DH 3rd floor SR

We examine the classical orthogonal polynomial ensembles using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. Equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Applications to sharp deviation inequalities on largest eigenvalues are discussed.

Taking a view common in the finite element analysis, we interpret

the first N terms of the usual Fourier series solution as the exact

solution of an approximating problem in a subspace spanned by the

eigenfunctions of the underlying Sturm Liouville problem. This view

leads to a consistent solution technique for the heat, wave and

Poisson's equation, and allows an analysis of the error caused by

truncating the Fourier series. Applications to a variety of problems

will be discussed to demonstrate that the analytic approach remains a

valuable complement to purely numerical methods.

The talk is intended for students with an interest in actually

solving partial differential equations. It assumes a standard

background in undergraduate mathematics but not necessarily prior

exposure to the subject. The goal is to show that there is more to

separation of variables than is apparent from standard texts on

engineering mathematics.

I will present two-sided estimates for the heat kernel of the elliptic

Schr

Using the dyadic parametrization of curves, and elementary theorems and

probability theory, examples are constructed of domains having bad properties on

boundary sets of large Hausdorff dimension (joint work with F.D. Lesley).