This is one of several talks that Raphael Rouquier is organising next week on
stability conditions on triangulated categories. I have been asked to point out
that the talk will be very similar to a talk given in Oxford, by the speaker,
last June.
I'll start with the definition of a stability condition on a triangulated
category and say a bit about the space of stability conditions.
Then I'll describe some known examples of these spaces. If I have time I'll
try to explain why mirror symmetry suggests that it should be possible to equp
these spaces with interesting geometric structures.
Sequential Monte Carlo Samplers are a class of stochastic algorithms for
Monte Carlo integral estimation w.r.t. probability distributions, which combine
elements of Markov chain Monte Carlo methods and importance sampling/resampling
schemes. We develop a stability analysis by functional inequalities for a
nonlinear flow of probability measures describing the limit behaviour of the
methods as the number of particles tends to infinity. Stability results are
derived both under global and local assumptions on the generator of the
underlying Metropolis dynamics. This allows us to prove that the combined
methods sometimes have good asymptotic stability properties in multimodal setups
where traditional MCMC methods mix extremely slowly. For example, this holds for
the mean field Ising model at all temperatures.