Cohomology, curvature, classifying spaces and symmetry

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Professor Leary will continue research on geometric group theory, and geometric topology, both alone and in collaboration. In particular, he will construct infinite groups having strong fixed point properties. This is expected to have application to the study of Kropholler's hierarchy of groups. He will also study a functor that he has developed (from simplicial complexes to locally CAT(0) cubicalcomplexes) that preserves homology. This work should have applications to topology, K-theory and group theory. He will also classify all spaces that can be built by gluing polygons in a certain way that have the `largest possible' groups of symmetries.A group is the mathematician's abstraction of the notion of symmetry:groups measure symmetry in the same way that numbers measure quantity.There are fascinating connections between group theory and geometry.Professor Leary aims to construct groups with surprising combinations of properties, which should lead to new results and insights within both geometry and group theory. He will also classify a family of geometrical objects which generalize the platonic solids, five symmetrical shapes that have been studied for at least four thousand years.