econometrics and finance courses need MAT201/202 instead. Students who complete 175 can continue in 202 if they wish.
*
J. Johnson*

MAT 191 An Integrated Introduction to Engineering, Mathematics, Physics (see EGR 191)

MAT 192 An Integrated Introduction to Engineering, Mathematics, Physics (see EGR 192)

MAT 199 Math Alive (see APC 199)

MAT 201 Multivariable Calculus Fall, Spring QR

Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields, and Stokes's theorem. Prerequisite: 104 or equivalent. Three classes.
* Staff*

MAT 202 Linear Algebra with Applications Fall, Spring QR

Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems.Three classes.
* Staff*

MAT 203 Advanced Vector Calculus Fall QR

Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent. Three classes.
*
C. Taylor*

MAT 204 Advanced Linear Algebra with Applications Spring QR

Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent. Three classes.
*
C. Taylor*

MAT 214 Numbers, Equations, and Proofs Fall QR

An introduction to classical number theory to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions, and quadratic reciprocity. There will be a topic from more advanced or more applied number theory such as p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the mathematics department and for non-majors interested in exposure to higher mathematics.
* Staff*

MAT 215 Honors Analysis (Single Variable) Fall, Spring QR

An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include the rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel theorem, the Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's theorem.
*
M. McConnell**,
A. Naor*

MAT 217 Honors Linear Algebra Spring QR

A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms.
* Staff*

MAT 218 Accelerated Honors Analysis II Spring QR

Continuation of MAT216, Accelerated Analysis I from the fall. A rigorous course in analysis with an emphasis on proof rather than applications. Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini. See the department website for details.
*
R. Gunning*

MAT 305 Mathematical Logic Not offered this year QR

A development of logic from the mathematical viewpoint, including propositional and predicate calculus, consequence and deduction, truth and satisfaction, the Goedel completeness and incompleteness theorems. Applications to model theory, recursion theory, and set theory as time permits. Some underclass background in logic or in mathematics is recommended.
* Staff*

MAT 306 Advanced Logic (see PHI 323)

MAT 320 Introduction to Real Analysis Fall QR

Introduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space and the theory of Fourier series. Prerequisite: MAT201 and MAT202 or equivalent.
*
V. Vicol*

MAT 322 Introduction to Differential Equations (see APC 350)

MAT 323 Topics in Mathematical Modeling (also APC 323) Not offered this year QR

Draws problems from the sciences & engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics. This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upper class undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.
* Staff*

MAT 325 Analysis I: Fourier Series and Partial Differential Equations Spring QR

Basic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Fast Fourier Transforms, Finite Fourier Series, Dirichlet Characters, and applications to properties of primes. Prerequisites: 215, 218, or permission of instructor.
*
E. Stein*

MAT 330 Complex Analysis with Applications Spring QR

The theory of functions of one complex variable, covering power series expansions, residues, contour integration, and conformal mapping. Although the theory will be given adequate treatment, the emphasis of this course is the use of complex analysis as a tool for solving problems. Prerequisite: MAT201 and MAT202 or equivalent.
*
M. Aizenman*

MAT 335 Analysis II: Complex Analysis Fall QR

Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. An overall view of Special Functions via the hypergeometric series. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters.
*
E. Stein*

MAT 345 Algebra I Fall QR

This course will cover the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions, and the representation theory of finite groups. Prerequisites: MAT204 or 217.
*
N. Katz*

MAT 346 Algebra II Spring QR

Continuation of MAT345. Further develop knowledge of algebraic structures by exploring examples that connect to higher mathematics. There will be opportunities for a student to explore an advanced topic in great depth, possibly for a junior project.
*
C. Skinner*

MAT 355 Introduction to Differential Geometry Spring QR

Introduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Gonnet formula, uniformization of surfaces, elementary notions of contact geometry. Prerequisite: MAT218 or 350 or equivalent.
*
L. Culler*

MAT 365 Topology Fall QR

Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisite: MAT202 or 204 or 218 or equivalent.
*
Z. Szabó*

MAT 375 Introduction to Graph Theory (also COS 342) Spring QR

The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or 204 or 217 or equivalent.
*
P. Seymour*

MAT 377 Combinatorial Mathematics (also APC 377) Fall QR

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements.
*
C. Liu*

MAT 378 Theory of Games Spring QR

Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisite: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended.
*
C. Liu*

MAT 380 Probability and Stochastic Systems (see ORF 309)

MAT 385 Probability Theory Spring QR

Sequence of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains. Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with the linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisite: MAT203 or 218 or equivalent.
*
Y. Sinai*

MAT 390 Introduction to Modern Applied Mathematics Not offered this year QR

Classical topics blended with modern topics involving numerical methods and discrete mathematics, including both theory and application. Symmetric linear equations, Fourier series and Laplace's equation, initial value problems, design and stability of difference methods, conjugate gradients, combinational optimization and network flows.
* Staff*

MAT 391 Mathematics in Engineering I (see MAE 305)

MAT 392 Mathematics in Engineering II (see MAE 306)

MAT 393 Mathematical Programming Not offered this year QR

Linear programs, duality, Dantzig's simplex method; theory of dual linear systems; matrix games, von Neumann's minimax theorem, simplex solution; algorithms for assignment, transport, flow; brief introduction to nonlinear programming.
* Staff*

MAT 407 Theory of Computation (see COS 487)

MAT 419 Topics in Number Theory Fall, Spring QR

Topics introducing various aspects of number theory, including analytic and algebraic number theory, L-functions, and modular forms. See *Course Offerings* listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent.
*
S. Zhang**,
C. Skinner*

MAT 425 Analysis III: Integration Theory and Hilbert Spaces Spring QR

The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier Transforms, and partial differential equations. Introduction to fractals. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters. Prerequisites: MAT215 or 218 or equivalent.
* Staff*

MAT 427 Ordinary Differential Equations Fall QR

Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. Topics include explicit solutions of some non-linear equations in two variables by separation of variables and integrating factors, explicit solution of simultaneous linear equations with constant coefficients, explicit solution of some linear equations with variable forcing term by Laplace transform methods, geometric methods (description of the phase portrait), and the fundamental existence and uniqueness theorem.
*
J. Mather*

MAT 429 Topics in Analysis Fall QR

Introduction to incompressible fluid dynamics. The course will give an introduction to the mathematical theory of the Euler equations, the fundamental partial differential equation arising in the study of incompressible fluids. We will discuss several topics in analysis that emerge in the study of these equations: Lebesgue and Sobolev spaces, distribution theory, elliptic PDEs, singular integrals, and Fourier analysis. Content varies from year to year. See *Course Offerings* listing for topic details.
*
I. Rodnianski*

MAT 449 Topics in Algebra Not offered this year QR

Topics in algebra selected from areas such as representation theory of finite groups and the theory of Lie algebras. Three classes. Prerequisite: MAT 345 or MAT 346.
* Staff*

MAT 459 Topics in Geometry Fall QR

Topics in geometry selected from areas such as differentiable and Riemannian manifolds, point set and algebraic topology, integral geometry. Prerequisite: departmental permission.
*
P. Yang*

MAT 473 Cryptography (see COS 433)

MAT 474 Introduction to Analytic Combinatorics (see COS 488)

MAT 481 Introduction to Nonlinear Dynamics (see CBE 448)

MAT 486 Random Processes Not offered this year QR

Wiener measure. Stochastic differential equations. Markov diffusion processes. Linear theory of stationary processes. Ergodicity, mixing, central limit theorem for stationary processes. If time permits, the theory of products of random matrices and PDE with random coefficients will be discussed. Prerequisite: MAT385.
* Staff*

MAT 493 Mathematical Methods of Physics (see PHY 403)