# Mathematics

The Department of Mathematics offers one master's degree in mathematics and one doctoral degree in mathematics. The areas of study for mathematics include algebra, algebraic geometry, real and complex analysis, differential geometry, and topology. Because it is difficult to make up coherent programs for students entering in the middle of the year, students are ordinarily admitted only in the fall.

When they first arrive, graduate students have the opportunity to share common concerns and to become acquainted. One of the most attractive features of our program is the friendly and supportive atmosphere that develops among our graduate students. Advanced courses in the Washington University Mathematics department can build on the common background shared by all students. As a result, these courses are richer and nearer to the level of PhD work than typical advanced courses.

Students typically complete the PhD program in five years, and those students may expect up to five years of support. Continuation of support each year is dependent upon normal progress toward the degree and the satisfactory performance of duties. A student who comes to Washington University with advanced preparation may finish in less time. On the other hand, some students find that it is advisable for them to take preparatory math courses before attempting the qualifying courses. In special cases, the time schedule may be lengthened accordingly. Each student should plan to develop a close relationship with their thesis advisor so that the advisor may have a realistic idea of the student's progress.

Graduate study in mathematics is not for everyone. Entering students usually find that the time and effort required to succeed goes well beyond anything they encountered as undergraduates. Success requires both ample mathematical ability and the determination to grapple with a subject for many days or weeks until the light of understanding shines through, and the experience can be daunting. Those who continue in their studies are largely those for whom the pleasure of attaining that understanding more than compensates for the required effort. For such persons, the life of a mathematician can be richly rewarding.

**Email:** Gregory Knese, Director of Graduate Studies, or Mary Ann Stenner

## Contact Info

Phone: | 314-935-6760 |

Website: | https://math.wustl.edu/graduate |

### Chair

**John Shareshian**

Professor

PhD, Rutgers University

Algebraic and topological combinatorics

### Director of Graduate Studies

**Gregory Knese**

Professor

PhD, Washington University

Complex function theory; operators; harmonic analysis

### Director of Undergraduate Studies

**Ari Stern**

Professor

PhD, California Institute of Technology

Geometric numerical analysis; computational mathematics

### Associate Director of Undergraduate Studies

**Blake Thornton**

Teaching Professor

PhD, University of Utah

Geometric topology

### Department Faculty

**Roya Beheshti Zavareh**

Professor

PhD, Massachusetts Institute of Technology

Algebraic geometry

**Alan Chang**

Assistant Professor

PhD, University of Chicago

Geometric measure theory; harmonic analysis

**Quo-Shin Chi**

Professor

PhD, Stanford University

Differential geometry

**Lawrence Conlon**

Emeriti Professor

PhD, Harvard University

Differential topology

**Aliakbar Daemi**

Assistant Professor

PhD, Harvard University

Gauge theory; low-dimensional topology; symplectic geometry

**Laura Escobar Vega**

Associate Professor

PhD, Cornell University

Combinatorics; algebraic geometry

**Renato Feres**

Professor

PhD, California Institute of Technology

Differential geometry; dynamical systems

**Steven Frankel**

Associate Professor

PhD, University of Cambridge

Geometric topology; dynamics

**Ron Freiwald**

Emeriti Professor

PhD, University of Rochester

General topology

**Andrew Walton Green**

William Chauvenet Postdoctoral Lecturer

PhD, Clemson University

Harmonic analysis; partial differential equations

**Gary R. Jensen**

Emeriti Professor

PhD, University of California, Berkeley

Differential geometry

**Silas Johnson**

Senior Lecturer

PhD, University of Wisconsin–Madison

Algebraic number theory; arithmetic statistics

**Matt Kerr**

Professor

PhD, Princeton University

Algebraic geometry; Hodge theory

**Steven G. Krantz**

Professor

PhD, Princeton University

Several complex variables; geometric analysis

**N. Mohan Kumar**

Emeriti Professor

PhD, Bombay University

Algebraic geometry; commutative algebra

**Wanlin Li**

Assistant Professor

PhD, University of Wisconsin–Madison

Number theory; arithmetic geometry

**Henri Martikainen**

Associate Professor

PhD, University of Helsinki, Finland

Harmonic analysis; geometric measure theory

**John E. McCarth**

Spencer T. Olin Professor of Mathematics

PhD, University of California, Berkeley

Analysis; operator theory; one and several complex variables

**Minh Nguyen**

Postdoctoral Lecturer

PhD, University of Arkansas

Gauge theory; low dimensional topology

**Charles Ouyang**

Assistant Professor

PhD, Rice University

(Higher) Teichmuller theory; Riemann surfaces; harmonic maps and minimal surfaces

**Martha Precup**

Associate Professor

PhD, University of Notre Dame

Applications of Lie theory to algebraic geometry and the related combinatorics

**Donsub Rim**

Assistant Professor

PhD, University of Washington

Applied mathematics

**Rachel Roberts**

Elinor Anheuser Professor of Mathematics

PhD, Cornell University

Low-dimensional topology

**Richard Rochberg**

Emeriti Professor

PhD, Harvard University

Complex analysis; interpolation theory

**Angel Roman**

Postdoctoral Lecturer

PhD, Pennsylvania State University

Representation theory; operator algebras

**Jesus Sanchez**

Postdoctoral Lecturer

PhD, Pennsylvania State University

Noncommutative index theory; cyclic cohomology; spin Riemannian geometry; high-dimensional gauge theory

**Karl Schaefer**

Lecturer

PhD, University of Chicago

Algebraic number theory

**Jack Shapiro**

Emeriti Professor

PhD, City University of New York

Algebraic K-theory

**Edward Spitznagel**

Emeriti Professor

PhD, University of Chicago

Statistics; statistical computation; application of statistics to medicine

**Yanli Song**

Associate Professor

PhD, Pennsylvania State University

Noncommutative geometry; symplectic geometry; representation theory

**Xiang Tang**

Professor

PhD, University of California, Berkeley

Symplectic geometry; noncommutative geometry; mathematical physics

**Joel Villatoro**

Postdoctoral Lecturer

PhD, University of Illinois at Urbana-Champaign

Differential geometry; Poisson geometry; singular spaces

**Brett Wick**

Professor

PhD, Brown University

Complex analysis; harmonic analysis; operator theory; several complex variables

**Mladen Victor Wickerhauser**

Professor

PhD, Yale University

Harmonic analysis; wavelets; numerical algorithms for data compression

**Edward N. Wilson**

Emeriti Professor

PhD, Washington University

Harmonic analysis; differential geometry

**David Wright**

Emeriti Professor

PhD, Columbia University

Affine algebraic geometry; polynomial automorphisms

**Jay Yang**

Postdoctoral Lecturer

PhD, University of Wisconsin–Madison

Commutative algebra; algebraic geometry

Visit online course listings to view semester offerings for L24 Math.

**L24 Math 501C Theoretical Physics**

The first part of a two-semester course reviewing the mathematical methods essential for the study of physics. Theory of functions of a complex variable, residue theory; review of ordinary differential equations; introduction to partial differential equations; integral transforms. Prerequisite: undergraduate differential equations (Math 217), or permission of instructor.

Same as L31 Physics 501

Credit 3 units.

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**L24 Math 5021 Complex Analysis I**

An intensive course in complex analysis at the introductory graduate level. Math 5021 and Math 5022 form the basis for the Ph.D. qualifying exam in complex analysis. Prerequisite: Math 4111, 4171 and 4181, or permission of the instructor.

Credit 3 units.

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**L24 Math 5022 Complex Analysis II**

Continuation of Math 5021. Prerequisite, Math 5021 or permission of intstructor.

Credit 3 units.

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**L24 Math 502C Methods of Theoretical Physics II**

Continuation of Phys 501. Introduction to function spaces; self-adjoint and unitary operators; eigenvalue problems, partial differential equations, special functions; integral equations; introduction to group theory. Prerequisite: Phys 501, or permission of instructor.

Same as L31 Physics 502

Credit 3 units.

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**L24 Math 5031 Algebra I**

An introductory graduate level course on the basic structures and methods of algebra. Detailed survey of group theory including the Sylow theorems and the structure of finitely generated Abelian groups, followed by a study of basic ring theory and the Galois theory of fields. Math 5031 and Math 5032 form the basis for the Ph.D. qualifying exam in algebra. Prerequisite: Math 430 or the equivalent, or permission of the instructor.

Credit 3 units.

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**L24 Math 5032 Algebra II**

Continuation of Math 5031. Prerequisite: Math 5031 or permission of instructor.

Credit 3 units.

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**L24 Math 5041 Geometry I**

Introductory graduate level course including differential calculus in n-space; differentiable manifolds; vector fields and flows; differential forms and calculus on manifolds; elements of Lie groups and Lie algebras; Frobenius theorem; elements of Riemannian geometry. Math 5041 and Math 5042 (or 5043) form the basis for the Ph.D. qualifying exam in geometry / topology. Prerequisites: Math 4121, 429, and 4181, or permission of the instructor.

Credit 3 units.

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**L24 Math 5042 Geometry II**

Continuation of Math 5041. Math 5042 and Math 5043 are offered in alternate spring semesters as a sequel to Math 5041. Prerequisite: Math 5041 or permission of instructor.

Credit 3 units.

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**L24 Math 5045 Geometry/Topology I: Algebraic Topology**

An introductory graduate-level course in algebraic topology, including fundamental groups, covering spaces, homology, and cohomology. Prerequisites: undergraduate courses in abstract algebra and point-set topology or permission from the instructor. Replaces 5043.

Credit 3 units.

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**L24 Math 5046 Geometry/Topology II: Differential Topology**

An introductory graduate-level course in the topology of smooth manifolds and vector bundles. Prerequisites: Math 5045 (GT I: Algebraic Topology) or permission from the instructor. Replaces 5041.

Credit 3 units.

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**L24 Math 5047 Geometry/Topology III: Differential Geometry**

An introductory graduate-level course in the geometry of smooth manifolds and vector bundles. Prerequisites: Math 5046 (Geometry/Topology II: Differntial Topology) or permission from the instructor. Replaces 5042.

Credit 3 units.

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**L24 Math 5051 Measure Theory and Functional Analysis I**

Introductory graduate level course including the theory of integration in Euclidean and abstract spaces, and an introduction to the basic ideas of functional analysis. Math 5051 and Math 5052 form the basis for the Ph.D. qualifying exam in real analysis. Prerequisites: Math 4111, 4171, and 4181, or permission of the instructor.

Credit 3 units.

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**L24 Math 5052 Measure Theory and Functional Analysis II**

Continuation of Math 5051. Prerequisite: Math 5051 or permission of instructor.

Credit 3 units.

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**L24 Math 510 Introduction to Fourier Series and Integrals**

The basic theory of Fourier series and Fourier integrals including different types of convergence. Applications to certain differential equations. Prerequisites: Math 4111 or permission of instructor.

Same as L24 Math 410

Credit 3 units. A&S IQ: NSM

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**L24 Math 5101 Introduction to Analysis**

The real number system and the least upper bound property; metric spaces (completeness, compactness, and connectedness); continuous functions (in R^n; on compact spaces; on connected spaces); C(X) (pointwise and uniform convergence; Weierstrass approximation theorem); differentiation (mean value theorem; Taylor's theorem); the contraction mapping theorem; the inverse and implicit function theorems. Prerequisite: Math 310 or permission of instructor.

Same as L24 Math 4111

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 5102 Introduction to Lebesgue Integration**

Riemann integration; measurable functions; measures; Lebesgue measure; the Lebesgue integral; integrable functions; L^p spaces; modes of convergence; decomposition of measures; product measures. Prerequisite: Math 4111 or permission of the instructor.

Same as L24 Math 4121

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 515 Theory of Partial Differential Equations I**

A rigorous mathematical study of topics in partial differential equations. Prerequisites: Math 5051 and Math 5052 or equivalent. Some knowledge of complex analysis will also be useful. No prior knowledge of partial differential equations is required.

Credit 3 units.

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**L24 Math 5160 Complex Variables**

Analytic functions, elementary functions and their properties, line integrals, the Cauchy integral formula, power series, residues, poles, conformal mapping and applications. Prereq: Math 310 and (Math 318 or Math 4111), or permission of instructor.

Same as L24 Math 416

Credit 3 units. A&S IQ: NSM Art: NSM

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**L24 Math 5201 Topology I**

An introduction to the most important ideas of topology. Course includes necessary ideas from set theory, topological spaces, subspaces, products and quotients, compactness and connectedness. Some time is also devoted to the particular case of metric spaces (including topics such as separability, completeness, completions, the Baire Caregory Theorem, and equivalents of compactness in metric spaces). Prerequisite: Math 4111 or permission of instructor.

Same as L24 Math 4171

Credit 3 units. A&S IQ: NSM Art: NSM

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**L24 Math 5202 Topology II**

A continuation of Math 4171 featuring more advanced topics in topology. The content may with each offering. Prerequisite: Math 4171, or permission of instructor.

Same as L24 Math 4181

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 523C Information Theory**

Discrete source and channel model, definition of information rate and channel capacity, coding theorems for sources and channels, encoding and decoding of data for transmission over noisy channels. Corequisite: ESE 520.

Same as E35 ESE 523

Credit 3 units. EN: BME T, TU

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**L24 Math 5301 Linear Algebra**

This course is an introduction to the linear algebra of finite-dimensional vector spaces. It includes systems of equations, matrices, determinants, inner product spaces, and spectral theory. Prerequisite: Math 310 or permission of instructor. Math 309 is not an explicit prerequisite, but students should already be familiar with such basic topics from matrix theory as matrix operations, linear systems, row reduction, and Gaussian elimination. (Material on these topics in early chapters of the text will be covered very quickly.)

Same as L24 Math 429

Credit 3 units. A&S IQ: NSM Art: NSM

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**L24 Math 5302 Modern Algebra**

Introduction to groups, rings, and fields. Includes permutation groups, group and ring homomorphisms, field extensions, connections with linear algebra. Prerequisite: Math 310, Math 429 or permission of the instructor.

Same as L24 Math 430

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 539 Topics in Algebraic Geometry**

Selected topics in algebraic geometry.

Credit 3 units.

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**L24 Math 541 Topics in Applied Mathematics**

Topic and prerequisites vary with each offering of the course.

Credit 3 units.

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**L24 Math 547 Topics in Geometry**

An introduction to Geometric Group Theory, concentrating on the theory of hyperbolic groups and group boundaries.

Credit 3 units.

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**L24 Math 550 Topics in Number Theory: Analytic Number Theory**

The course will develop analytic methods for problems which occur in algebraic number theory and algebraic geometry. We will consider Riemann zeta function, Dirichlet L-functions, multiple zeta functions, multiple DirichetL-functions (according to Manin), polylogarithms, reciprocity laws on curves and surfaces and multiple Dedekind zeta functions. Prerequisite: Permission of Instructor

Credit 3 units.

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**L24 Math 5501 Numerical Applied Mathematics**

Computer arithmetic, error propagation, condition number and stability; mathematical modeling, approximation and convergence; roots of functions; calculus of finite differences; implicit and explicit methods for initial value and boundary value problems; numerical integration; numerical solution of linear systems, matrix equations, and eigensystems; Fourier transforms; optimization. Various software packages may be introduced and used. Prerequisites: Math 217 or 312, Math 309, Math 310 and CSE 131 (or other computer background with permission of the instructor).

Same as L24 Math 449

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 5502 Topics in Applied Mathematics**

Topic may vary with each offering of the course. Prerequisite: CSE 131 and, Math 449, or permission of the instructor.

Same as L24 Math 450

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 5560 Topics in Financial Mathematics**

An introduction to the principles and methods of financial mathematics, with a focus on discrete-time stochastic models. Topics include no-arbitrage pricing of financial derivatives, risk-neutral probability measures, the Cox-Ross-Rubenstein and Black-Scholes-Merton options pricing models, and implied volatility. Prerequisites: Math 233, Math 3200, Math 310 or permission of instructor.

Same as L24 Math 456

Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM

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**L24 Math 597 Teaching Seminar**

Principles and practice in the teaching of mathematics at the college and university level. Prerequisite: graduate standing, or permission of instructor.

Credit 1 unit.

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**L24 Math 598 Mathematical Professional Development**

This course includes topics on professional development and responsible conduct of research. Prerequisites: none.

Credit 1 unit.

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