Mathematical Sciences: Algebraic Problems Arising Out of Cohomology of Groups, Fixed point Theory and Shape

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Professor Geoghegan's areas of interest are topology and related group theory. He will investigate problems in geometric and homological group theory for which topological methods, coming from shape theory, fixed point theory and infinite- dimensional topology, are appropriate. More specifically he will investigate various questions involving finiteness properties and cohomological properties of groups which can be interpreted as questions about "ends" of covering spaces. He will continue his collaboration with Professor K.S. Brown of Cornell University on infinite-dimensional finitely presented discrete groups. On his own he will continue his investigation of the topological meaning of a well known conjecture of H. Bass about the Hattori-Stallings trace of an idempotent matrix with group ring entries. Groups are ubiquitous in mathematics and the natural sciences as codifiers of symmetry, and one can never have too many approaches to discovering their properties.