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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