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Series: Stochastics Seminar

A Markov intertwining relation between two Markov processes X and Y is a weak similitude relation G\Lambda = \Lambda L between their generators L and G, where \Lambda is a transition kernel between the underlying state spaces. This notion is an important tool to deduce quantitative estimates on the speed of convergence to equilibrium of X via strong stationary times when Y is absorbed, as shown by the theory of Diaconis and Fill for finite state spaces. In this talk we will only consider processes Y taking as values some subsets of the state space of X. Our goal is to present extensions of the above method to elliptic
diffusion processes on differentiable manifolds, via stochastic
modifications of mean curvature flows. We will see that Pitman's theorem about the
intertwining relation between the Brownian motion and the Bessel-3
process is curiously ubiquitous in this approach. It even serves as an inspiring guide to construct couplings associated to finite
Markov intertwining relations via random mappings, in the spirit of the
coupling-from-the-past algorithm of Propp and Wilson and of the evolving sets of Morris and Peres.

Series: Geometry Topology Seminar

The h-principle is a powerful tool in differential topology which is used to study spaces of functionswith certain distinguished properties (immersions, submersions, k-mersions, embeddings, free maps, etc.). Iwill discuss some examples of the h-principle and give a neat proof of a special case of the Smale-HirschTheorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. Finally, I willdefine and discuss totally convex immersions and discuss some h-principle statements in this context.

Series: Analysis Seminar

Series: PDE Seminar

Series: Geometry Topology Seminar

Novikov homology was introduced by Novikov in the
early 1980s motivated by problems in hydrodynamics. The Novikov
inequalities in the Novikov homology theory
give lower bounds for the number of critical points of a Morse closed 1-form on a compact
differentiable manifold M. In the first part of my talk I shall
survey the Novikov homology theory in finite dimensional setting
and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my
talk
I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group
associated to a cohomology
class of a closed 1-form on M and its application to obtaining
new lower bounds for the number of
critical points of a Morse 1-form.

Series: Stochastics Seminar

We shall prove that a certain stochastic ordering defined in terms of
convex symmetric sets is inherited by sums of independent symmetric
random vectors. Joint work with W. Bednorz.

Series: School of Mathematics Colloquium

Series: PDE Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar