Speaker: E.B. Saff, Vanderbilt University, Nashville, Tennessee , Host: Lubinsky
Title: Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds

Abstract: For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere. The research is joint with D. Hardin.

Speaker: Prof. Rodrigo Bañuelos, Department of Mathematics, Purdue University, Host: Houdré
Title: Generalized Isoperimetric Inequalities

Abstract: In 1973, J.M. Luttinger proved that multiple convolutions of multiple functions are nondecreasing if each function is replaced by its symmetric decreasing rearrangement. This rearrangement inequalities and their extensions by H.S. Brascamp, E.H. Lieb, and Luttinger, provide, via what Luttinger termed "generalized isoperimetric inequalities for heat kernels," a powerful and elegant method for proving many of the classical geometric and physical isoperimetric inequalities for regions of fixed volume in \mathbb R_n. These include, amongst others, the classical isoperimetric inequality, the Rayleigh-Faber-Krahn inequality for the lowest eigenvalue and Polya's inequalities for the torsional rigidity and electrostatic capacity. The key to these applications is the representation of the Dirichlet heat kernel in terms of multiple integrals of Gaussian kernels. These techniques apply not only to Brownian motion but also to other Levy processes with radial symmetric decreasing transition prob-abilities such as the "\alpha-symmetric stable processes" and the "relativistic Brownian motion." After discussing some of these classical results, we will present versions of rearrangement inequalities for multiple integrals where instead of fixing the volume of the region we fix its diameter or in radius. These yield new sharp inequalities for Dirichlet heat kernels which have applications to open problems of current interest such as the spectral gap conjecture of M. van den Berg (problem #44 in Yau's 1990 "open problems in geometry") an the "hot-spots" conjecture for conditioned Brownian motion.

This talk is designed for a general audience. We will show some pictures and discuss some results. Technicalities will be kept to a minimum.

Speaker: Jim Geelen, University of Waterloo , Host: Thomas
Title: Extremal problems in matroid theory

Abstract: Mader proved that if G is a simple graph with no K_n-minor, then |V(G)|\le \alpha_n |E(G)|, where \alpha_n is a constant depending only on n. We extend this result to the class of matroids representable over any given finite field. The proof relies on an extension of a well-known extremal result of Erdös and Pósa to matroids. This is joint work with Bert Gerards and Geoff Whittle. This talk is intended to be accessible; all necessary background material will be introduced.

Speaker: Robert V. Kohn, Courant Institute of Mathematical Sciences, New York University, Host: Gangbo
Title: The Mathematics of Coarsening

Abstract: I will discuss surface-energy-driven coarsening of two-phase microstructures. Such coarsening is observed in many physical systems. Experiments and simulations suggest that solutions are in some sense statistically self-similar. There is, however, virtually nothing known with mathematical rigor. I will introduce this topic, then present recent joint work with Felix Otto (Comm. Math. Phys. 2002). Our main accomplishment is an upper bound on the coarsening rate, consistent with the conjectured self-similar behavior. Our work is also interesting for its viewpoint, which is new and potentially applicable to many other problems. I will close with one such application, to epitaxial growth -- joint work with Xiaodong Yan (Comm. Pure Appl. Math. 2003).

Speaker: Vitaly Bergelson, Department of Mathematics Ohio State University, Host: Houdré
Title: Multiple recurrence and the properties of large sets

Abstract: Many familiar theorems in various areas of mathematics have the following common feature: if A is a large set, then the set of its differences (or, sometimes, the set of distances between its elements) is VERY large. For example:

(i) If A is a set of reals having positive Lebesgue measure, then there exists a positive real a, so that A-A contains the interval (-a,a). (ii) If A is a set of natural numbers having positive upper density, then for any polynomial p(n) having integer coefficients and zero constant term, the set A-A contains infinitely many integers of the form p(n). (iii) If F is an infinite algebraic field and G is a subgroup of finite index in the multiplicative group F*, then G-G = F.

In this talk we shall discuss these and other similar results from the perspective of Ergodic Ramsey Theory. This discussion will lead us to new interesting results and conjectures. In particular we will see the foregoing as a special case of the appearance of rather arbitrary finite configurations inside sufficiently large sets.

The talk is intended for a general audience.

Fall 2002
Spring 2003

Seminars | CDSNS Colloquium | CDSNS/ACELab | Colloquium | Combinatorics | Geometry | Geometry/Topology/Algebra | Nonlinear Science | Research Horizons | Stochastics | Topology |