The Geometry/Topology RTG

Discrete groups in algebraic geometry and Lie theory: My current major projects are (1) the classify the possible root lattices for Lorentzian Kac-Moody algebras, and (2) Investigate the conjectural appearance of the monster sporadic finite simple group as the fundamental group of an specific natural orbifold obtained from complex hyperbolic space.

The first problem revolves around Coxeter groups acting on hyperbolic space (my favorite topic) and the second involves an interplay between algebraic geometry, finite group theory, braid groups and complex hyperbolic geometry (my other favorite topics). I'm also interested in related problems, such as the topology of hyperplane complements arising from reflection groups.

I work in the interface between representation theory, algebraic geometry and mathematical physics. The field of representation theory is devoted to the abstract study of symmetries and the possible ways in which they can manifest. Algebraic geometry, on the other hand, is an abstraction of the familiar structures of geometry which is flexible enough to apply to problems that a priori do not seem particularly geometric (in particular in number theory). My specialty, geometric representation theory, combines these two fields to search for universal geometric explanations of the most fundamental symmetries. Mathematical physics is not only a source of phenomena involving symmetry but a source of powerful conceptual frameworks to organize these observations.

I work primarily on problems in stable homotopy theory and more specifically the study of arithmetic and geometric phenomena as measured by algebraic invariants of structured ring spectra. Most notably, I am very interested in the algebraic K-theory of ring spectra and related linearized invariants (e.g., topological Hochschild homology). This work is closely related to the study of the homotopy theory of stable categories (e.g., the spectral or DG-derived category of a scheme). I also work on problems related to the foundations of the theory of Thom spectra.

One of my main area of research is optimal transport: given a mass distribution, find the most effective way of a moving this mass from one place to another, minimizing the transportation cost. Optimal transport theory has already shown to be extremely useful for many geometric applications, for instance for defining a good notion of Ricci curvature bound on metric measure spaces.

Recently, the study of the regularity theory for Monge-Ampere type equations related to optimal transport led to the discovery of a new type of curvature, called Ma-Trudinger-Wang curvature. This notion of curvature, which is still not well understood, has strong relations with the geometry of the cut-locus of the underlying manifold on which the optimal transport problem take place. To illustrate this with an example, together with Rifford and Villani we have been able to exploit this relation to show that all injectivity domains of smooth perturbations of the round sphere are uniformly convex.

I work on problems in geometry and topology, often with an eye towards quantum field theory and more recently topological invariants of condensed matter systems. I am also deeply interested in the mathematical structure of quantum field theories.

I am an algebraic geometer; most of my work has been about moduli spaces (spaces that parameterize geometric objects) and birational geometry (the natural surgery transformations in algebraic geometry). Most recently I have been working at the joint interface of moduli spaces, birational geometry, and mirror symmetry.

I am a geometric analyst. The motivations for my research program are to find and classify optimal geometries for manifolds by using geometric evolution equations like Ricci flow, Kaehler-Ricci flow, and cross curvature flow. Broad goals of this program include a deeper analytic understanding of the nonlinear partial differential equations governing curvature flows, applications to new questions about the geometry and topology of manifolds, insights into parallels between various geometric flows, and applications of curvature flows to problems motivated by materials science and physics.

I work on geometric problems which arise naturally in quantum field theory and string theory. Often this means using the tools of physics to formulate geometric conjectures. Most recently I have been focused on the hyperkahler geometry of integrable systems, which arises naturally in supersymmetric field theory. In the not-too-distant past I worked on the topological string. In the slightly-more-distant past I studied the quasinormal frequencies of black holes.

Low-dimensional topology, especially smooth 4-manifolds; symplectic topology; Floer cohomology and TQFT techniques. Interaction between symplectic and low-dimensional topology occurs not just because some 4-manifolds are symplectic, but because the main tool of the symplectic topologist's trade - the theory of pseudo-holomorphic curves - can be adapted to provide invariants of 3- and 4-manifolds. I have a particular interest in symplectic Lefschetz fibrations and their generalizations, structures which can be used to describe 4- manifolds as families of surfaces. That description leads to a "genetic" encoding of 4-manifolds via sequences of circles on surfaces, and I am interested in how one can understand 4-manifold invariants from these sequences.

Most of my work concerns the topology and dynamical properties of tiling spaces. Tiling theory lies at the crossroads of discrete geometry, combinatorics, and dynamical systems, with additional applications to material science (quasicrystals), number theory (Diophantine approximations), theoretical computer science (automatic sequences), and even neurobiology (grid cells). My other major research interest is in phases and phase transitions in dense random graphs. In addition, I have a catholic interest in problems relating geometry and physics, ranging from adiabatic quantum transport to sphere packing.

During my years as a research mathematician, I have worked on a variety of geometric partial differential equations. In the last few yeras I have spent a great deal of time on integrable systems, with a particular interest in how they might relate to quantum cohomology. Most recently I have renewed my interest in gauge theoretic questions. There are still number of new and old differential equations formulated by physicists which are not well understood. While they may have other significance in mathematics, these equations also pose a challend to mathematicians purely as partial differential equations, and are a souce of new techniques and ideas.