# Seminars & Events for 2013-2014

##### Some wonderful conjectures (but very few theorems) at the boundary between analysis, combinatorics and probability

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. Many problems in combinatorics, statistical mechanics, number theory and analysis give rise to power series (whether formal or convergent) of the form f(x,y) = sum_{n \ge 0} a_n(y) x^n, where a_n(y) are formal power series or analytic functions satisfying a_n(0) \neq 0 for n = 0,1 and a_n(0) = 0 for n \ge 2.

##### Length and volume in contact three-manifolds

We give an introduction to a theorem (joint with Dan Cristofaro-Gardiner and Vinicius Ramos) that relates the volume of a contact three-manifold to the lengths of certain collections of closed orbits of the Reeb vector field. This implies that the Reeb vector field always has at least two closed orbits.

##### An L log L bound on the vorticity in the 3D NSE

The goal of this lecture is to present a spatially localized L log L bound on the vorticity in the 3D Navier-Stokes equations, assuming a very mild, _purely geometric_ condition.

##### How not to define cylindrical contact homology

We consider the problem of defining cylindrical contact homology, in the absence of contractible Reeb orbits, using "classical" methods. The main technical difficulty is failure of transversality of multiply covered cylinders. One can fix this difficulty by using S1-dependent almost complex structures, but at the expense of introducing another difficulty which we will explain.

##### Arithmetic invariant theory and arithmetic statistics

**PLEASE NOTE SPECIAL DAY AND LOCATION. **I will give an overview of some recent work in the subjects of "arithmetic invariant theory" and "arithmetic statistics." The first has to do with using representations of groups to study moduli spaces of arithmetic or geometric objects.

##### Isotropic curvature, macroscopic dimension Filling radius of contractible loops and fundamental group

I will discuss the proof of a conjecture of Gromov's to the effect that manifolds with uniformly positive isotropic curvature (and bounded geometry) are macroscopically 1-dimensional on the scale of the isotropic curvature. One of the main techniques involved is modeled on Donaldson's version of H\"ormander technique to produce (almost) holomorphic sections.

##### Geometric view of conformal PDEs

**THIS SEMINAR HAS BEEN MOVED FROM OCTOBER 25 TO OCTOBER 18.**

##### Higher Singular Integrals

There is a very natural way to extend Calder\'{o}n's calculations which generated his commutators and the Cauchy integral on Lipschitz curves, to include operators of multiplication with functions of arbitrary polynomial growth. The plan of the talk is to describe some examples of such calculations, and to focus on a particular case that goes beyond the classical Calder\'{o}n program."

##### The statistical price to pay for computational efficiency

The ever increasing size of current datasets has made computation an essential aspect of the design of statistical procedures.

##### Foliations and Serre-Tate parameters

Serre and Tate defined "canonical coordinates" locally around the moduli point of a polarized ordinary abelian variety. In this talk we will describe two kind of generalizations (joint work with Ching-Li Chai):

-- describe global coordinates;

##### Stable small-data shock formation for wave equations in 3D

PLEASE NOTE SPECIAL DAY AND TIME. CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. I will present some preliminary results obtained in collaboration with G. Holzegel, S. Klainerman, and W. Wong. Our main result is a proof of stable shock formation in solutions to a class of nonlinear wave equations in three spatial dimensions.

##### Dynamics near criticality

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT. Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs.

##### War and Peace in Modern Mathematics

Wars, both actual and metaphorical, are omnipresent in the history of mathematics. From sixteenth century arithmetic publishers who promoted the value of trigonometry in out-ballisticking one's foes to the game theorists and fluid dynamicists of the Cold War, mathematicians have both actively participated in warfare and actively appropriated the goals, values, and images of warfare to their ow

##### The Cayley Plane and String Bordism

I will describe how an affinity between projective spaces and bordism rings extends further than previously known.

##### Universality in interface growth models

Over the past few years, there has been growing evidence, at the heuristic, the mathematically rigorous, and even the experimental level, that models of one-dimensional interface growth exhibit a "universal" behaviour at large scales.

##### Some Results in Complex Hyperbolic Geometry

**PLEASE NOTE SPECIAL DAY AND LOCATION. **In this talk, I present a new approach to the study of cusped complex hyperbolic manifolds through their compactifications. Among other things, I give effective bounds on the number of complex hyperbolic manifolds with given upper bound on the volume.

##### Gromov-Uhlenbeck Compactness and the Atiyah-Floer Conjecture

Let M be a symplectic manifold with a hamiltonian group action by G. We introduce an analytic framework that relates holomorphic curves in the symplectic quotient of M to gauge theory on M. As an application of these ideas, we discuss the relation between instanton Floer homology and Lagrangian Floer homology of representation varieties.

##### On kinetic Delaunay triangulations: a near-quadratic bound for unit speed motions

Let P be a collection of n points in the plane, each moving along some straight line and at unit speed. Three points of P form a Delaunay triangle if their circumscribing circle contains no further point of P. These triangles form the famous Delaunay triangulation, denoted by DT(P).

##### The local Gan-Gross-Prasad conjecture for tempered representations of unitary groups

Let $E/F$ be a quadratic extension of $p$-adic fields. Let $V_n\subset V_{n+1}$ be hermitian spaces of dimension $n$ and $n+1$ respectively. For $\sigma$ and $\pi$ smooth irreducible representations of $U(V_n)$ and $U(V_{n+1})$ set $m(\pi,\sigma)=dim\; Hom_{U(V_n)}(\pi,\sigma)$.

##### Enumeration of real rational curves

The classical problem of enumerating rational curves in projective spaces is solved using a recursion formula for Gromov-Witten invariants. In this talk, I will describe a similar relation for real Gromov-Witten invariants with conjugate pairs of constraints.