Wochenplan des Fachbereichs Mathematik und Informatik

I will discuss an approach to the Dwyer-Weiss-Williams index theorems for topological and for smooth manifold bundles, which is based on a formalism for bivariant theories and known results about cobordism categories. In the smooth case, the theorem implies that a canonical transformation from stable homotopy to the algebraic K-theory of spaces is natural with respect to transfer maps. I will then present some results on the analogous question for Waldhausen's splitting map from the algebraic K-theory of spaces to stable homotopy. This splitting map gives a splitting of the algebraic K-theory of spaces into stable homotopy and the Diff-Whitehead space. If time permits, I will present some results on the homotopy type of the h-cobordism category and discuss its connection with the Diff-Whitehead space. (This is joint work with W. Steimle.)

The nuclear dimension of a C*-algebra, introduced by Winter and Zacharias, is a non-commutative generalisation of the covering dimension of a topological space.
Whilst any non-negative integer or infinity can be realised as the nuclear dimension of some commutative C*-algebra, the nuclear dimension of a simple C*-algebra must be either 0,1 or infinity. This trichotomy is just one application of my joint work on the Toms--Winter Conjecture with Castillejos, Tikuisis, White, and Winter. In this talk, I will outline the results, their application to classification theory, and the new ideas at the heart of our work.
I will then discuss the recent developments on the nuclear dimension of extensions, including the work on the Cuntz--Toeplitz algebras undertaken during the Glasgow Summer Project 2019.

We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity of the canonical trace. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact.
Joint work with Mehrdad Kalantar.

Kolloquium über Geschichte und Didaktik der Mathematik

In this talk I will discuss two specific questions in the theory of homogenization. First I will talk about recent progress on qualitative homogenization of degenerate elliptic equations and related probabilistic question of invariance principle for random walk in random environment. The second is more quantitative, and is related to the generalization of the multipole expansion to the case of a uniformly elliptic heterogeneous media. This is done by analyzing spaces of growing and decaying harmonic functions and relation between them.