# Seminars & Events for 2008-2009

##### A Giambelli formula for isotropic Grassmannians

The structure of the cohomology ring of the Grassmannian has been studied for well over a century, but the analogous questions for the non-maximal isotropic Grassmannians of the symplectic and orthogonal groups are rather unexplored.

##### Current large deviations in stochastic systems

Using the hydrodynamic limit theory, we will review the large deviations of the heat current through a diffusive system maintained off equilibrium by two heat baths at unequal temperatures. In particular, we will discuss the occurrence of dynamical phase transitions which may occur for some models and the structure of the long range correlations in systems maintained off equilibrium.

##### Cremona transformations and homeomorphisms of topological surfaces

##### Embedded Surfaces in 4-Manifolds

Given a 2-dimensional homology class in a closed 4-manifold, you can try to represent it by an embedded, closed, orientable surface. What is the minimum genus of such a surface? Come find out.

##### On the Singular Probability of Random Discrete Matrices

Let $n$ be a large integer and $M_n$ be an $n$ by $n$ random matrix whose entries are independent (but not necessarily identically distributed) random variables. The main goal of this paper is to prove a general upper bound for the probability that $M_{n}$ is singular.

##### Weight Cycling and Serre-type Conjectures

Suppose that $\rho$ is a three-dimensional modular mod p Galois representation whose restriction to the decomposition groups at p is irreducible and generic. If $\rho$ is modular in some (Serre) weight, then a representation-theoretic argument shows that it also has to be modular in certain other weights (we can give a short list of possibilities).

##### The singular set of $C^{1}$ smooth surfaces in the Heisenberg group

##### TBA

##### A mechanical model for Fourier's law for heat conduction

Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics remains thus a major unsolved problem.

##### Symmetric functions of a large number of variables

##### Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period $[0, T/\epsilon]$ for initial data of form $\epsilon\Psi$, where $T$ depends only on $\Psi$. We show that for such data there exists a unique solution for a time period $[0, e^{T/{\epsilon}}]$.

##### Subtle invariants and Traverso's conjectures for p-divisible groups

Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of positive characteristic $p$. We will first define several subtle invariants of $D$ which have been introduced recently and which are crucial for any strong, refined classification of $D$. Then we will present our results on them.

##### On the Number of Solutions to Asymptotic Plateau Problem

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique.

##### Vanishing and torsion-free theorems for the log minimal model program

We will discuss Ambro's formulation of Kollár's injectivity, torsion-free, and vanishing theorems. It is indispensable for the study of the log minimal model program for log canonical pairs.

##### On quantum, stationary, non-equilibrium states

In this talk I will describe recent results on existence and dynamical stability of stationary, non-equilibrium states in certain models of quantum statistical mechanics. This is a joint work with Marco Merkli and Matthias Mueck.

##### Fundamental lemma and Hitchin fibration

The fundamental lemma is an identity of orbital integrals on p-adic reductive groups which was stated precisely by Langlands and Shelstads as a conjecture in the 80's. We now have a proof due to the efforts of many peoples with many ingredients. I will only explain how a certain particular type of geometry like affine Springer fibers and Hitchin were helpful in this proof.

##### Minimal intersection and self-intersection of curves on surfaces

Consider the set of free homotopy classes of directed closed curves on an oriented surface and denote by V the Z-module generated by this set. Goldman discovered a Lie algebra structure on this module, obtained by combining the geometric intersection of curves with the usual loop product.

##### Faltings' height of CM cycles and Derivative of $L$-functions

In this talk, we first describe a systematic way to construct `automorphic Green functions' for Kudla's special divisors on a Shimura variety of orthogonal type $(n, 2)$. We then give an explicit formula for their values at a CM cycle. This formula suggests a direct relation between the Faltings' height of these CM cycles with the central derivative of some Rankin-Selberg $L$-function.

##### Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

Complete Riemannian manifolds with nonnegative Ricci curvature have been well studied. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a rigidity theorem which characterizes hyperbolic manifolds.