University of Aberdeen

Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. At the end we will mention explicit computations for some arithmetic groups and mapping class groups in terms of class numbers and special values of zeta functions. This is all joint with Wolfgang Lück and Stefan Schwede.

I shall describe recent work with Srikanth Iyengar, Henning 
Krause and Julia Pevtsova on the representation theory and cohomology
of finite group schemes and finite supergroup schemes. Particular emphasis 
will be placed on the role of generic points, detection of projectivity
for modules, and detection modulo nilpotents for cohomology.

 

Working together with the Blue Brain Project at the EPFL, I'm trying to develop new topological methods for neural modelling. As a mathematician, however, I'm really motivated by how these questions in neuroscience can inspire new mathematics. I will introduce new work that I am doing, together with Kathryn Hess and Ran Levi, on brain plasticity and learning processes, and discuss some of the topological and geometric features that are appearing in our investigations.