# Seminars & Events for 2012-2013

##### Scalar equations as asymptotic models for internal waves in Oceanography

We propose to derive rigorously scalar asymptotic models for the propagation of gravity waves at the interface between two layers of immiscible fluids of different densities (modeling fresh and salt water interface).

##### Robust Subspace Modeling

Consider a dataset of vector-valued observations that consists of a modest number of noisy inliers, which are explained well by a low-dimensional subspace, along with a large number of outliers, which have no linear structure. We describe a convex optimization problem that can reliably fit a low-dimensional model to this type of data. When the inliers are contained in a low-dimensional subspace we provide a rigorous theory that describes when this optimization can recover the subspace exactly. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Traps and Random Walks in Random Environments on a Strip

PLEASE NOTE SPECIAL DAY (TUESDAY, NOV. 13)

##### Towards crystallization in Coulomb systems

We are interested in the statistical mechanics of (classical) two-dimensional Coulomb gases and one-dimensional log gases in a confining potential. We connect the Hamiltonian to the "renormalized energy", a way to compute the total Coulomb interaction of an infinite jellium, and whose minimum is expected to be achieved by the triangular lattice in 2D, and is achieved by the lattice Z in 1D.

##### MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures

We report on joint work in progress with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular:

(--) We will give a "modular interpretation" for all minimal models of M.

##### Open problems in mathematical general relativity

Introduced by Einstein in 1915, the theory of general relativity is probably one of the most well-tested theories in physics. And yet a basic mathematical understanding of this theory is lacking (for no other

reason than being difficult). I would like to introduce some of the major open questions.

##### How long does it take to catch a drunk miscreant?

We discuss the answer to a question of Churchley who asked how long it will take a cop to catch a drunk robber who moves randomly. We begin by discussing other variants of the cop-robber paradigm. This is

joint work with Alex Scott, Colin McDiarmid, and Ross Kang. We rely heavily on work of Komarov and Winkler.

##### On Littlewood’s conjecture in Diophantine approximation

The purpose of my talk will be to present some of the results that have been established for Littlewood’s conjecture. In the first part of the discussion I will give an expository overview of what is currently known about the conjecture and what are some of the questions that naturally arise from it.

##### Deformations of periodic frameworks

A d-periodic bar-and-joint framework is an abstraction (and generalization to arbitrary dimension d) of an atom-and-bond crystal structure. We describe a deformation theory for this type of frameworks

##### Galois representations for regular algebraic cuspidal automorphic forms

To any essentially self-dual, regular algebraic (ie cohomological) automorphic representation of GL(n) over a CM field one knows how to associate a compatible system of l-adic representations. These l-adic representations occur (perhaps slightly twisted) in the cohomology of a Shimura variety.

##### On the degeneracy of optimal transport

It is a well known result of Caffarelli that an upper and lower bound on the Monge-Amp{\`e}re measure of a convex function u implies the function must actually be strictly convex.

##### Abstract analogues of flux as symplectic invariants

This talk is part of a circle of ideas that one could call "categorical dynamics". We look at how objects of the Fukaya category move under deformations prescribed by fixing an odd degree quantum cohomology class. This is an analogue of moving Lagrangian submanifolds under non-Hamiltonian deformations.

##### Global results for linear waves on expanding Schwarzschild de Sitter cosmologies

In this talk I will present recent results for solutions to the linear wave equation on Schwarzschild de Sitter black hole spacetimes. We focus here on the expanding region of the spacetime, and exhibit a stability mechanism which manifests itself in a global redshift effect. I shall describe how this can be combined with earlier results to obtain a global linear stability result.

##### How to Design Simple Efficient Mechanisms that are also Composable

SPECIAL JOINT SEMINAR WITH COMPUTER SCIENCE. E-commerce applications require simple, and well-designed systems, and systems that work well even if users participate in multiple mechanisms (and the value of each player overall is a complex function of their outcomes). Traditional mechanism design has considered such mechanisms only in isolation, and the mechanisms it proposes tend to be complex and impractical. In contrast, players typically participate in various mechanisms, mechanisms that are run by different principals (e.g. different sellers on eBay or different ad-exchange platforms) and coordinating them to run a single combined mechanism is infeasible or impractical. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Rationality in families of threefolds

In a joint work with Tommaso de Fernex, we prove that in a family of projective threefolds deﬁned over an algebraically closed ﬁeld, the locus of rational ﬁbers is a countable union of closed subsets of the locus of separably rationally connected ﬁbers.

##### On a class of super-critical quasi-geostrophic type models in fluid dynamics

I will discuss some recent results on a class of active scalars with non-local transport and super-critical dissipation. Part of the talk is based on some joint work with Hongjie Dong.

##### Dynamics of a Cytokine Storm

Six volunteers experienced severe inflammatory response during the Phase I clinical trial of a monoclonal antibody that was designed to stimulate a regulatory T cell response. Soon after the trial began, each volunteer experienced a "cytokine storm", a dramatic increase in cytokine concentrations. (CLICK ON SEMINAR TITLE FOR FULL ABSTRACT.)

##### Enumeration of singular curves with tangency conditions

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions.

##### On Fluid Mechanics and its Difficulties

I'll spend the first half (but hopefully less) of the talk giving an abridged introduction to the Euler and Navier-Stokes equations, covering the basic derivations and some of the known results. Afterwards, I will provide two examples to illustrate some of the difficulties in analyzing these equations.

##### A Szemeredi-Trotter theorem in R^4

The Szemeredi-Trotter theorem states that m points and n lines in the plane can have at most O(m^{2/3}n^{2/3}+m+n) incidences. This theorem has seen a number of generalizations, including a theorem of Toth that obtains the same result for (complex) points and lines in the complex plane. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.