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# Christophe Pittet

## Geometric and spectral methods in group theory

### Программа курса

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