9504989 Miller Several directions of research in Algebraic Topology are funded by this grant. Professors Miller and Hopkins are carrying forward their investigation of the homotopy theory of "elliptic spectra." This includes both arithmetic and analytic aspects, connecting to the arithmetic of elliptic curves and to index theory and conformal field theory. Professor Liu's work is related; he is studying the implications of the modular properties of elliptic genera. Professor Hesselholt is pursuing his analysis of algebraic K-theory via a range of homotopy-theoretic tools associated with topological Hochschild homology. Professor Garoufalidis's research focuses on the construction and analysis of 3-manifold invariants analogous to the Vassiliev invariants of knots. Despite the diverse nature of these research projects, they are all reflections of a single great tide that is currently lifting large parts of Mathematics. Inspired largely by Physics, many fields are in the process of moving from a study of phenomena accessible from a "one-dimensional" perspective to an attack using "two-dimensional" tools. In Number Theory it is this new approach which led to the solution of the 350-year-old Fermat Conjecture. The final outcome remains to be seen in Topology; but already it has yielded a much deeper understanding of large parts of homotopy theory, as well as the celebrated constructions of new invariants for knots and higher-dimensional geometric structures. ***
