Georgia Institute of Technology College of Sciences

Ergodic theory of randomly forced space-time homogeneous Burgers equation in noncompact setting has been developed in a recent paper by Eric Cator , Kostya Khanin, and myself. The analysis is based on first passage percolation methods that allow to study coalescing one-sided action minimizers and construct the global solution via Busemann functions. i will talk about this theory and its extension to the case of space-continuous kick forcing. In this setting, the minimizers do not coalesce, so for the ergodic program to go through, one must use new soft results on their behavior to define generalized Busemann functions along appropriate subsequences.

It is well known that the Horn inequalities characterize the relationship of eigenvalues of Hermitian matrices A, B, and A+B. At the same time, similar inequalities characterize the relationship of the sizes of the Jordan models of a nilpotent matrix, of its restriction to an invariant subspace, and of its compression to the orthogonal complement. In this talk, we provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_0 (such operator can be thought of as the infinite dimensional generalization of matrices, that is an operator will be annihilated by an H-infinity function), of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. Our approach also explains why the same combinatorics solves the eigenvalue and the Jordan form problems. This talk is based on the joint work with H. Bercovici.

In this talk we will begin by discussing the problem of understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. Recently much progress has occurred in this topic. We will then look at an application of the theory of operads to this problem in the case when the underlying manifold is an n-sphere. In the case when n>2, this space is a homotopy commutative, homotopy associative H-space. In particular, we show that it admits an action of the little n-disks operad. Via theorems of Stasheff, Boardman, Vogt and May, this allows us to demonstrate that the path component of this space containing the round metric, is weakly homotopy equivalent to an n-fold loop space.

I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.