Topology of manifolds, abstract homotopy theory, algebraic K-theory
Much of Tom Goodwillie's work is connected with the circle of ideas called the calculus of functors. In one version, it seeks to describe the homotopy type of a function space, such as the space E(M,N) of embeddings of one manifold M in another manifold N, by using the overall structure of a functor E(-,N) whose domain is a category of subspaces of M. In another version, the objects of study are (values of) rather general functors in some larger setting for example, functors from the category of spaces to itself. In the latter version, the functors that take homotopy pushout diagrams to homotopy pullback diagrams play a key role; approximation of general functors by functors of this kind is analogous to the approximation of functions by linear functions. The first applications have been to Waldhausen's K-theory functor and to the identity functor. Goodwillie is currently exploring a larger pattern that includes these two examples.
NSF, Calculus of Functors. Grant amount - $335,250. (2002-2007)
