Christophe Pittet

Geometric and spectral methods in group theory

Three or four lectures on isoperimetric inequalities in infinite groups (October)

1) The general picture.: Heat diffusion and return probabilities of random walks are related to isospectral inequalities.; Nash inequalities are useful tools to estimate the rate of decay of return probabilities in term of the spectral geometry of the space (isospectral profile).
2) Cheeger inequality and Folner couples.: The isospectral profile can be estimate with purely geometric tools: isoperimetric inequalities and Folner couples.
3) Isoperimetric inequalities.: The Coulhon-Saloff-Coste isoperimetric inequality is related to the rate of growth of balls.; The Erchler-Gromov isoperimetric inequality for wreath products is an entropic inequality.

Three lectures on the spectral distribution near zero of Laplace operators on infinite groups (end of November begining of December)

1) Von Neumann traces and Novikov-Shubin invariants
2) A simple formula for the spectral distribution near zero of Laplace operators
3) Examples of groups and computations of spectral distributions