Calculus of Functors and Applications

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

DMS-0307069 Gregory Z. Arone The main goal of this project is to apply calculus of functors, especially the "orthogonal calculus" version developed by M. Weiss, to the study of spaces of embeddings. In more detail, let M, N be smooth manifolds. The PI would like to study the space of embeddings Emb(M,N) by taking the cartesian product of one or two of these manifolds with a generic Euclidean space, and investigating the properties of the obtained functor of the Eucliedan space. The general theory of calculus associates with such a functor a sequence of "derivatives", where the n-th derivative is a spectrum with an action of the orthogonal group O(n), and a "Taylor tower" - a sequence of approximations by polynomial functors. The study of the derivatives of the embedding functor leads one to consider a rather beautiful array of topological constructions, some classical, some new. The classical constructions that one encounters include (a generalization of) the Fulton-McPherson compactification, partition posets and spaces of trees. The PI hopes that this project will yield significant new insights into the topology of spaces of embeddings and automorphisms of manifolds. Manifolds are among the basic objects of study in mathematics. Manifolds come in different dimensions. One dimensional manifolds are curves, two dimensional manifolds are surfaces, and high dimensional manifolds are suitable extensions of these concepts. One of the basic questions about manifolds is: given a manifold M, what are the possible symmetries (diffeomorphisms) of M? It was T. Goodwillie who came up with the very striking idea that rather than approach such questions "one manifold at at a time", one should study systematically how the space of symmetries (or whatever it is we want to study) changes, as one varies the manifold. This results in a theory analogous to the classical differential calculus, where functions are studied via their derivatives, Taylor polynomials and so forth. This idea provides one with a powerful and beautiful framework for studying manifolds (and other objects of interest in mathematics, especially topology), subsumes a fair amount of classical techniques, and leads one to discover beautiful new constructions in topology.