Mathematical Sciences: Homotopy Fixed Point Problems in K-Theory

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The investigator will work on the algebraic K-theory of group rings of infinite discrete groups, using the homotopy fixed point set of the action of the group on a "bounded K-group." He hopes to extend the class of groups to which the methods apply to all fundamental groups of non-positively curved manifolds with boundary, to S-arithmetic groups, and to discrete subgroups of p- adic groups. In addition, he plans to study the Dennis trace map from the stable pseudoisotopy space of a manifold M to the S1- homotopy fixed point set of its free loop space, the singular loci in representations of the symmetric group, and orbit spectra in equivariant stable homotopy theory. K-theory is one of the standard techniques for finding topological invariants of spaces and for determining properties of these invariants. Sophisticated algebra is involved. Typically one relates geometric objects to algebraic ones, calculates with the algebraic objects, and then translates the results back into geometry.