# Seminars & Events for 2014-2015

##### Singular moduli spaces and Nakajima quiver varieties

The aim of this talk is to study a class of singularities of moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations.

##### Mirror symmetry & Looijenga's conjecture

A cusp singularity is an isolated surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In the 1980's Looijenga conjectured that a cusp singularity is smoothable if and only if the minimal resolution of the dual cusp is the anticanonical divisor of some rational surface.

##### On the Gromov width of polygon spaces

After Gromov’s foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold \((M, \omega)\) is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in \((M, \omega)\).

##### Decouplings and applications

We describe a new Fourier analytic method for estimating a wide variety of exponential sums. The talk will mainly focus on the applications to number theory and PDEs. This is joint work with Jean Bourgain.

##### Set Oriented Numerical Methods for Dynamical Systems and Optimization

Over the last two decades so-called *set oriented* numerical methods have been developed in the context of the numerical treatment of dynamical systems. The basic idea is to cover the objects of interest - for instance *invariant **sets* or *invariant measures *- by outer approximations which are created via multilevel subdivision techniques. At the beginning of this

##### Beauville's splitting principle for Chow rings of projective hyperkaehler manifolds

Being the natural generalization of K3 surfaces, hyperkaehler varieties, also known as irreducible holomorphic symplectic varieties, are one of the building blocks of smooth projective varieties with trivial canonical bundle. One of the guiding conjectures in the study of algebraic cycles of such varieties is Beauville's splitting principle.

##### Fluctuations of the stationary Kardar-Parisi-Zhang equations

Up to a random height shift, two-sided Brownian motion is invariant for the Kardar-Parisi-Zhang equation.

##### Elliptic genera of Pfaffian-Grassmannian double mirrors

For an odd integer n>3 the data of generic n-dimensional subspace of the space of skew bilinear forms on an n-dimensional vector space define two different Calabi-Yau varieties of dimension n−4.

##### The p-adic uniformization of some curves

The classical uniformization theorem classifies Riemann surfaces according to their universal covers, and thus provides an important tool for the study of Riemann surfaces (curves over complex numbers). In the p-adic world, similar results hold for some classes of curves.

##### Few distinct distances and perpendicular bisectors

Let d(n) be the smallest number of distinct distances determined by any set of n points in the real plane. For n sufficiently large, is each set of n points that determines d(n) distances the intersection of an equalateral triangular lattice with a convex set? Is there at least a line that contains n^ε points, for some ε>0?

##### Act globally, compute locally: group actions, fixed points, and localization

Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know that the global integral is zero, we conclude that the sum of the local integrals is zero.

##### All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups

By the inspirational works of Peter Sarnak and Ralph Phillips (ACTA 1985), we know that all classical Schottky groups (dim n >3) must have Hausdorff dimension strictly bounded away from dim n-1. Later Peter Doyle (ACTA 1988) showed that it is also true for n=3.

##### Representations of finite groups and applications

In the first part of the talk we will survey some recent results on representations of finite (simple) groups. In the second part we will discuss applications of these results to various problems in number theory and algebraic geometry.

##### Thin knotted vortex tubes in stationary solutions to the Euler equation

In this talk we will discuss the proof of the existence of thin vortex tubes for stationary solutions to the incompressible Euler equation in R^3.

##### C^0-characterization of symplectic and contact embeddings

Symplectic and anti-symplectic embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of C^0-rigidity of symplectic embeddings, and in particular, diffeomorphisms.

##### Shear-free asymptotically hyperboloidal initial data in general relativity

One of the most useful tools for studying isolated gravitational systems is conformal compactification, which transforms ``infinity'' into a finite boundary by multiplying the metric by a suitable smooth function.

##### Asymptotic geometry and large isoperimetric regions

**This is an additional talk on this date. **In spite of the long history of the isoperimetric problem, the list of manifolds where isoperimetric regions are well understood is remarkably short.

##### TBA - Nahmod: CANCELLED - NEW DATE TBD

##### The physical and mathematical structure of images

Images are both maps of continuous physical phenomena and discrete mathematical objects. While Shannon established the fundamental relationship between physical and mathematical images over half a century ago, considerable further progress in understanding this relationship has been achieved in the past quarter century through the development of wavelets and compressive measurement.