Michael J. Hopkins, Lars Hesselhot, and Haynes R. Miller
Professor Hesselholt, working with Ib Madsen and Thomas Geisser, has made deep advances in understanding the cyclotomic trace spaces associated to a ring. These spaces were invented to describing the diffeomorphism groups of high dimensional manifolds, and Hesselholt's work is a significant step in that program. Central to Hesselholt's work is a connection he descovered between the cyclotomic trace spaces and the de Rham-Witt complex of p-adic arithmetic algebraic geometry. This has led to a generalization of the de Rham-Witt complex to a mixed characteristic situation, where it is related to the sheaf of p-adic vanishing cycles. This relationship further suggests the existence of a motivic theory behind the cyclotomic trace spaces. Under the support of this grant, Hesselholt will pursue this striking relationship between p-adic arithmetic algebraic geometry and the computations in topology relevant to understanding diffeomorphism groups. Professor Hopkins, working with Matthew Ando and Charles Rezk has recently uncovered a new approach to congruences between modular forms using algebraic topology, which he plans to pursue with an eye on several open problems in the area. In another collaborative effort, Professors Hopkins, Dan Freed and Constantin Teleman discovered a topological expression for the Verlinde algebra--an algebraic structure arising in mathematical physics and in the theory of representations of loop groups. The three of them plan to explore this connection more deeply, with the long term ambition of discovering the topological sources of topological quantum field theories. Hopkins will also continue his work with Isadore Singer on their theory of ``differential function spaces,'' which offers a refinement of algebraic topology especially suited for meeting the demands of mathematical physics. Professor Miller will continue his work on Landweber exact and elliptic cohomology theories, his work with Bill Dwyer on topological Hochschild homology and the geometry of free loop spaces, and his study of the homotopy theory of algebraic structures arising in knot theory and the modern theory of Hopf algebras.
The field of algebraic topology arose in the late 19th century as mathematicians noticed deep similarities between very different looking areas of research. By the beginning of the 20th century, the basic theory of algebraic topology was in place, and it was being used to explain, codify and reveal many qualitative aspects of geometry. During the next hundred years algebraic topology advanced at an amazing pace, and by now it seems as if every branch of contemporary mathematics draws on some form of algebraic topology when undertaking to articulate its qualitative aspects. The algebraic topology group at MIT runs a large and diverse program emphasizing the many connections between algebraic topology, algebraic geometry, geometry, and mathematical physics.
