## Department of Mathematics

#### Chair

David Gabai

#### Associate Chair

Christopher M. Skinner

#### Departmental Representative

Christopher M. Skinner

Jennifer M. Johnson

#### Director of Graduate Studies

Alexandru D. Ionescu

Nicolas P. Templier

#### Professor

Michael Aizenman, also Physics

Manjul Bhargava

Sun-Yung Alice Chang

Peter Constantin, also Applied and Computational Mathematics

John H. Conway

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

Edward Nelson

Peter S. Ozsváth

Igor Y. Rodnianski

Peter C. Sarnak

Paul D. Seymour, also Applied and Computational Mathematics

Yakov G. Sinai

Christopher M. Skinner

Zoltán Szábo

Gang Tian

Paul C. Yang

Shou-Wu Zhang

#### Associate Professor

Amit Singer, also Applied and Computational Mathematics

#### Assistant Professor

Szu-Yu Chen

Zeev Dvir, also Computer Science

Rupert Frank

Choonghong Oh

Sucharit Sarkar

Alexander Sodin

Claus M. Sorensen

Nicolas P. Templier

Stefan H.M. van Zwam

Vlad Vicol

Micah W. Warren

Anna K. Wienhard

#### Instructor

Stefanos Aretakis

Costante Bellettini

Jeffrey S. Case

Michael Damron

Jonathan Fickenscher

Aurel Mihai Fulger

Penka V. Georgieva

David Geraghty

Tasho Kaletha

Victor Lie

Yu-Han Liu

Niels Møller

Luc L. Nguyen

Zsolt Patakfalvi

Oana Pocovnicu

Claudiu C. Raicu

Nicholas Sheridan

Kevin F. Tucker

Bart Vandereycken

Fang Wang

#### Senior Lecturer

Jennifer M. Johnson

#### Associated Faculty

John P. Burgess, Philosophy

René A. Carmona, Operations Research and Financial Engineering

Bernard Chazelle, Computer Science

Erhan Çınlar, Operations Research and Financial Engineering

Hans Halvorson, Philosophy

Philip J. Holmes, Mechanical and Aerospace Engineering

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.

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.

Students who are not prepared to begin with 103 may take 100, a rigorous precalculus/prestatistics refresher, which may be followed by 102 or 103; 102 is a one-semester survey of selected topics from 103 and 104 for students who do not intend to take further calculus courses. Students who will later need 104 should not take 102.

Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)-102-175, but 103-175 is strongly preferred; 175 covers selected topics from 104, 201, and 202, with biology and economics applications in mind. Other students will need the standard 103-104-201-202 sequence instead, especially recommended for those who plan to continue with 300-level mathematics courses and/or go on to graduate school in economics or finance.

Prospective mathematics majors must take at least one course introducing formal mathematical argument and rigorous proofs. For many, this will be 215, but 214 and 217 are more algebraic alternatives. One recommended sequence for prospective majors is 215-217-218, especially suitable for those who already have some experience with constructing proofs. Other possible sequences for prospective majors include 203-214-217 and 203-204-215, although the latter is relatively rare. Note that 215 and 217 may be taken in either order, as can 203 and 204.

**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 a 5 on the Advanced Placement Calculus AB examination or a 4 on the Advanced Placement Calculus BC examination. To qualify for placement into 201 or 202, a student should have a score of 5 on the Advanced Placement Calculus BC examination. Students who possess in addition a particularly strong interest in mathematics and an SAT Mathematics score of at least 750 may instead opt for 203, 214, or 215. 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 or 214 and either 217 or 204 and either 218 or 203 are required for admission to the department. Prospective mathematics majors should consult the department as early as possible and plan a program that includes as much of the 215-217-218 sequence as possible. Most majors begin taking courses at the 300 level by the second semester of the sophomore year. Further information for prospective majors is available on the department home page.

#### Program of Study

Students concentrating in mathematics must complete the following requirements:

1. one course in real analysis from the 320s or 420s or 385;

2. one course in complex analysis from the 330s;

3. one course in algebra from the 340s or 440s;

4. one course in geometry or topology from the 350s or 450s or 360s or 460s or, alternatively, one course in discrete mathematics from the 370s or 470s;

5. an additional four courses at the 300 level or higher, up to three of which may be cognate courses outside the Department of Mathematics, with permission from the junior or senior advisers or departmental representative.

The departmental grade (the average grade on 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, which is 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, but may alternatively consist of reading in a special subject (e.g., a topic in Fourier analysis, representation theory, or Galois theory), and writing a paper based on that reading. The independent work in the senior year centers on writing a senior thesis.

#### 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.
* Staff*

MAT 102 Survey of Calculus Spring 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. Students who need to take further calculus courses like MAT104 or MAT175 should take MAT103 instead. Three classes.
* Staff*

MAT 103 Calculus I Fall 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. 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 in calculus and linear algebra as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, improper integrals and rates of growth, partial derivatives, gradient, Lagrange multipliers, linear systems, determinants, matrix inversion, eigenvalues and eigenvectors, as time permits. Prerequisite: MAT103 or equivalent. Three classes.
* Staff*

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.
* Staff*

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.
* Staff*

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 Analysis in a 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. Damron**,
R. Gunning**,
L. Nguyen*

MAT 217 Honors Linear Algebra Fall, 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, linear systems of differential equations, the spectral theorem for normal transformations, bilinear and quadratic forms.
*
T. Kaletha**,
M. Damron**,
R. Gunning*

MAT 218 Analysis in Several Variables Fall, Spring QR

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's theorem, change of variables theorem, and the theorems of Green, Gauss, and Stokes. Prerequisites: MAT 215 and MAT 217, or equivalent.
*
P. Georgieva*

MAT 305 Mathematical Logic Spring 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.
*
J. Conway*

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.
*
M. Warren*

MAT 322 Introduction to Differential Equations (see APC 350)

MAT 323 Topics in Mathematical Modeling (also APC 323) Fall 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.
*
P. Holmes*

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.
*
J. Fickenscher*

MAT 330 Complex Analysis with Applications Fall, 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.
*
J. Case**,
V. Vicol*

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: MAT202 or 204 or 217.
*
C. Raicu*

MAT 346 Algebra II Spring QR

A continuation of Algebra I, this course will cover the basics of ring theory, module theory, field theory, and Galois theory. Topics include structure theorems for modules, polynomial rings, field extensions, and solvability of polynomial equations by radicals. Prerequisite: MAT345 or equivalent.
*
Z. Patakfalvi*

MAT 355 Introduction to Differential Geometry Fall QR

Introduction to the Riemannian geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Gonnet Theorem, uniformization of surfaces. Prerequisite: MAT218 or equivalent.
*
G. Tian*

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

Introduction to combinatorics, a fundamental mathematical discipline as well as an essential component of many mathematical areas. While in the past many of the basic combinatorial results were at first obtained by ingenuity and detailed reasoning, modern theory has grown out of this early stage and often relies on deep, well-developed tools. Topics include Ramsey Theory, Turan Theorem and Extremal Graph Theory, Probabilistic Argument, Algebraic Methods and Spectral Techniques. Showcases the gems of modern combinatorics. Prerequisites: MAT202 or equivalent.
*
S. van Zwam*

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.
*
S. van Zwam*

MAT 380 Probability and Stochastic Systems (see ORF 309)

MAT 385 Probability Theory Fall 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 Spring QR

Course on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and L-functions. Prerequisites: MAT 214 and 345 or equivalent.
*
S. Zhang*

MAT 425 Analysis III: Integration Theory and Hilbert Space 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.
*
A. Ionescu*

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 Not offered this year QR

Topics in analysis selected from areas such as functional analysis, operator theory, and harmonic analysis. Content varies from year to year.
* Staff*

MAT 449 Topics in Algebra Not offered this year QR

Topics in algebra selected from areas such as the analytic and algebraic theory of numbers and algebraic geometry. Three classes. Prerequisite: MAT 322 or MAT 323.
*
N. Katz*

MAT 459 Topics in Geometry Not offered this year QR

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

MAT 473 Cryptography (see COS 433)

MAT 481 Introduction to Nonlinear Dynamics (see CBE 448)

MAT 486 Random Processes Spring 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.
*
Y. Sinai*

MAT 493 Mathematical Methods of Physics (see PHY 403)