Swarthmore BA 1960, Princeton PhD 1964
I am interested in a variety of topics in and around algebraic topology, with recent focus primarily on equivariant stable homotopy theory in general and equivariant infinite loop space theory in particular.
The calculational parts of algebraic topology nowadays tend to focus on stable homotopy theory, which includes all of homology and cohomology theory. That area changed drastically in the late 1990s with the introduction here and elsewhere of categories of spectra ("stable spaces") in which one "can do algebra". In the last decade, equivariant stable homotopy theory, pioneered here already in the 1980's, has come to the forefront due to its unexpected applications to the solution of nonequivariant problems, most notably the Hill, Hopkins, Ravenel solution of the Kervaire invariant problem in all but one case.
The equivariant theory is now well developed theoretically, but it is still in a primitive state calculationally. These new developments have also led back to interest in the calculation of stable homotopy groups of spheres, which is where I started. It is essential to understanding of both the general structure of the stable homotopy category (chromatic homotopy theory) and the pivotal specific problem of solving the last open case of the Kervaire invariant problem. The equivariant versions of chromatic homotopy theory and the homotopy groups of spheres are virtually unexplored territory.
Infinite loop space theory nonequivariantly led directly to higher category theory, to many related developments, and to a wealth of specific calculations. The equivariant version, especially its multiplicative elaboration, is vastly more difficult and also much more categorically intensive, requiring new category theory as well as new algebraic topology. The idea is to build interesting highly structured \(G\)-spectra from highly structured \(G\)-spaces or, more deeply, \(G\)-categories. For example, such \(G\)-spectra are expected to play a large role in developing equivariant algebraic \(K\)-theory.