I am interested in large-scale topological invariants of infinite groups. A finitely generated group with a specified generating set carries a natural metric called the word metric. This turns the algebraic object into a geometric one. Since different choices of generating sets yield distinct metrics, the correct approach is to consider all such metrics equivalent and to focus on the topological properties of the space that are invariant of small changes. I am particularly interested in properties relating to large-scale coverings and notions of dimension. I work on asymptotic dimension and related notions as well as G. Yu's property A.
I am interested in algebraic groups, a field that lies at the intersection of algebra, number theory, and algebraic geometry. More specifically, I study arithmetic groups which can be roughly thought of as discrete subgroups of linear (matrix) groups. I try to understand how combinatorial properties of such a discrete subgroup considered as an abstract group affect its arithmetic and geometric properties.
I am interested in the representation theory of infinite groups, in particular how groups act (or fail to act) on spaces of non-positive curvature. The class of non-positively curved spaces includes hyperbolic spaces, as well as infinite dimensional vector spaces. I am in particular interested in studying the cohomology of such group actions, as the study embodies a large class of interesting properties, such as Kazhdan's Property (T), that have both geometric and measure-theoretic consequences.