Georgia Institute of Technology College of Sciences

In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type. The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p. Let D be a compact smooth domain with boundary $\partial D$ contained in $T^N \times ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then this is also the case on the interior of D. There are several variants of this result in different settings. To make the result accessible to the layman, we will explain the method on the function distance to a closed subset of an Euclidean space. This example contains all the ideas of the general case.

These are two half an hour talks.Evgeny's abstract: The most useful approach to a classication of 3-manifolds is the complexity theory foundedby S. Matveev. Unfortunately, exact values of complexity are known for few infinite seriesof 3-manifold only. We present the results on complexity for two infinite series of hyperbolic3-manifolds with boundary.Andrei's abstract: We define coordinates on virtual braid groups. We prove that these coordinates are faithful invariants of virtual braids on two strings, and present evidence that they are also very powerful invariants for general virtual braids.The talk is based on the joint work with V.Bardakov and B.Wiest.

This is the second of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

The area was essentially originated by the general question: How many zeros of a random polynomials are real? Kac showed that the expected number of real zeros for a polynomial with i.i.d. Gaussian coefficients is logarithmic in terms of the degree. Later, it was found that most of zeros of random polynomials are asymptotically uniformly distributed near the unit circumference (with probability one) under mild assumptions on the coefficients. Thus two main directions of research are related to the almost sure limits of the zero counting measures, and to the quantitative results on the expected number of zeros in various sets. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by various bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane, and quantify this convergence. Random coefficients may be dependent and need not have identical distributions in our results.