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 our Faculty as Invited Speakers page 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 the Events area of our site for a list of department hosted events.

For a quick overview of the mathematics faculty and research areas, see Our People landing page. A more in-depth and picturesque survey can be found on the alphabetic lists of faculty and faculty-by-title pages.

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.

- Workshop on Higher-Order Asymptotics and Post-Selection Inference (Todd Kuffner, NSF). WHOA-PSI focuses on one of the most active and exciting frontiers of statistics. The workshop brings together researchers working in three areas: (i) theory and methodology of statistical inference post model selection; (ii) higher-order asymptotic properties of statistical procedures; and (iii) application areas where both (i) and (ii) are needed. The goals of the workshop are to promote valid and accurate post-selection inference, facilitate collaboration, publicize new results, identify important applications requiring new results, and elevate junior researchers. Within post-selection inference, the workshop includes speakers from a broad spectrum of paradigms, such as high-dimensional inference, data splitting, selective inference, simultaneous inference, and Bayesian post-selection inference. The routes to higher-order accuracy considered include analytic refinements of first-order asymptotic theory, as well as resampling-based refinements.

• International Workshop on Operator Theory and Applications 2016 (Greg Knese, NSF). IWOTA has two main scientific goals. First, it serves as a meeting place for mathematicians in the field of operator theory. Many of the main speakers of the conference are experts in operator theory with recent innovations to share with the broader community. The conference has numerous special sessions to allow subgroups within operator theory (such as free analysis/probability, applied harmonic analysis, and systems and function theory) to collaborate. Second, IWOTA has always had the goal of increasing the interaction of pure operator theorists, applied mathematicians, and engineers. IWOTA 2016 intends to continue this tradition through its main speakers and special sessions.

• CAREER: Bridging High-Frequency Data Analysis and Continuous-time Features of Levy Models (Jose Figueroa-Lopez, NSF). Motivated by recent theoretical findings in the field, the investigator identifies some key open problems of the asymptotic behavior of Levy processes in short time and connects them to two important statistical problems commonly appearing in applications: parametric estimation and change-point detection for Levy models. For finite random samples, methods such as maximum likelihood estimation and cumulative sum (CUSUM) sequential rules are known to be optimal for dealing with the two previously mentioned problems. Although one expects that optimality would be preserved when the time span between consecutive observations of a Levy process shrinks to zero, there exist important examples showing this not always to be the case. The mystery behind these counterintuitive results is closely connected to the "fine" distributional properties of Levy processes in short time. Rather than directly attacking the two proposed problems in continuous time, the investigator builds on the well-studied analogous problems in discrete time and fill in the infinite time continuum by analyzing their evolution when the time span between consecutive observations is made increasingly small. This bottom-up approach is not only appealing but also useful since in practice one would like to determine the performance of statistical methods for high-frequency observations rather than for continuous-time observations, which are arguably never available. The focus on Levy processes is motivated by the fact that the latter are the simplest stochastic models displaying abrupt changes while still preserving the parsimonious statistical properties of their increments. Extensions to other multi-factor stochastic models driven by Levy processes are also contemplated.

• CAREER: An Integrated Proposal Based on the Corona Problem (Brett Wick, NSF). The research objective of this project is to conduct a deeper study of the Corona Problem, using tools and techniques developed in interrelated areas of analysis, with a goal of settling open and important questions. The Corona Problem can be phrased as a question about left invertibility of matrices in particular algebras of analytic functions. Additionally, it has formulations in the areas of operator theory, complex differential geometry, functional analysis, and commutative algebra. The Corona Problem has served as an impetus for research in four main areas of analysis: complex analysis, function theory, harmonic analysis, and operator theory. Additionally, it arises in real-world applications through the use of control theory to engineering questions. This research program utilizes knowledge and techniques from these broad areas of analysis to provide an array of tools with which to approach the challenging questions raised in this project. The proposed research is based on recent, significant contributions made by the principal investigatorr and focuses on key questions connected to the Corona Problem. In particular, the principal investigator will address questions that relate the Corona Problem to complex differential geometry via the curvature of canonical vector bundles associated with the problem. The Corona Problem will additionally be studied for more general multiplier algebras of analytic functions. Finally, the connection with the Corona Problem and control theory will be explored via the computation of the stable rank of rings of analytic functions.

• Geometry of Moduli Spaces of Rational Curves (Roya Beheshti-Zavareh, Simons). We study four inter-related projects in these lines of research: (a) The geometry of spaces of rational curves on general complete intersections: the PI investigates some open questions on the geometry of spaces of rational curves on general complete intersections in projective space (and more generally in homogeneous spaces). (b) Rational curves on special hypersurfaces: the PI investigates under what conditions the space of rational curves in every smooth hypersurface of a given dimension and degree has the expected dimension. (c) Rational surfaces contained in hypersurfaces: this is the continuation of a collaboration of the PI with Jason Starr which studies families of rational surfaces which sweep out smooth hypersurfaces of index 1 in projective space; the original motivation for this study comes from the question of unirationality of such hypersurfaces. (d) Uniruledness of spaces of rational curves: the goal of this project is to investigate for which hypersurfaces there are sweeping components of spaces of rational curves which are uniruled.

• Applications of Harmonic Analysis to Function Theory and Operator Theory (Brett Wick, NSF). This research project will conduct a study of fundamental questions in function theory and operator theory using the tools and techniques of harmonic analysis. The project will address important questions now open to exploration because of recent advances made by the principal investigator and his collaborators. Resolution of these problems raised will find applications in function theoretic operator theory and yield new tools and techniques that can be adopted by the larger analysis community.

• Topological, Enumerative, and Algebraic Combinatorics (John Shareshian, NSF). The current main goal of this project is to show that the Frobenius characteristic of a representation of the symmetric group on the cohomology of a regular semisimple Hessenberg variety is a refinement of the chromatic symmetric function of a graph naturally associated to the variety.

• Conference on Finite Simple Groups and their Applications (John Shareshian, NSA).

• Conference on Algebraic, Enumerative, and Topological Combinatorics (John Shareshian, NSF). This conference will provide an opportunity for established experts and early career mathematicians to share ideas and results in the field of algebraic, enumerative and topological combinatorics. The conference will feature talks on topics at the forefront of current research in the field. Among the likely topics of discussion both in talks and informal discussions are connections between combinatorics and various other areas, including algebraic geometry, topology, commutative algebra, representation theory and probability.

• The Mathematics of Michelle Wachs - Miami 2015 (John Shareshian, NSA).

• Foliations, Flows, and Contact Structures (Rachel Roberts, Simons). This is the study of flows, codimension-one foliations, and contact structures in 3-manifolds. Studies of actions of 3-manifold groups are also done. In recent work with Will Kazez, we show that the techniques of Eliashberg and Thurston extend to taut oriented continuous foliations. Any contact structure sufficiently close to such a foliation is therefore weakly symplectically fillable and hence universally tight. This allows applications of smooth foliation theory to contact topology to be generalized and extended to constructions of continuous foliations. Implicit in this work is the important role of dominating 2-forms, or equivalently (after fixing a Riemannian metric), of volume preserving flows. In recent work with Tao Li and Tejas Kalelkar, we give constructions of taut, oriented continuous foliations. The existence of these foliations has ramifications in the study of L-spaces and also in the study of actions of 3-manifold groups. In recent work with Sergio Fenley, we demonstrate the existence of infinitely many hyperbolic manifolds each of which contains no R-covered foliation but has fundamental group which acts nontrivially and faithfully on R. This answers a question of Gabai and Thurston.

• Harmonic Analysis and Spaces of Analytic Functions in Several Variables (Greg Knese, NSF). The specific topics this project will develop further are (1) characterizations of when a positive trigonometric polynomial can be factored as a single square, (2) weak factorization of Hardy spaces on the polydisk, and (3) operator theoretic models for bounded analytic functions. Applications: multivariate moment problems.

• Noncommutative Geometry and Index Theory (Xiang Tang, NSF). The principal investigator will pursue three research projects in noncommutative geometry and index theory. (1) He will study a long-standing index problem, raised by Alain Connes, about the index of a groupoid elliptic differential operator on a general Lie groupoid. (2) He will continue his program of using the joint force of noncommutative geometry and symplectic topology to study a duality conjecture, inspired from string theory in physics, about gerbes on orbifolds. (3) He will investigate an interesting index problem, motivated by multivariate operator theory, about some essentially normal Hilbert modules using the recent developments in complex analysis and noncommutative geometry.

• Collaborative Research in Hodge Theory, Moduli, and Representation Theory (Matthew Kerr, NSF). The project will develop Hodge theory and apply it to problems in algebraic geometry, number theory and representation theory. The researchers intend to focus on four related topics: (1) Mumford-Tate (MT) domains, (2) moduli spaces, (3) algebraic cycles and the Hodge conjecture, and (4) mixed Hodge modules. (1) MT domains are classifying spaces of Hodge structures, and, roughly speaking, the boundary components of Mumford-Tate domains parametrize degenerations of Hodge structures. The PIs intend to advance number theory, representation theory and algebraic geometry by studying Mumford-Tate domains and their boundary components.

• 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 (John 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.

• Spaces of Rational Curves in Projective Varieties (Roya Beheshti-Zavareh, NSF). In the proposed research, some open questions on the dimension, irreducibility, Kodaira dimension, and several other aspects of the geometry of spaces of rational curves on complete intersections in projective space and other homogeneous varieties are investigated.

• 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.

• Hodge theory, algebraic cycles, and arithmetic (Matt Kerr, 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.

• Problems in Function Theory and Operator Theory (Richard Rochberg, 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, 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, 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, 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, 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.

A list of past research projects can be found on our Faculty Research Projects (Cont.) page.

*Pictured above: Profs. Richard Rochberg and Nets Katz.*