A significant measure of outside recognition of the mathematical activities at Washington University is the grant support held by its faculty. In the past five years, several dozen research and research-related projects conducted by Washington University faculty were funded by both federal and private agencies. These have included research grants, grants to purchase computing equipment, grants to run conferences, grants to develop educational projects, grants to develop software and algorithms, and grants to develop collaborations with engineering, statistics, the medical school, and other departments.

Also, Washington University mathematics faculty members are frequent speakers and visitors at other universities around the world. You may visit the PEOPLE area of our website to view the Faculty as Invited Speakers page (under Faculty) for details on where you may hear and meet one of our professors on travel. Our faculty members conduct seminars, host colloquia, and organize conferences having to do with their research interests. See our list of department hosted EVENTS for details.

For a quick overview of the mathematics faculty and research areas, visit the PEOPLE area of our website. A more in-depth and picturesque survey can be found on the alphabetic lists of faculty pages, also in the PEOPLE area.

A list of recent publications by members of our faculty can be found here.

We now summarize just a few of the research activities recently supported by grants from or contracts with the National Science Foundation or similar agencies.

- Collaborative research in Hodge Theory, Moduli, and Representation Theory (Matthew Kerr, NSF).
- Collaborative research in geometric numerical analysis (Ari Stern, Simons). The cross-disciplinary topic of geometric numerical analysis lies at the intersection of geometry, analysis, and computation. This project proposes to use insights from geometry in order to develop novel numerical methods for differential equations, along with techniques to analyze them.
- Multivariable Operator Theory and Applications (McCarthy, NSF). Holomorphic functions of a single variable can easily be applied to matrices whose spectrum lies in the domain of a function, and this has developed into a very successful and important theory. McCarthy is studying functions of several variables applied to matrices. The theory immediately splits into the commutative and non-commutative cases. In the former, he is investigating when functions of several variables preserve the natural order structure on matrices, and what happens when the size of the matrices becomes infinite (the operator case). Conversely, he is using tools from operator theory to understand functions of several variables, such as characterizing asymptotic expansions of functions in the Pick class (this is called the Hamburger problem in one variable). In the non-commutative case, he is investigating how non-commutative holomorphic functions can be obtained as limits of non-commutative polynomials. He is also working on applying analytic techniques to improve ultrasound imaging.
- Geometry of projective varieties (Roya Beheshti-Zavareh, Simons). The focus of this project is to investigate questions on moduli spaces of rational curves contained in smooth projective varieties in particular smooth hypersurfaces. The study of geometric properties of these moduli spaces sheds light on the arithmetic and geometric properties of hypersurfaces, and plays an important role in modern enumerative geometry as well as in birational and higher dimensional geometry.
- Noncommuntative Geometry of Orbifolds (Xiang Tang, NSA). Orbifolds are interesting geometric objects showing up naturally in the study of many branches of Mathematics. In this project, we will use noncommutative geometry tools to study orbifolds. In particular, we will study the deformation theory of orbifold algebras and its connections to the geometry and topology of orbifolds.
- Multivariable Operator Theory and Applications (McCarthy, NSF). Holomorphic functions of a single variable can easily be applied to matrices whose spectrum lies in the domain of a function, and this has developed into a very successful and important theory. McCarthy is studying functions of several variables applied to matrices. The theory immediately splits into the commutative and non-commutative cases. In the former, he is investigating when functions of several variables preserve the natural order structure on matrices, and what happens when the size of the matrices becomes infinite (the operator case). Conversely, he is using tools from operator theory to understand functions of several variables, such as characterizing asymptotic expansions of functions in the Pick class (this is called the Hamburger problem in one variable). In the non-commutative case, he is investigating how non-commutative holomorphic functions can be obtained as limits of non-commutative polynomials. He is also working on applying analytic techniques to improve ultrasound imaging.
- Geometry of projective varieties (Roya Beheshti-Zavareh, SIMONS). The focus of this project is to investigate questions on moduli spaces of rational curves contained in smooth projective varieties in particular smooth hypersurfaces. The study of geometric properties of these moduli spaces sheds light on the arithmetic and geometric properties of hypersurfaces, and plays an important role in modern enumerative geometry as well as in birational and higher dimensional geometry.
- Hodge theory, algebraic cycles, and arithmetic (Matt Kerr, Algebra and Number Theory Program, NSF). This project studies aspects of the theory of period domains and generalized algebraic cycles related to Calabi-Yau varieties and degenerations thereof. It has a particular focus on problems with applications to number theory and physics: automorphic cohomology; irrationality proofs; string dualities; and asymptotics of instanton numbers.
- Symplectic and Spectral Theory of Integrable Systems (Alvaro Pelayo, Geometric Analysis Program, NSF). Investigation of classical and quantum semitoric systems: verifying that the semiclassical joint spectrum of a quantum semitoric system determines completely the system (this is the Spectral Conjecture, widely considered the most spectacular problem in the area); furthering the study of semitoric systems in a more general context, in particular as it regards to the study of the convexity and connectivity properties of the singular Lagrangian fibrations which semitoric systems induce, and which are of special interest in mirror symmetry.
- Problems in Function Theory and Operator Theory (Richard Rochberg, Analysis Program, NSF). This group is working on specific problems in the operator theoretic function theory of spaces of holomorphic functions which are subspaces of potential spaces.
- Operator Theory and Complex Analysis (John McCarthy, Analysis Program, NSF). This group studies the interaction between operator theory and complex analysis in one and several variables. Operator theory can shed light on problems like the boundary behavior of holomorphic functions. Conversely, one can use a combination of function theory and operator theory to help analyze functions that are matrix monotone or matrix convex.
- Noncommutative Geometry: Its Applications to Geometry and Analysis (Xiang Tang, Geometric Analysis Program, NSF). This group studies various problems in geometry using noncommutative geometric tools. This includes index theory on singular spaces, duality of gerbes on orbifolds, Rankin-Cohen deformations and Hopf algebras.
- Algebraic, topological and enumerative combinatorics (John Shareshian, Algebra, number theory, and combinatorics Program, NSF). This group examines a class of lattices appearing in a conjecture that specializes both Shareshian's topological conjecture that there is some finite lattice L such that there is no finite group G whose subgroup lattice contains an interval isomorphic with L, and a combinatorial conjecture of M. Aschbacher.
- Statistical Aggregation in Massive Data Environments (Nan Lin, Statistics Program, NSF). This group develops statistically sound compression and aggregation methods for advanced statistical analysis of data cubes and data streams, uses the compression-then-aggregation strategy to improve computational efficiency of certain statistical analyses, and develops associated asymptotic theories.