Homotopy Theory and Its Applications

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9803428 Miller In earlier work, Professors Hopkins and Miller have shown how the purely algebraic theory of elliptic curves can be embedded into homotopy theory. The result is a construction of a new object, the spectrum TMF of topological modular forms, which is a deep enrichment of the classical ring of modular forms. Professor Hopkins hopes to show that this object can be used in several ways. It should receive a generalized "Witten genus," assigning sophisticated algebraic invariants to certain geometric manifolds. On the other hand, it should account for deep arithmetic congruences satisfied by certain theta-functions (conjectured earlier by Professor Hopkins and proven by R. Borcherds). Hopkins' program is to extend the construction of TMF to a construction of suitably defined "topological theta-functions." Professor Miller intends to pursue earlier work with Hopkins on higher analogues of real K-theory; there is an infinite family of these, of which only essentially one has been investigated. He hopes to use the theory of formal groups to give conceptual and generalizable proofs of recent elaborate computations of certain homotopy groups. He will also attempt to extend the relationship between K-theory and the spectral theory of a manifold to an elliptic analogue. Professor Hesselholt intends to continue his research into the relationship between a homotopy-theoretically defined "linear" invariant of rings (topological cyclic homology) and a deep arithmetic "nonlinear" invariant (K-theory), and to extend his computation of TC to more difficult cases. The work of Professors Hopkins, Miller, and Hesselholt stands at the forefront of the application of homotopy theory to other parts of mathematics, notably, arithmetic and conformal field theory. Homotopy theory offers a vast enlargement of combinatorics, in which one keeps track not only of equivalence classes but also of the ways in which various objects are equivalent to each other. The w ork of these investigators serves as a formal background against which one may organize the search for objects with meaning in these other areas, in addition to possessing great intrinsic complexity and beauty. ***