Bonn University

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex or mod-$\ell$) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress including joint work with Kaletha and Spice that introduces a twist to the story, and outline some applications.

Stochastic quantisation is, broadly speaking, the use of a stochastic differential equation to construct a given probability distribution. Usually this refers to Markovian Langevin evolution with given invariant measure. However we will show that it is possible to construct other kind of equations (elliptic stochastic partial differential equations) whose solutions have prescribed marginals. This connection was discovered in the '80 by Parisi and Sourlas in the context of dimensional reduction of statistical field theories in random external fields. This purely probabilistic results has a proof which depends on a supersymmetric formulation of the problem, i.e. a formulation involving a non-commutative random field defined on a non-commutative space. This talk is based on joint work with S. Albeverio and F. C. de Vecchi.

 I will talk about optimal estimates for stochastic integrals
in the case when both rough paths and martingales play a role.

This is an ongoing joint work with Peter Friz (TU Berlin).

We derive generalized lower Ricci bounds in terms of signed measures. And we prove associated gradient estimates for the heat flow with Neumann boundary conditions on domains of metric measure spaces obtained through „convexification“ of the domains by means of subtle time changes. This improves upon previous results both in the case of non-convex domains and in the case of convex domains.
 

TBC

In this talk, I will describe how to use the partial hodograph-Legendre transformation to show the analyticity of the free boundary in the elliptic thin obstacle problem. In particular, I will discuss the invertibility of this transformation and show that the resulting fully nonlinear PDE has a subelliptic structure. This is based on a joint work with Herbert Koch and Arshak Petrosyan.

Abstract:In this talk we consider a continuous time random walk $X$ on $\mathbb{Z}^d$ in an environment of random conductances taking values in $[0, \infty)$. Assuming that the law of the conductances is ergodic with respect to space shifts, we present a quenched invariance principle for $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme. Under the same conditions we also present a local limit theorem. For the proof some Hölder regularity of the transition density is needed, which follows from a parabolic Harnack inequality. This is joint work with J.-D. Deuschel and M. Slowik.

Abstract: The boundary Harnack principle states that the ratio of any two functions, which are positive and harmonic on a domain, is bounded near some part of the boundary where both functions vanish. A given domain may or may not have this property, depending on the geometry of its boundary and the underlying metric measure space.

In this talk, we will consider a scale-invariant boundary Harnack principle on domains that are inner uniform. This has applications such as two-sided bounds on the Dirichlet heat kernel, or the identification of the Martin boundary and the topological boundary for bounded inner uniform domains.

The inner uniformity provides a large class of domains which may have very rough boundary as long as there are no cusps. Aikawa and Ancona proved the scale-invariant boundary Harnack principle on inner uniform domains in Euclidean space. Gyrya and Saloff-Coste gave a proof in the setting of non-fractal strictly local Dirichlet spaces that satisfy a parabolic Harnack inequality.

I will present a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces that satisfy a parabolic Harnack inequality. This result applies to fractal spaces.