# Seminars & Events for Department Colloquium

##### Open-closed Gromov-Witten theory

Lagrangian Floer theory studies boundary valued problem for the maps from bordered Riemann surface with boundary condition given Lagrangian submanifold. Gromov-Witten theory studies pseudo-holomorphic maps from Riemann surface. Various relations between them is found and is expected to be useful both in symplectic geometry and the study of Mirror symmetry. I want to discuss some of such applications and several basic constructions we need for it.

##### Renaissance of the h-principle in symplectic topology

Flexible and rigid methods coexisted in Symplectic Topology since its inception. The developmnet during last 20 years was dominated by rigid results, with accidental flexible breakthroughs (such as Donaldson's theory of almost holomorphic sections). In the talk I will discuss some recently found new remarkable examples of symplectic flexibility.

##### From the Jones polynomial to Khovanov homotopy

In the early 1980's, Jones introduced a new knot invariant, now called the Jones polynomial. Roughly ten years ago, Khovanov gave a refinement -- or categorification -- of the Jones polynomial; this refinement is now called Khovanov homology. In this talk we will sketch definitions of the Jones polynomial and Khovanov homology, and mention some of their most spectacular applications. We will then discuss a recent space-level refinement of Khovanov homology, and some applications of it; this refinement is joint work with S. Sarkar.

##### In search of a Langlands transform

Class field theory expresses Galois groups of abelian extensions of a number field F in terms of harmonic analysis on the multiplicative group of locally compact topological ring, the adèle ring, attached to F. Among the most far-reaching predictions of the Langlands program is the existence of a vast generalization of class field theory, in the form of a correspondence between n-dimensional representations of the Galois group of F and automorphic representations, which arise in the harmonic analysis on a homogeneous space for GL(n) of the adèle ring of F. In some cases, the cohomology of Shimura varieties provides a procedure for transforming automorphic representations to Galois representations. I will outline the scope and limitations of this partial realization of the hypothetical Langlands correspondence.

##### Asymptotics of Eigenvalues and Eigenfunctions in Periodic Homogenization

Understanding asymptotics of eigenvalues and eigenfunctions in homogenization is an important problem and it has attracted a great deal of attentions. One of the practical reasons is that it may be applied in the studies of some inverse problems and problems of uniform boundary controllability. In this talk, I shall describe a recent joint work with C.Kenig and Zhongwei Shen on the Uniform (and optimal) estimates for

eigenvalues and eigenfunctions in elliptic periodic homogenization.

##### Categorification at a prime root of unity

Quantization parameter q becomes a grading shift after categorification. When q specializes to a root of unity, categorification becomes more subtle. We'll discuss an approach to categorification at a prime root of unity p via p-differentials in characteristic p.

##### Derangements

A fixed point free permutation of a set is called a derangement. It is an old result of Jordan that if G is a transitive permutation group on a finite set, then derangements exist. In fact, some anaylsis of this problem predates group theory going back to the early 1700's. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Hidden structures in large matrices

Consider the problem of estimating the entries of a large matrix, when most of the entries are either hidden from us or blurred by noise. Of course, one needs to assume that the matrix has some structure for this estimation to be possible. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### Hydrodynamic turbulence as a problem in non-equilibrium statistical mechanics

The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### Riemann-Roch for graphs and applications

We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### The energy-critical NLS in the exterior of a convex obstacle

We will discuss the induction on energy technique in the study of nonlinear dispersive equations using the problem described in the title as our primary model case. To the extent time allows, we will then

describe some of the new difficulties connected with this particular problem and how they were overcome in recent joint work with Monica Visan and Xiaoyi Zhang.

##### From Nash to Onsager: funny coincidences across differential geometry and the theory of turbulence

The incompressible Euler equations were derived more than 250 years ago by Euler to describe the motion of an inviscid incompressible fluid. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE TITLE.

##### Decay estimates for scalar waves and Maxwell flows on relativistic black hole backgrounds

Decay estimates for wave equations are crucial in the study of a broad range of nonlinear problems. In this talk I will describe recent and ongoing work on such estimates for linear wave equations on relativistic black hole space-times. The emphasis will be both on scalar waves and on spin systems such as the Maxwell equations.

##### Regularity of Boltzmann equation in convex domains

We establish Sobolev regularity for solutions to the Boltzmann equation in a convex domain with diffuse boundary condition.

##### Riemannian Hyperbolization

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT. The strict hyperbolization process of R. Charney and M. Davis produces a large and rich class of negatively curved spaces (in the geodesic sense).