This talk focuses on asymptotic properties
of geometric branching processes on hyperbolic spaces
and manifolds. (In certain aspects, processes on hyperbolic spaces are
simpler than on Euclidean spaces.)
The first paper in this direction was
by Lalley and Sellke (1997) and dealt with a homogenous branching diffusion on a hyperbolic (Lobachevsky) plane).
Afterwards, Karpelevich, Pechersky and Suhov (1998) extended
it to general homogeneous branching processes on
hyperbolic spaces of any dimension. Later on, kelbert and Suhov
(2006, 2007) proceeded to include non-homogeneous branching
processes. One of the main questions here is to calculate
the Hausdorff dimension of the limiting set on the absolute.
I will not assume any preliminary knowledge of hyperbolic
geometry.

The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as `local fields' are approximately gaussian in the presence of a `cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method.

Abstract: (Thanks to work of Abel Klein and others) it is understood how
to represent the quantum Ising model in terms of a certain classical model
of stochastic geometry called the `continuum random-cluster model'. In the
regime of large external field, this geometrical model is subcritical. By
developing bounds for its `ratio weak' mixing rate, one obtains estimates
involving the reduced density matrix of the quantum Ising model. The
implications of these estimates for entanglement do not appear to be best
possible, but they are at least robust for disordered systems. [Joint work
with Tobias Osborne and Petra Scudo.]

I shall give an overview of reaction-diffusion fronts in
random flows, especially the variational formula of front speeds of
Kolmogorov-Petrovsky-Piskunov reactions. Large deviation of the random
flows is essential to the formula and the analysis of front
speed asymptotics.

The various concepts of volatility (realized, local, stochastic, implied), well defined or depending on a given model and/or statistical estimates, will be discussed. Backward and forward equations for call-option payoffs (Black-Scholes and Dupire equations) will be revisited. We will show that, besides the Black-Scholes model with constant volatility, fast mean reverting stochastic volatility models can reconcile local and implied volatilities. If time permits we will also look at the relation between volatility and correlation in the multidimensional case.
The talk is addressed to a general audience in Probability without any particular deep background in financial mathematics.