This talk is devoted to the "Kac-Rice formula", which is an explicit way to compute
the expected number of zeroes of a random series with independent Gaussian coefficients.
We will discuss the original proofs of Kac and Rice (1940's),
an elegant geometrical proof due to Edelman and Kostlan (1995), some interesting examples,
and extensions to complex zeroes and eigenvalues of random matrices.
Here is a title and abstract for the lunch seminar:
Rigidity sequences for weakly mixing transformations
Abstract: I will present a recent result of Bassam Fayad
and Jean-Paul Thouvenot that shows that any rigidty
sequence for an irrational rotation is also a rigidity
sequence for some weakly mixing transformation.
We will review a Breiman type theorem for Gibbs measures due to Gurevich and Tempelman. For a translation invariant Gibbs measure on a suitable translation invariant configuration set X \subset S^G, where G is an amenable group and S is a finite set, we will prove the convergence of the Shannon-McMillan-Breiman ratio on a specific subset of "generic" configurations. Provided that the above Gibbs measure exists, we also prove the convergence in the definition of pressure and the fact that this Gibbs measure is an equilibrium state.