Let X be a Gorenstein orbifold and Y a crepant resolution of
X. Suppose that the quantum cohomology algebra of Y is semisimple. We describe joint work with Iritani which shows that in this situation the genus-zero crepant resolution conjecture implies a higher-genus version of the crepant resolution conjecture. We expect that the higher-genus version in fact holds without the semisimplicity hypothesis.
In this talk I will present recent results on ergodicity of Markov semigroups in large dimensional spaces including interacting Levy type systems as well as some R-D models.
In this talk, the convergence analysis of a class of weak approximations of
solutions of stochastic differential equations is presented. This class includes
recent approximations such as Kusuoka's moment similar families method and the
Lyons-Victoir cubature on Wiener Space approach. It will be shown that the rate
of convergence depends intrinsically on the smoothness of the chosen test
function. For smooth functions (the required degree of smoothness depends on the
order of the approximation), an equidistant partition of the time interval on
which the approximation is sought is optimal. For functions that are less smooth
(for example Lipschitz functions), the rate of convergence decays and the
optimal partition is no longer equidistant. An asymptotic rate of convergence
will also be presented for the Lyons-Victoir method. The analysis rests upon
Kusuoka-Stroock's results on the smoothness of the distribution of the solution
of a stochastic differential equation. Finally, the results will be applied to
the numerical solution of the filtering problem.