Professor Leary will continue research on geometric group theory, geometric topology and L2-cohomology, both alone and in collaboration. In particular, he will construct groups for which the classifying space for proper actions has surprising finiteness properties. He will also develop and study a functor that preserves homology but replaces an arbitrary topological space by a space admitting a non-positively curved metric. This work should have applications to topology, algebraic K-theory and group theory. Thirdly, he will contribute to the study of L2-cohomology, by computing this invariant for a large class of topological spaces, including the complements of hyperplane arrangements.
A group is the mathematician's abstraction of the notion of symmetry: groups measure symmetry in the same way that numbers measure quantity. Some of the most interesting groups arise from geometry, such as the collection of symmetries of a tiling of the plane. Professor Leary aims to construct groups that share many of the properties of groups coming from tilings, but that cannot arise from any `tiling' of any space. In a different direction, he hopes to show that some aspects of geometry can be modeled (in a certain precise sense) within the theory of groups. This should lead to new insights within both geometry and group theory.
