Hyper-Kähler manifolds

I will give an overview of my perspective on the central role of hyper-Kähler manifolds in algebraic geometry both as the natural higher dimensional analogues of K3 surfaces and in terms of their relationship to cubic fourfolds.

Random permutations, partitions and PDEs

We start with a seemingly innocuous question - what do large random permutations look like? Focusing on the structure of their "increasing subsequences" we encounter some remarkable mathematics related to symmetric functions (e.g. Schur and Macdonald), random matrices, and stochastic PDEs. No prior knowledge of any of this will be assumed. (slides)

Multiple Dirichlet Series

The Riemann zeta function and the Dirichlet L-functions are examples of Dirichlet series in one complex variable that were introduced to count primes.The Langlands L-function (associated to a certain infinite dimensional representation) can be thought of as a super generalization of the Riemann zeta function. But it is still a Dirichlet series in one complex variable. Multiple Dirichlet series are Dirichlet series in several complex variables that retain many of the properties of the Langlands L-function such as analytic continuation and functional equations. In this talk I will introduce a family of multiple Dirichlet series associated to root systems of classical Lie groups.

Symplectic flexibility

Symplectic manifolds often have interesting invariants coming from J-holomorphic curves that can be used to construct `exotic' symplectic manifolds, which are diffeomorphic but not symplectomorphic. I will describe recent developments in symplectic flexibilty - that for certain classes of symplectic manifolds diffeomorphic actually implies symplectomorphic- and explain how this phenomenon put constraints on certain J-holomorphic curve invariants.

Singularity formation in geometric flows

Geometric evolution equations like the Ricci flow and the mean curvature flow have played a central role in differential geometry. The main problem is to understand singularity formation. In other words, we want to know what happens in regions where the curvature becomes large. From one point of view, singularities are undesirable in that they can prevent us from continuing the solution. On the other hand, singularities tend to have a very deep special structure, which only reveals itself as the curvature becomes large. I will discuss some of the broad ideas underlying this theory, and indicate some recent developments.

The Optimal Transport Problem

The spherical Hecke algebra

The spherical Hecke algebra is a wonderful confluence of representation theory, algebraic geometry, combinatorics, and number theory. In this talk I will introduce the three natural bases of the algebra, and discuss how the transition matrices between these bases encode rich and deep information. The talk will be based on concrete examples.

Heegaard Floer homology

In the early 2000s Peter Ozsváth and Zoltan Szabó constructed Heegaard Floer homology, a package of powerful invariants of smooth 3- and 4-manifolds and knots. Over the last decade, it has become a central tool in low-dimensional topology and used extensively to study important questions concerning unknotting number, slice genus, knot concordance, Dehn surgery, taut foliations, contact structures, and smooth 4-manifolds. I will give an overview of the construction of Heegaard Floer homology, provide concrete examples and discuss some of the topological applications.

Some examples from the zoo of modular forms: From Maass forms and indefinite theta functions to automorphic forms on exceptional groups

I will give a survey of some rather disparate results on modular forms. I will begin discussing modular forms, in particular theta functions, from a very classical and concrete point of view, and will eventually move to the language of automorphic representations. The first portion of the talk will concentrate on joint work with Matt Krauel and Larry Rolen, and the latter part of the talk will be on joint work with Gordan Savin.

Conformally invariant random structures

In many natural situations, random objects arising as scaling limits are expected to be invariant under rotation and scaling. They can then be expected to be invariant under transformations that are locally to first order compositions of rotation and scaling, i.e. conformal maps. In two dimensions, there is an abundance of such conformal maps, which puts strong constraints on conformally invariant random structures. We'll discuss some examples and recent developments in that direction.

Nonlinear Geometric flows

We will give an overview on the development of extrinsic Nonlinear Geometric Flows, emphasizing both the analytical and geometric point of views. Special emphasis will be given to the Mean curvature flow (an example of quasilinear diffusion), the Inverse Mean curvature flow (an example of ultra-fast diffusion) and to the Gauss curvature flow (an example of slow-diffusion).

Diagrammatic Hecke category

The Hecke category has played an important role in geometric representation theory since its very beginning. More recently, a diagrammatic version of this category has emerged. I will describe this diagrammatic Hecke category and discuss some applications to representation theory.

Stratification of schemes

In this talk I will introduce the affine statification number on schemes. I will go over some examples and properties, and prove a bound on its size. The results are based on a joint paper by Mike Roth and Ravi Vakil.

Floer theory and the Arnol'd conjecture