Extra Lectures
Wednesday, May 2nd, 2007, 10:00 a.m., rm 1180 IPAM
"Slicing, Rectifiability, and Scans"
Bob Hardt (Rice University)
ABSTRACT: Around 2000, B. White and L.Ambrosio-B.Kirchheim gave beautiful new proofs of the important rectifiability closure theorems of Geometric Measure Theory using 0 dimensional slicing. Their works gave generalizations to, respectively, chains with coefficients in various groups and to currents in certain metric spaces. In work with T. De Pauw (UCLouvain?) we combine and generalize these ideas to cover all cases using the new notion of a scan. We'll start with some remarks about slicing currents.
Friday, May 4th, 2007, 11:00 a.m., rm 1200 at IPAM
"Convergence of Gibbs Measures Absolutely Continuous with Respect to Hausdorff Measures"
Bob Hardt (Rice University)
ABSTRACT: Simulated annealing is a stochastic optimization algorithm that mimics the physical process of a thermodynamic system settling into a state of minimal energy while lowering the 'temperature'. In applications such as to modeling micromagnetism one considers a nonnegative function F on a state space and the probability measures P_c obtained by normalizing the measures exp(-cF(x))dx. As c approaches infinity, a subsequence of these measures approaches a "Gibbs measure" concentrated on the zero set of F. In joint work with D.Cox (Rice) and Petr Kloucek (Neuchatel) we describe various non-degenerate cases when this limiting measure can be explicitly described in terms of Hausdorff measure and the Hessian of F on its zero set. Also in case F(x)=dist(x,A)2, E. Samansky has studied this for A being a self-similar fractal or A being a real (possibly singular) semi-algebraic set.