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.
An M-dependent process X(n) on the integers, is a process for which every event concerning with X(-1),X(-2),... is independent from every event concerning with X(M),X(M+1),...
Such processes play an important role both as scaling limits of physical systems and as a tool in approximating other processes.
A question that has risen independently in several contexts is:
"is there an M dependent proper colouring of the integer lattice for some finite M?"
Let G be an infinite connected graph. For each vertex of G we decide randomly and independently: with probability p we paint it blue and with probability 1-p we paint it yellow. Now, consider the subgraph of blue vertices: does it contain an infinite connected component?
There is a critical probability p_c(G), such that if p>p_c then almost surely there is a blue infinite connected component and if pp_c or p<p_c.
We will focus on planar graphs, specifically on the triangular