Algebraic Topology

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DMS-0306429Matthew Ando and Randy McCarthyAndo and McCarthy will study the relationship between stable homotopytheory and mathematical physics and number theory mediated by ellipticcohomology, and the relationship between stable homotopy theory andalgebra mediated by K-theory. Ando will use elliptic cohomology toinvestigate what string theory and the theory of elliptic curves haveto say about the topology of compact manifolds. He is particularlyinterested in bringing recent work on open string theories andD-branes to bear on problems in elliptic cohomology. McCarthy willdevelop a theory of Witt structures for bimodules and a theory ofsmoothness and De Rham cohomology for commutative ring spectra. Theformer offers new insight and substantial generalizations ofcalculations of Hesselholt and Madsen, while the latter offers thepossibility of developing a crystalline Chern character and soextending to commutative ring spectra the work of Bloch.The last few years have seen an astonishing proliferation ofunexpected new structure in topology, involving new interactions withother areas of mathematics and physics. Ando and McCarthy areparticularly interested in articulating these new connections.Ando's research seeks to give explicit form to the deep and surprisingrelationship between topology, high energy physics, and number theorysignaled by elliptic cohomology. The traditional paradigm ofalgebraic topology is the use of algebra to model phenomena intopology. McCarthy's research turns this paradigm upside down, usingthe deep structure of topological spaces to tackle hard problems inalgebra.