Mathematical Sciences: K-Theory of Group Rings, a Conjectureof Hambleton-Taylor-Williams, and Isospectral Deformations of Nilmanifolds

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This project is concerned with problems in algebraic K- theory and geometric analysis. The principal investigator will study the G-theory of group rings of finite groups. He will study a conjecture of Hambleton, Taylor and Williams for which he has already established several special cases. He will also work on determining to what extent the geometry of a Riemannian manifold is determined by the spectrum of the Laplace operator. This research is concerned with algebraic K-theory. In a broad sense algebraic K-theory concerns the evolution of concepts from linear algebra such as basis and vector space. This work has implications for algebraic geometry and geometric analysis, and promises to make interesting connections between a number of different areas of mathematics.