Georgia Institute of Technology College of Sciences

In this talk,we study weighted L^p-norm inequalities for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems follow from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels. These results are applicable to spectral multipliersfor group invariant Laplace operators acting on Lie groups of polynomialgrowth and elliptic operators on compact manifolds. This is joint work with Adam Sikora and Lixin Yan.

It is well known that a needle thrown at random has zero probability of intersecting any given irregular planar set of finite 1-dimensional Hausdorff measure. Sharp quantitative estimates for fine open coverings of such sets are still not known, even for such sets as the Sierpinski gasket and the 4-corner Cantor set (with self-similarities 1/4 and 1/3). In 2008, Nazarov, Peres, and Volberg provided the sharpest known upper bound for the 4-corner Cantor set. Volberg and I have recently used the same ideas to get a similar estimate for the Sierpinski gasket. Namely, the probability that Buffon's needle will land in a 3^{-n}-neighborhood of the Sierpinski gasket is no more than C_p/n^p, where p is any small enough positive number.

We will introduce a Bargmann transform from the space of square integrable functions on the n-sphere onto a suitable Hilbert space of holomorphic functions on a null quadric. On base of our Bargmann transform, we will introduce a set of coherent states and study their semiclassical properties. For the particular cases n=2,3,5, we will show the relation with two known regularizations of the Kepler problem: the Kustaanheimo-Stiefel and Moser regularizations.

I will review recent and classical results concerning the asymptotic properties (as N --> \infty) of 'ground state' configurations of N particles restricted to a d-dimensional compact set A\subset {\bf R}^p that minimize the Riesz s-energy functional \sum_{i\neq j}\frac{1}{|x_{i}-x_{j}|^{s}} for s>0. Specifically, we will discuss the following (1) For s < d, the ground state configurations have limit distribution as N --> \infty given by the equilibrium measure \mu_s, while the first order asymptotic growth of the energy of these configurations is given by the 'transfinite diameter' of A. (2) We study the behavior of \mu_s as s approaches the critical value d (for s\ge d, there is no equilibrium measure). In the case that A is a fractal, the notion of 'order two density' introduced by Bedford and Fisher naturally arises. This is joint work with M. Calef. (3) As s --> \infty, ground state configurations approach best-packing configurations on A. In work with S. Borodachov and E. Saff we show that such configurations are asymptotically uniformly distributed on A.