# Seminars & Events for 2008-2009

##### Global Schrodinger maps: small data in the critical Sobolev spaces

I will discuss recent work on the global regularity of the Schrodinger map initial-value problem with small data, in all dimensions $d\geq 2$.

##### Heat conduction and Non-Equilibrium Statistical Mechanics: some considerations

I will present some standard models of Non-Equilibrium Statistical Mechanics focusing mainly on heat conduction. I'll discuss their origin and significance and present some of the available analytical and numerical results. The talk will be at an introductory level.

##### A general unique continuation theorem for the Einstein equations

I will discuss a unique continuation result for the vacuum Einstein equations across bifurcate horizons. The main result uses a recent Carleman estimate of Ionescu and Klainerman, together with some geometric gauge constructions. More broadly I will indicate how Carleman estimates for the wave operator can be used to derive unique continuation for the Einstein equations.

##### Unitary representations of simple Lie groups

By 1950, work of Gelfand and others had led to a general program for "non-commutative harmonic analysis": understanding very general mathematical problems (particularly of geometry or analysis) in the presence of a (non-commutative) symmetry group G. A first step in that program is classification of unitary representations - that is, the realizations of G as automorphisms of a Hilbert space.

##### Ergodic Theory

I will be discussing two basic invariants of ergodic theory, entropy and ergodicity, through the problems they were invented to solve. This talk will be heavy on examples and low on rigour.

##### The Maximum Number of Colorings of Graphs of Given Order and Size

Wilf asked in the 1980s about $f(n,m,l)$, the maximum number of $l$-colorings that a graph with $n$ vertices and $m$ edges can have. We essentially solve this problem for $l=3$, in particular proving, for all large $n$, the conjecture of Lazebnik (1989) that if $m\le n^2/4$ then the maximum number of 3-colorings is achieved by a semi-complete biparite graph.

##### Khovanov homology, open books, and tight contact structures

I will discuss a construction modeled on Khovanov homology which associates to a surface, $S$, and a product of Dehn twists, $\Phi$, a group $Kh(S,\Phi)$. The group $Kh(S,\Phi)$ may sometimes be used to determine whether the contact structure compatible with the open book $(S,Phi)$ is tight or non-fillable.

##### Dynamical Mordell-Lang problems

The Mordell-Lang conjecture, proved by Faltings and Vojta, states that a finitely generated subgroup of a semiabelian variety intersects any subvariety of that semiabelian variety in a union of finitely many translates of subgroups.

##### The space of positive scalar curvature metrics on the three-sphere

In this talk we will discuss a proof of the path-connectedness of the space of positive scalar curvature metrics on the three-sphere. The proof uses the Ricci flow with surgery and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental.

##### Group representation patterns in digital signal processing

In the lecture we will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing.

##### Leaves in moduli spaces in characteristic p

We try to understand the geometry of the moduli space of polarized abelian varieties in characteristic p. E.g. the phenomenon that Hecke orbits blow up and down in a rather unpredictable way. Choose a point $x$, corresponding to a polarized abelian variety. We study $C(x)$ consisting of all moduli points of polarized abelian varieties which have the same $p$-adic and $\ell$-adic invariants.

##### Thermostats: equivalence and thermodynamic limit

An ongoing question is whether the isokinetic, isoenergetic or other thermostat models are acceptable as model of thermostats in nonequilibrium statistical mechanics as opposed to thermostats which are made of infinite Hamiltonian systems in thermal equilibrium.

##### Three conjectures in arithmetic geometry

We discuss the Manin-Mumford conjecture (about the closure of any set of torsion points in an abelian variety), the André-Oort conjecture (about the closure of any set of CM-points in a moduli space) and the Hecke Orbit Conjecture (about the closure of the Hecke orbit of a moduli point). These conjectures, on the borderline of geometry and arithmetic, seem to be (have been) solved.

##### Differential Galois Theory

Algebraic groups, differential equations, and Galois theory: who isn't scared of at least one of those? It being the last GSS talk before Halloween, I'll do my best to mention all three. The ostensible purpose of doing so will be to explain why certain indefinite integrals cannot be written in terms of elementary functions. No knowledge of elementary functions will be presupposed.

##### Eliminating cycles in the torus via isoperimetric inequalities

Let $G_{\infty}=(C_m^d)_{\infty}$ denote the graph whose set of vertices is $\{1,\ldots ,m\}^d$, where two distinct vertices are adjacent iff they are either equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the graph on the same set of vertices in which two vertices are adjacent iff they are adjacent in one coordinate in $C_m$ and equal in all others.

##### On Simons conjecture for knots

Let $K$ be a non-trivial knot in $S3$. Simon's Conjecture asserts that $\pi_1(S3\setminus K)$ surjects only finitely many distinct knot groups. We discuss the proof of this for a class of small knots that includes 2-bridge knots.

##### The coefficients of harmonic Maass forms and combinatorial applications

The subject of partition theory has long been an excellent source of combinatorial hypergeometric q-series that are also automorphic forms.

##### On the Tate and Langlands--Rapoport conjectures for Shimura varieties of Hodge type

Let $p$ be a prime. Let $F$ be an algebraic closure of the finite field $F_p$ with $p$ elements. An integral canonical model $N$ of a Shimura variety $Sh(G,X)$ of Hodge type is a regular, closed subscheme of a suitable pull back of the Mumford moduli tower $M$ over $Z_{(p)}$.

##### Spectral-element and adjoint methods in computational seismology

We provide an introduction to the use of spectral-element and adjoint methods in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss how these equations may be solved numerically based upon the spectral-element method (SEM) to address the forward problem in seismology.