Algebraic Topology

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

DMS-0306429 Matthew Ando and Randy McCarthy Ando and McCarthy will study the relationship between stable homotopy theory and mathematical physics and number theory mediated by elliptic cohomology, and the relationship between stable homotopy theory and algebra mediated by K-theory. Ando will use elliptic cohomology to investigate what string theory and the theory of elliptic curves have to say about the topology of compact manifolds. He is particularly interested in bringing recent work on open string theories and D-branes to bear on problems in elliptic cohomology. McCarthy will develop a theory of Witt structures for bimodules and a theory of smoothness and De Rham cohomology for commutative ring spectra. The former offers new insight and substantial generalizations of calculations of Hesselholt and Madsen, while the latter offers the possibility of developing a crystalline Chern character and so extending to commutative ring spectra the work of Bloch. The last few years have seen an astonishing proliferation of unexpected new structure in topology, involving new interactions with other areas of mathematics and physics. Ando and McCarthy are particularly interested in articulating these new connections. Ando's research seeks to give explicit form to the deep and surprising relationship between topology, high energy physics, and number theory signaled by elliptic cohomology. The traditional paradigm of algebraic topology is the use of algebra to model phenomena in topology. McCarthy's research turns this paradigm upside down, using the deep structure of topological spaces to tackle hard problems in algebra.