Oberseminare und sonstige Vorträge

We show that the connected component of the embedding space of links in R^3 containing the split union of m Hopf links and n unknots is homotopy equivalent to a parametrised 'round' subspace, extending work of Brendle Hatcher on the unknot case. We discuss potential generalisations and applications.This is joint work with Corey Bregman.

"In this talk the geodesic equation in symmetric spaces is considered as a boundary value problem. We show that in extrinsic symmetric spaces this problem can be reduced to solving a matrix equation. Extrinsic symmetric spaces are symmetric spaces that admit a special isometric embedding into a Euclidean vector space. Some geometric properties of these spaces will also be mentioned."

Ricci solitons are solutions to the Ricci flow modulo a diffeomorphism, and arise as singularity models for the Ricci flow. In this talk, we will outline the proof of the uniqueness of the conjugate heat kernel on non-compact shrinking Ricci solitons, without any a priori curvature or volume assumptions. We then use the uniqueness of the conjugate heat kernel to prove that Perelman's celebrated Entropy functional has a unique minimiser on shrinking Ricci solitons. Consequently, this shows that the Riemann tensor is bounded with quadratic decay and that the injectivity radius is bounded from below. These geometric conditions imply shrinking solitons have a unique tangent cone at infinity.

We discuss a realizationwise correspondence between a Brownian excursion (conditioned to reach height one) and a triple consisting of (1) the local time profile of the excursion, (2) an array of independent time-homogeneous Poisson processes on the real line, and (3) a fair coin tossing sequence, where (2) and (3) encode the ordering by height respectively the left-right ordering of the subexcursions. The three components turn out to be independent, with (1) giving a time change that is responsible for the time-homogeneity of the Poisson processes. By the Ray-Knight theorem, (1) is the excursion of a Feller branching diffusion; thus the metric structure associated with (2), which generates the so-called lookdown space, can be seen as representing the genealogy underlying the Feller branching diffusion. We will relate our approach also to earlier work of Aldous and Warren on Brownian excursions conditioned on their local time profile. The lecture is based on joint work with Stephan Gufler and Goetz Kersting.

I will first introduce scalar-valued and operator-valued Schur multipliers and their partially defined versions, and will provide a Grothendieck-type characterisation of operator-valued Schur multipliers. After a brief introduction to the matrix setting, I will talk about the positive extension problem of Schur multipliers and characterise its affirmative solution in terms of structures on an operator system associated with the domain of the multipliers. This talk will be based on joint papers with R. Levene and I. G. Todorov.