Algebraic Topology

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DMS-0071482Randy McCarthy (PI)Matthew Ando (co-PI)A fundamental idea in algebraic topology is to study topological spaces by attaching to them algebraic structures which are invariant under deformations. A good structure will often produce profound relationships with other areas of mathematics. Ando and McCarthy propose to investigate problems involving three suchstructures, elliptic cohomology, algebraic K-theory, and the calculus of functors. Ando proposes several projects involving elliptic cohomology, which provides a relationship between topology, algebraic geometry, and string theory. For example, he proposes a new approach to therigidity theorems for elliptic genera, based on the algebraic theory of theta functions. McCarthy plans to further pursue his already successful program of using the calculus of functors to betterunderstand algebraic K-theory. For example, he hopes that ideas from the calculus of functors will clarify Neeman's K-theory of triangluated categories. Ando and McCarthy also plan a joint project toinvestigate the role of the calculus of functors in the new homotopy theory of schemes of Voevodsky and others.