We consider random Taylor series and the random $\dzeta$-functions. We prove non-continuation results for both, in case of independent random variables. Also, if the series defined by a stationary process can be continued beyond the radius of convergence we show that the process is deterministic.

We consider a nonlinear Schr\"odinger equation (NLS) with random
coefficients, in a regime of separation of scales corresponding to
diffusion approximation. The primary goal is to propose and
study an efficient numerical scheme in this framework. We use a
pseudo-spectral splitting scheme and we establish the order of the
global error. In particular we show that we can take an integration step
larger than the smallest scale of the problem, here the correlation
length of the random medium. We study
the asymptotic behavior of the numerical solution in the diffusion
approximation regime.

All physical systems in equilibrium obey the laws of
thermodynamics. In other words, whatever the precise nature of the
interaction between the atoms and molecules at the microscopic level,
at the macroscopic level, physical systems exhibit universal behavior in
the sense that they are all governed by the same laws and formulae of
thermodynamics.

The speaker will recount some recent history of universality ideas in
physics starting with Wigner's model for the scattering of neutrons
off large nuclei and show how these ideas have led mathematicians to
investigate universal behavior for a variety of mathematical systems.
This is true not only for systems which have a physical origin, but also
for systems which arise in a purely mathematical context such as the
Riemann hypothesis, and a version of the card game solitaire called
patience sorting.

Let G be a transient graph, and flip a fair coin at each vertex.
This gives a distribution P. Now start a random walk from a vertex v, and
retoss the coin at each visited vertex, this time with probability 0.75
for heads and probability 0.25 for tails. The eventual configuration of
the coins gives a distribution Q. Are P and Q absolutely continuous w.r.t.
each other? are they singular? (i.e. can you tell whether a random walker
had tampered with the coins or not?) In the talk I'll answer to this
question for various graphs and various types of random walk. Based on
joint work with Y. Peres.