My research is in geometry and topology with an emphasis on manifolds and groups acting on them. It tends to revolve around aspherical manifolds such as locally symmetric spaces. Many interesting examples of such spaces (for instance the space of flat, volume one n-tori) are non-compact, and a good chunk of my time has been spent coming to grips with that.
One focus of my research is understanding ends of non-compact aspherical manifolds [papers 8,9,10,11 and 15]. For instance, in joint work with Tam, we've shown that many of the qualitative features of ends of locally symmetric spaces are valid for general nonpositively curved manifolds.
Another is producing various sorts of homology in aspherical manifolds and using it to obstruct group actions [8,14,16,17].
A third is finding efficient ways to thicken aspherical complexes to manifolds. This has interesting and surprising connections to classical conjectures about L^2 Betti numbers, as well as more recent questions about homology growth and fibering [1,2,4,7,12,13].
Recently, I have also been trying to understand the topology of some 2-complexes [3,5,6].
A short description of some things I am currently interested in appears here and a more elaborate one here (and the versions from a few years ago are here and here).