## Department of Mathematics

#### Chair

Sun-Yung Alice Chang

#### Associate Chair

Christopher M. Skinner

#### Departmental Representative

Christopher M. Skinner

Jennifer M. Johnson

#### Director of Graduate Studies

David Gabai

#### Professor

Michael Aizenman, also Physics

Manjul Bhargava

William Browder

Sun-Yung Alice Chang

Peter Constantin, also Applied and Computational Mathematics

John H. Conway

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

Edward Nelson

Andrei Okounkov

Peter S. Ozsváth

Rahul V. Pandharipande

Igor Y. Rodnianski

Peter C. Sarnak

Paul D. Seymour, also Applied and Computational Mathematics

Yakov G. Sinai

Christopher M. Skinner

Elias M. Stein

Zoltán Szábo

Gang Tian

Andrew J. Wiles

Paul C. Yang

Shou-Wu Zhang

#### Associate Professor

Amit Singer, also Applied and Computational Mathematics

#### Assistant Professor

John A. Baldwin

Szu-Yu Chen

Zeev Dvir, also Computer Science

Rupert Frank

Gustav H. Holzegel

Choonghong Oh

Claus M. Sorensen

Nicolas P. Templier

Stefan H.M. van Zwam

Micah W. Warren

Anna K. Wienhard

Xinji Yuan

#### Instructor

Costante Bellettini

Jeffrey S. Case

Michael Damron

Jonathan Fickenscher

Penka V. Georgieva

David Geraghty

Tasho Kaletha

Kai-Wen Lan

Victor Lie

Yu-Han Liu

Luc L. Nguyen

Zsolt Patakfalvi

Claudiu C. Raicu

Kevin F. Tucker

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

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

The usual underclass sequences for mathematics courses are 103-104-201-202, which emphasizes applications, or 103-104-203-204, which provides a mixture of theory and application. For students who are not prepared to begin with 103 there is a two-semester sequence, 101-102, providing an introduction to calculus, which may be followed by 104 and higher courses. For economics majors who wish to take only one semester of sophomore-level mathematics, there is a sequence 103-104-200; however those students who may wish to continue with 300-level mathematics courses should not take this option.

For students with a very high aptitude in mathematics there are honors courses: 211, 214, 215, 217, and 218. Prospective majors are required to take at least one course introducing them to formal mathematical arguments and rigorous proofs; 211 provides this introduction through analysis-based applied mathematics, 214 through number theory, and 215 through one-variable analysis. Students need take only one of these three courses, but those not planning to become mathematics majors are encouraged to take more than one. Those students who have taken 211 or 215 follow this by 204 or 217, for linear algebra, and 218 or 203, for multivariable calculus (211 students need instructor approval for 218). Those students who have taken 214 follow this by 203, for multivariable calculus, and 217 or 204, for linear algebra. The courses in 203 and 204 may be taken in either order.

**Placement. **Students with little or no background in calculus are initially placed in 103, or in 101 if their SAT (mathematics aptitude) test scores indicate that they have insufficient preparation for 103. To qualify for placement in 104 a student should score either a 4 or better on the AB Advanced Placement Examination or should have completed a year of high school calculus and scored 700 or better on the SAT (mathematics aptitude) test. To qualify for placement into 200 or 201 a student should have a 4 or better on the BC Advanced Placement Examination. To place into 203 a 4 on the BC Advanced Placement Examination and a 750 on the SAT (mathematics aptitude) test are required. Students who have a very strong mathematics aptitude and achieve a 5 on the AB or BC Advanced Placement Examination and 760 or better on the SAT (mathematics aptitude) test qualify for placement into 215 or 214.

#### Advanced Placement

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

#### Prerequisites

Except in unusual circumstances, admission to the department requires the student to take, before the junior year, one course from 211, 214, or 215, one course from 217 or 204, and one course from 218 or 203. Prospective mathematics majors should consult their mathematics instructors or the departmental representative about their programs as early as possible, particularly if they enter with advanced placement, and should plan their program from the honors sequence if possible.

#### Program of Study

Students concentrating in mathematics must complete the following requirements:

1. one course in real analysis: 303, 304, 314, 330, 332, 350, 390, or a more advanced course;

2. one course in complex analysis: 317, 331, or a more advanced course;

3. one course in algebra: 322, 323, or a more advanced course;

4. either one course in geometry (325, 326, 327, 328, or a more advanced course), or alternatively one course in discrete mathematics (306, 307, or 308);

5. a choice of an additional four courses at the 300 level or higher to be counted as "departmental courses.'' Up to three of these may be cognate courses outside the mathematics department, with permission from the departmental representative; the remainder are upper-division courses in the mathematics department.

The departmental grade, the average grade on the eight departmental courses, together with grades and reports on independent work are 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 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. The mathematics courses must include the three courses required for the departmental students. For example, some students may be interested in combining work in computer science with a major in mathematics. For those interested in theoretical computer science, a suitable group of courses would be MAT 302, 306, 314, and 317 and COS 423, 487, and 524. For those students interested in applications, COS 217, 318, and 320 would be suitable. COS 425, 426, and 506 would also be appropriate for a mathematics major with an emphasis on computer science.

Excellent computing facilities are available to all students through the University Computing Center and in Fine Hall.

#### 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 101 Calculus Fall

Basic concepts and methods of the differential and integral calculus of elementary functions of one variable, including a review of the necessary techniques from algebra, analytic geometry, and trigonometry. Emphasis on intuitive and graphical understanding, illustrated by problems in many fields. Three classes.
* Staff*

MAT 102 Calculus Spring QR

Continuation of the differential and integral calculus of elementary functions in one variable, including applications. Three classes.
* Staff*

MAT 103 Calculus Fall QR

Basic concepts, methods, and applications of differential and integral calculus of elementary functions of one variable. Three classes.
* Staff*

MAT 104 Calculus Fall, Spring QR

Further techniques and applications of differential and integral calculus. Sequences and series, including Taylor's series with remainder. Complex numbers and functions. Elementary differential equations. Three classes. Prerequisite: 102 or 103 or one term's advanced placement.
*
L. Nguyen*

MAT 151 Introduction to Mathmatical Modeling (see APC 151)

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 200 Linear Algebra and Multivariable Calculus for Economists Fall, Spring QR

This course provides the student with the foundations to pursue studies in economic analysis. Topics include: systems of linear equations, matrices, determinants, Gaussian elimination, Euclidean space, differential vector calculus, the implicit function theorem, quadratic forms, and first- and second-order conditions for unconstrained and constrained optimization. The course stresses the computational aspects of these principles. Three classes. Prerequisite: 104 or instructor's permission.
* Staff*

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. Three classes. Prerequisite: 104, 217, or instructor's permission.
*
J. Kollár*

MAT 202 Linear Algebra with Applications Fall, Spring QR

Euclidean spaces, vector spaces, systems of linear equations, matrices and linear transformations, determinants, eigen values and applications to systems of differential equations, symmetric matrices, and quadratic forms. Differentiable vector functions, the chain rule, inverse and implicit functions, maxima and minima. MAT 104 or instructor's permission. Three classes.
*
Y. Liu*

MAT 203 Advanced Multivariable Calculus Fall QR

Calculus of vector functions in space, gradients, chain rule, curvilinear coordinates, multiple integrals, Stokes' theorem, and applications. Emphasis on both theoretical aspects and problem solving. Recommended for mathematically inclined scientists and engineers. Three classes. Prerequisite: 104 or 217.
* Staff*

MAT 204 Advanced Linear Algebra with Applications Spring QR

Vector spaces, linear transformations, matrices, determinants and systems of linear equations, eigenvalues, inner product spaces, symmetric matrices, and quadratic forms. Applications to calculus in n-dimensional space and systems of differential equations. Three classes. Prerequisite: 203.
* 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 nonmajors interested in exposure to higher mathematics. Three classes.
*
D. Geraghty*

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. Three classes.
*
M. Damron*

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. Three classes.
*
T. Kaletha*

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. Three classes. Prerequisites: 215 and 217, or instructor's permission.
* Staff*

MAT 301 Mathematics in Engineering I (see MAE 305)

MAT 302 Mathematics in Engineering II (see MAE 306)

MAT 303 Ordinary Differential Equations Fall QR

Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. Topics include linear equations with constant coefficients, the Laplace transform, separation and comparison theorems, power series solutions, matrix methods, and stability theorems. Three classes. Prerequisite: 202, 204, or 217.
* Staff*

MAT 304 Introduction to Partial Differential Equations QR

Introduction to the techniques necessary for the formulation and solution of problems involving partial differential equations in the natural sciences and engineering, with detailed study of the heat and wave equations. Topics include method of eigenfunction expansions, Fourier series, the Fourier transform, inhomogeneous problems, the method of variation of parameters. Three classes. Prerequisite: 202, 204, or 218.
*
J. Mather*

MAT 305 Mathematical Programming Fall 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. Three classes. Prerequisite: 200, 202, 204, or 217.
* Staff*

MAT 306 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: 202, 204, or 217, or instructor's permission. Three classes.
*
P. Seymour*

MAT 307 Combinatorial Mathematics (also APC 307) Fall

Introduction to combinatorics, a fundamental mathematical discipline and an essential component of many mathematical areas. While 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. Covers several important areas and techniques such as Ramsey Theory, Turan Theorem and Extremal Graph Theory, Probabilistic Argument, Algebraic Methods and Spectral Techniques. Showcases the gems of modern combinatorics. Two 90-minute classes.
*
R. Calderbank**,
J. Fox*

MAT 308 Theory of Games (also ECO 318) Spring

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. Three classes. Prerequisite: 305 or instructor's permission. Offered in alternate years.
*
G. Todorov*

MAT 309 Probability and Stochastic Systems (see ORF 309)

MAT 310 Mathematical Statistics

The statistical problems of estimation, testing, and decision making will be formulated theoretically, especially in those situations where optimal solutions exist. Conventional and Bayesian methods will be compared. Broadening the usual assumptions leads to robust methods of estimation and testing. Three classes. Prerequisite: 309.
* Staff*

MAT 311 Introduction to Modern Applied Mathematics Spring 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. Three classes.
* Staff*

MAT 312 Mathematical Logic Spring QR

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

MAT 313 Advanced Logic (see PHI 323)

MAT 314 Introduction to Real Analysis Fall QR

Introduction to analysis of n-dimensional space, the Riemann-Stieltjes integral, Lebesgue theory of measure and integration on the line, Fourier series. Three classes. Prerequisite: 218, or 204 and 215, or permission of instructor.
*
D. Beliaev*

MAT 317 Complex Analysis with Applications Fall, Spring QR

Study of functions of a complex variable, with emphasis on techniques and applications. Analytic functions and complex integration, power series, residues with applications to evaluation of integrals, and conformal mapping. Three classes. Prerequisite: 202, 204, or 218.
*
M. Aizenman*

MAT 322 Algebra with Galois Theory Fall QR

Group theory, field extensions, splitting fields, the main theorem of Galois theory, cyclotomic extensions, Kummer extensions, solvability by radicals. Two 90-minute classes. Prerequisites: 202, 204, or 217.
* Staff*

MAT 323 Algebra Fall QR

Group theory: subgroups and quotient groups, examples including permutation groups and linear groups, and the Sylow theorems. Ring theory: ideals, fields of quotients, congruences, Fermat's theorem, modules over principal domains or Euclidean rings; applications to coding theory. Two 90-minute classes. Prerequisite: 202, 204, or 217.
*
R. Calderbank*

MAT 325 Topology Spring QR

Introduction to point-set topology; the characterization and properties of topological spaces, and the study of the fundamental group of a space and of covering spaces. Three classes. Prerequisite: 202, 204, or 218.
* Staff*

MAT 326 Algebraic Topology Fall

Study of homology and homotopy groups of topological spaces, and of selected topics in algebraic topology. Three classes. Prerequisites: MAT 322 or MAT 323, and MAT 325.
*
Z. Szabó*

MAT 327 Introduction to Differential Geometry Fall QR

Differential geometry of curves and surfaces in three-dimensional space. Intrinsic geometry, geodesics, curvature, Gauss-Bonnet theorem. Three classes. Prerequisite: 202, 204, or 218. Offered in alternate years.
*
P. Yang*

MAT 328 Differential Geometry Spring QR

Manifolds, vector bundles, geometric structures, Riemann geometry and applications. Prerequisite: 327.
* Staff*

MAT 330 Analysis I: Fourier Series and Partial Differential Equations

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. Three classes.
*
S. Klainerman*

MAT 331 Analysis II: Complex Analysis Fall

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. Prerequisites: 215, 218, or permission of instructor. Three classes.
*
E. Stein*

MAT 332 Analysis III: Integration Theory and Hilbert Space Spring

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: 215, 218, or permission of instructor. Three classes.
* Staff*

MAT 350 Introduction to Differential Equations (see APC 350)

MAT 351 Topics in Mathematical Modeling (also APC 351) Fall QR

Draws on problems from the sciences and engineering for which mathematical models have been developed to describe, understand, and predict natural and man-made phenomena. Topics vary from year to year, ranging over the physical sciences and biology, including cognitive science and neurobiology. Model-building strategies are described, including level of detail and selection of appropriate mathematical "languages." Analytical and computational results and their implications are described and the question of how applications motivate mathematical developments is addressed. Two 90-minute classes.
*
P. Holmes*

MAT 390 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, percolation theory. Two 90-minute lectures. Prerequisite: 203, 218, or instructor's permission.
*
Y. Sinai*

MAT 391 Random Processes

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. Two 90-minute lectures. Prerequisite: 390.
*
Y. Sinai*

MAT 407 Mathematical Methods of Physics (see PHY 403)

MAT 424 Topics in Algebra Fall, Spring

Selected topics such as: algebraic number theory, algebraic geometry, Noetherian rings, Wedderburn theory, representations of finite groups. Three classes. Prerequisite: MAT 322 or MAT 323.
*
N. Katz*

MAT 433 Analysis IV: Special Topics in Analysis Fall

Selected topics in analysis. Course content may vary from year to year. Topics may include complex variables, Hilbert Space Theory, basic partial differential equations. Prerequisites: 331 and 332, or permission of instructor. Two 90-minute classes.
*
C. Fefferman*

MAT 443 Cryptography (see COS 433)

MAT 447 Theory of Computation (see COS 487)

MAT 448 Introduction to Nonlinear Dynamics (see CBE 448)

MAT 451 Advanced Topics in Analysis Fall

Topics in analysis selected from areas such as functional analysis, operator theory, and harmonic analysis. Content varies from year to year. Three classes. Prerequisite 314 or instructor's permission.
*
E. Lieb*

MAT 452 Advanced Topics in Analysis

Topics in analysis selected from areas such as functional analysis, operator theory, and harmonic analysis. Content varies from year to year. Three classes. Prerequisite 314 or instructor's permission.
* Staff*

MAT 453 Advanced Topics in Algebra Fall

Course on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and L-functions. Prerequisites: MAT 214, MAT 322 or instructor's permission.
*
C. Skinner*

MAT 454 Advanced Topics in Algebra

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 455 Advanced Topics in Geometry

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

MAT 456 Advanced Topics in Geometry

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