Homotopy Theory and Applications

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DMS-0306519Michael J. Hopkins, Lars Hesselhot, and Haynes R. MillerProfessor Hesselholt, working with Ib Madsen and Thomas Geisser, hasmade deep advances in understanding the cyclotomic trace spacesassociated to a ring. These spaces were invented to describing thediffeomorphism groups of high dimensional manifolds, and Hesselholt'swork is a significant step in that program. Central to Hesselholt'swork is a connection he descovered between the cyclotomic trace spacesand the de Rham-Witt complex of p-adic arithmetic algebraicgeometry. This has led to a generalization of the de Rham-Witt complexto a mixed characteristic situation, where it is related to the sheafof p-adic vanishing cycles. This relationship further suggests theexistence of a motivic theory behind the cyclotomic tracespaces. Under the support of this grant, Hesselholt will pursue thisstriking relationship between p-adic arithmetic algebraic geometry andthe computations in topology relevant to understanding diffeomorphismgroups. Professor Hopkins, working with Matthew Ando and Charles Rezkhas recently uncovered a new approach to congruences between modularforms using algebraic topology, which he plans to pursue with an eyeon several open problems in the area. In another collaborativeeffort, Professors Hopkins, Dan Freed and Constantin Telemandiscovered a topological expression for the Verlinde algebra--analgebraic structure arising in mathematical physics and in the theoryof representations of loop groups. The three of them plan to explorethis connection more deeply, with the long term ambition ofdiscovering the topological sources of topological quantum fieldtheories. Hopkins will also continue his work with Isadore Singer ontheir theory of ``differential function spaces,'' which offers arefinement of algebraic topology especially suited for meeting thedemands of mathematical physics. Professor Miller will continue hiswork on Landweber exact and elliptic cohomology theories, his workwith Bill Dwyer on topological Hochschild homology and the geometry offree loop spaces, and his study of the homotopy theory of algebraicstructures arising in knot theory and the modern theory of Hopfalgebras.The field of algebraic topology arose in the late 19th century asmathematicians noticed deep similarities between very differentlooking areas of research. By the beginning of the 20th century, thebasic theory of algebraic topology was in place, and it was being usedto explain, codify and reveal many qualitative aspects of geometry.During the next hundred years algebraic topology advanced at anamazing pace, and by now it seems as if every branch of contemporarymathematics draws on some form of algebraic topology when undertakingto articulate its qualitative aspects. The algebraic topology groupat MIT runs a large and diverse program emphasizing the manyconnections between algebraic topology, algebraic geometry, geometry,and mathematical physics.