Cohomology, Finiteness Conditions and Classifying Spaces for Families

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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 classifyingspace for proper actions has surprising finiteness properties. Hewill also develop and study a functor that preserves homology butreplaces an arbitrary topological space by a space admitting anon-positively curved metric. This work should have applications totopology, algebraic K-theory and group theory. Thirdly, he willcontribute to the study of L2-cohomology, by computing this invariantfor a large class of topological spaces, including the complements ofhyperplane arrangements.A group is the mathematician's abstraction of the notionof symmetry: groups measure symmetry in the same way thatnumbers measure quantity. Some of the most interestinggroups arise from geometry, such as the collection ofsymmetries of a tiling of the plane. Professor Learyaims to construct groups that share many of the propertiesof groups coming from tilings, but that cannot arise fromany `tiling' of any space. In a different direction, hehopes 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 andgroup theory.