# Seminars & Events for 2013-2014

##### 2D Coulomb Gases: A Statistical Mechanical Approach to Abrikosov Vortex Lattices

##### Ill-posedness results for some of the equations of hydrodynamics

We will begin by presenting some new instability and ill-posedness results for the incompressible Euler equations. These ill-posedness results are based upon some special exact solutions to the 3D Euler equations which bring out both the non-locality and the non-linear nature of the vortex stretching term.

##### Naturality in sutured monopole homology

Kronheimer and Mrowka defined a version of monopole Floer homology which assigns to any balanced sutured manifold a module up to isomorphism. In this talk, I will discuss how to replace “a module up to isomorphism” with something more natural, by showing that different choices made in the construction are related by canonical isomorphisms which are well-defined up to multiplication by a unit.

##### The Landau-Siegel zero and spacing of zeros of L-functions

Let χ be a primitive real character. We first establish a relationship between the existence of the Landau-Siegel zero of L(s,χ) and the distribution of zeros of the Dirichlet L-function L(s,ψ), with ψ belonging to a set Ψ of primitive characters, in a region Ω. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### PDEs of Monge-Ampere type

A considerable amount of research activity in recent years has been devoted to the study of nonlinear partial differential equations of Monge-Ampere type (MATEs) in connection with their applications to conformal geometry, optimal transportation and geometric optics.

##### Witten spinors on nonspin manifolds

Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator.

##### Formation of shocks for quasilinear wave equations

For any nonzero real constant $c_0$, we exhibit a family of smooth initial data for $$\big(-1 + c_0 (\partial_t \varphi)^2\big)\partial_t^2 \varphi + \triangle \varphi = 0 $$ and show that shocks form in the future. No symmetry condition is assumed. The work combines ideas from fluid mechanics, e.g. shock formation for Euler equations, and from general relativity, e.g.

##### The cover number of a matrix and its applications

The (epsilon)-cover number of a matrix A with entries in [-1,1] is the minimum number of aligned cubes of edge-length epsilon that are needed to cover the convex hull of the columns of A.

##### Flag Hilbert Schemes

Just like the punctual Hilbert scheme of a smooth surface parametrizes ideals supported at a given point, the flag Hilbert scheme parametrizes such ideals together with a full flag to the structure sheaf.

##### Construction and Analysis of a hierarchical massless QFT

I will discuss how rigorous renormalization group methods can be used to construct a massless QFT over a space with hierarchical structure. The model under study is a phi-4 perturbation of a three dimensional fractional Gaussian Free Field and is meant to capture some of the behavior of the Wilson-Fisher 4-epsilon fixed point.

##### Lambda-Rings

There are many operations that act on representations of a group, such as Sym^n, Wedge^n, and more exotic beasts. We can express these in terms of a simple, elegant set of operations, that involves, surprisingly, elementary number theory! I will explain how this works, prove some fun theorems about it, and, hopefully, answer the best generals question of all time.

##### New results on zeroes of stationary Gaussian functions

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity.

##### 3-Manifold Mutations Detected by Heegaard Floer Homology

Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology.

##### Pairs of p-adic L-functions for elliptic curves at supersingular primes

Iwasawa Theory for elliptic curves/modular forms has been traditionally in better shape at ordinary primes than at supersingular ones. After sketching the ordinary theory, we will indicate what makes the supersingular case more complicated, and then introduce ***pairs*** of objects that that are as simple as their ordinary counterparts.

##### Positive loops and orderability in contact geometry

**PLEASE NOTE DIFFERENT TIME. ** Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold Σ. We know, for instance, that if Σ admits a 2-subcritical Stein filling, it must be non-orderable.

##### Critical metrics on connected sums of Einstein four-manifolds

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. PLEASE NOTE DIFFERENT LOCATION. I will discuss a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds.

##### TBA - Viaclovsky

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. I will discuss a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds.

##### Yamabe flow, its singularity profiles and ancient solutions

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. PLESE NOTE DIFFERENT LOCATION AND TIME. We will discuss conformally flat complete Yamabe flow and show that in some cases we can give the precise description of singularity profiles close to the extinction time of the solution. We will also talk about a construction

##### TBA - Sesum

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. We will discuss conformally flat complete Yamabe flow and show

##### Well-posedness for Euler 2D in non-smooth domains

The well-posedness of the Euler system has been of course the matter of many works, but a common point in all the previous studies is that the boundary is at least $C^{1,1}$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.