My research interests include algebraic K-theory and L-theory, cobordisms, modular representation theory.
One of my current projects is an application of the K-theoretic machinery to the manifold topology. Namely, we propose to approach the so called cut-and-paste invariants of manifolds by constructing a K-theory spectrum that recovers the classical SK-group as its zeroth homotopy group. The goal of the project is to investigate what kind of information is contained in higher homotopy groups and to relate the constructed spectrum to the classical spectra, for instance Madsen-Tillmann spectra and the symmetric L-theory.
My second project focuses on investigating G-theory of integral group rings for finite groups and is closely related to modular representation theory. The goal of the project is to derive the version of the Hambleton-Taylor-Williams decomposition of G_n(ZG) that would hold for all finite groups G. As a part of the project I am investigating certain identities for the narrow class groups. The project mostly has an algebraic flavour, but requires the tools of algebraic topology, number theory, and modular representation theory.