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. 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
Brett Wick
Robert G. and Maxine W. Scheibe Professor of Mathematics
PhD, Brown University
Complex analysis; harmonic analysis; operator theory; several complex variables
Director of Graduate Studies
Matt Kerr
Professor
PhD, Princeton University
Algebraic geometry; Hodge theory
Director of Undergraduate Studies
Xiang Tang
Professor
PhD, University of California, Berkeley
Symplectic geometry; noncommutative geometry; mathematical physics
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
Aliakbar Daemi
Associate Professor
PhD, Harvard University
Gauge theory; low-dimensional topology; symplectic geometry
Parker Evans
William Chauvenet Postdoctoral Lecturer
PhD, Rice University
Differential geometry
Renato Feres
Professor
PhD, California Institute of Technology
Differential geometry; dynamical systems
Ben Foster
William Chauvenet Postdoctoral Lecturer
PhD, Stanford University
Elliptic PDEs; harmonic analysis; geometric measure theory
Steven Frankel
Associate Professor
PhD, University of Cambridge
Geometric topology; dynamics
Ron Freiwald
Professor Emeritus
PhD, University of Rochester
General topology
Gary R. Jensen
Professor Emeritus
PhD, University of California, Berkeley
Differential geometry
Silas Johnson
Senior Lecturer
PhD, University of Wisconsin–Madison
Algebraic number theory; arithmetic statistics
Gregory Knese
Professor
PhD, Washington University
Complex function theory; operators; harmonic analysis
Steven G. Krantz
Professor Emeritus
PhD, Princeton University
Several complex variables; geometric analysis
N. Mohan Kumar
Professor Emeritus
PhD, Bombay University
Algebraic geometry; commutative algebra
Carl Lian
Assistant Professor
PhD, Columbia University
Algebraic geometry; combinatorics
Hsin-Chieh Liao
William Chauvenet Postdoctoral Lecturer
PhD, University of Miami
Algebraic, enumerative and topological combinatorics
Henri Martikainen
Associate Professor
PhD, University of Helsinki, Finland
Harmonic analysis; geometric measure theory
John E. McCarthy
Spencer T. Olin Professor of Mathematics
PhD, University of California, Berkeley
Analysis; operator theory; one and several complex variables
Ilyas Mustapha
Postdoctoral Lecturer
PhD, Kansas State University
Computational and theoretical mechanics; numerical analysis; scientific machine learning; variational inequalities
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
Karl Schaefer
Senior Lecturer
PhD, University of Chicago
Algebraic number theory
Jack Shapiro
Professor Emeritus
PhD, City University of New York
Algebraic K-theory
John Shareshian
Professor
PhD, Rutgers University
Algebraic and topological combinatorics
Yanli Song
Associate Professor
PhD, Pennsylvania State University
Noncommutative geometry; symplectic geometry; representation theory
Ari Stern
Professor
PhD, California Institute of Technology
Geometric numerical analysis; computational mathematics
Brandon Sweeting
Postdoctoral Lecturer
PhD, University of Cincinnati
Harmonic analysis; operator theory
Mladen Victor Wickerhauser
Professor
PhD, Yale University
Harmonic analysis; wavelets; numerical algorithms for data compression
Edward N. Wilson
Professor Emeritus
PhD, Washington University
Harmonic analysis; differential geometry
David Wright
Professor Emeritus
PhD, Columbia University
Affine algebraic geometry; polynomial automorphisms
MATH 5011 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring
MATH 5012 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring
MATH 5015 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.
Credit 3 units. A&S IQ: NSM
Typical periods offered: Fall
MATH 5016 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring
MATH 5021 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Fall
MATH 5022 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring
MATH 5031 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.)
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Fall, Spring
MATH 5032 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring
MATH 5051 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).
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Fall
MATH 5052 Topics in Applied Mathematics
Topic may vary with each offering of the course. Prerequisite: CSE 131 and, Math 449, or permission of the instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Fall, Spring
MATH 5056 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.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Fall
MATH 5057 The Mathematics of Quantum Theory
An introduction to the mathematical foundations of quantum theory aimed at advanced undergraduate/beginning graduate students in Mathematics and Engineering, although students from other disciplines are equally welcome to attend. Topics include: the mathematical postulates of quantum theory and simple physical systems, spectral theory of self-adjoint operators, rudiments of Lie groups, Lie algebras and unitary group representations, elements of quantum probability and quantum information theory. Prerequisites: Linear algebra at the level of Math 429 or equivalent, multivariate calculus at the level of Math 318, and basic probability theory at the undergraduate level such as Math 493 or instructor's permission.
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring
MATH 5090 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.
Typical periods offered: Fall, Spring
MATH 5095 Mathematical Professional Development
This course includes topics on professional development and responsible conduct of research. Prerequisites: none.
Credit 1 unit.
Typical periods offered: Fall
MATH 5121 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.
Typical periods offered: Fall
MATH 5122 Complex Analysis II
Continuation of Math 5021. Prerequisite, Math 5021 or permission of intstructor.
Credit 3 units.
Typical periods offered: Spring
MATH 5151 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.
Typical periods offered: Fall
MATH 5152 Measure Theory and Functional Analysis II
Continuation of Math 5051. Prerequisite: Math 5051 or permission of instructor.
Credit 3 units.
Typical periods offered: Spring
MATH 5190 Topics in Analysis
This course will focus on the interplay between operator theory and complex analysis, in one and several variables.
Credit 3 units.
Typical periods offered: Fall, Spring
MATH 5193 Topics in Complex Variables
Selected topics in complex variables
Credit 3 units.
Typical periods offered: Fall
MATH 5195 Harmonic Analysis
Math 519 will be an advanced course in harmonic analysis. Topics covered include the basics of the theory of Calderon-Zygmund operators, Maximal function, and Littlewood-Paley Theory. Special emphasis will be placed upon the connections and differences between one parameter and multiparameter harmonic analysis.
Credit 3 units.
Typical periods offered: Fall
MATH 5197 Functional Analysis
Course description TBD.
Credit 3 units.
Typical periods offered: Fall
MATH 5221 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.
Typical periods offered: Fall
MATH 5222 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.
Typical periods offered: Spring
MATH 5223 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.
Typical periods offered: Fall
MATH 5290 Topics in Geometry
An introduction to Geometric Group Theory, concentrating on the theory of hyperbolic groups and group boundaries.
Credit 3 units.
Typical periods offered: Fall, Spring
MATH 5293 Topics in Riemannian Geometry
A selection of topics on the geometry and dynamics of low-dimensional manifolds, including the Thurston norm and the interaction between flows and foliations.
Credit 3 units.
Typical periods offered: Fall
MATH 5295 Topics in Topology
Course description TBD.
Credit 3 units.
Typical periods offered: Fall, Spring
MATH 5331 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.
Typical periods offered: Fall
MATH 5332 Algebra II
Continuation of Math 5031. Prerequisite: Math 5031 or permission of instructor.
Credit 3 units.
Typical periods offered: Spring
MATH 5333 Algebraic Geometry
Introduction to affine and projective algebraic varieties, the Zariski topology, regular and rational mappings, simple and singular points, divisors and differential forms, genus, the Riemann-Roch theorem.
Credit 3 units.
Typical periods offered: Spring
MATH 5390 Topics in Algebra
Selected topics in algebra vary by semester.
Credit 3 units.
Typical periods offered: Fall
MATH 5393 Topics in Algebraic Geometry
Selected topics in algebraic geometry.
Credit 3 units.
Typical periods offered: Fall, Spring
MATH 5480 Topics in Statistics
Topics vary semester to semester
Credit 3 units. A&S IQ: NSM Art: NSM
Typical periods offered: Spring, Summer
MATH 5510 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.
Typical periods offered: Spring
MATH 5590 Topics in Applied Mathematics
Topic and prerequisites vary with each offering of the course.
Credit 3 units.
Typical periods offered: Fall
MATH 5595 Topics in Applied Mathematics
Course description TBD.
Credit 3 units.
Typical periods offered: Fall
MATH 5910 Research
Independent Research for Credit.
Credit 3 units.
Typical periods offered: Fall, Spring, Summer
MATH 6000 Master's Continuing Student Status
Course description TBD.
Credit 0 units.
Typical periods offered: Fall, Spring
MATH 6010 Master's Nonresident
Course description TBD.
Credit 0 units.
Typical periods offered: Fall, Spring
MATH 6020 Master's Resident
Course description TBD.
Credit 0 units.
Typical periods offered: Fall, Spring
MATH 8000 Doctoral Continuing Student Status
Course description TBD.
Credit 0 units.
Typical periods offered: Fall, Spring
MATH 8010 Doctoral Nonresident
Course description TBD.
Credit 0 units.
Typical periods offered: Fall, Spring
MATH 8020 Doctoral Resident
Course description TBD.
Credit 0 units.
Typical periods offered: Fall, Spring