Math 711 — Fall 2001 Seminar in Analysis: Analysis and Potential Theory on Manifolds

The aim of this course is to present some of the basic techniques used in the study of the solutions of the Laplace equation (i.e., harmonic functions) and the heat diffusion equation on manifolds. We will study the celebrated iteration technique of Jürgen Moser, both in the elliptic and parabolic cases.

Functional inequalities are important tools and we will discuss Poincar\'e and Sobolev type inequalities. Applications to heat kernel estimates and spectral problems will be presented. The emphasize will be on techniques that show the stability of the results under reasonable perturbations such as quasi-isometries. If time permits, we will discuss discretization techniques which play an important role in this context.

Some familiarity with functional analysis is required. Familiarity with PDE and/or Riemannian geometry is a plus, but not a prerequisite.

References:

I. Chavel, Riemannian geometry: a modern introduction, Cambridge University Press, 1993.

E.B. Davies, Heat kernels and spectral theory, Cambridge University Press, 1989.

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bulletin of the AMS, 36, 1999, 135-249.

L. Saloff-Coste, Aspect of Sobolev type inequalities, London Math. Soc. Lect. Note Series, Cambridge University Press, to appear in 2001.

N. Varopoulos et al, Analysis and geometry on groups, Cambridge University Press, 1992.