Arone is interested in the interplay between calculus of functors, operads and geometric topology. He is investigating the Taylor approximations to spaces of smooth embeddings, and he has found formulas for these approximations in terms of spaces of trees and spaces of graphs. He also intends to show (in a joint work with P. Lambrechts and I. Volic) that the formality of the little cubes operad implies quite strong rational splitting results for spaces of embeddings of a manifold into a Euclidean space. These splitting results generalize to general embedding spaces some major theorems of knot theory (e.g., the collapse of the Vassiliev spectral sequence). In longer term, Arone would like to extend this work to the study of diffeomorphisms of manifolds. Roughly speaking, the first derivative of the space of diffeomorphisms is given by Waldhausen's A-theory, or alternatively by topological cyclic homology (TC), and Arone would like to find the higher analogues of TC, corresponding to the higher derivatives. The role played by the circle group in the definition of TC should in higher degrees be played by a category of graphs a fixed homotopy type, perhaps related to the outer space of graphs that has been used to study the groups of automorphisms of a free group. In a different vein, Arone is also interested in developing further the general theory of calculus of functors. For instance, he is interested in developing a theory of polynomial functors that would generalize Goodwillie's theory of homogeneous functor. Arone's understanding of the subject was given a boost by the work of M. Ching, which made clear the relevance of operads to this question. Arone is also interested in applying calculus of functors to mainstream homotopy theory. In a recent work with K. Lesh, he discovered a new filtration of complex K-theory and a rather surprising relationship between this filtration and the calculus of functors. The paper ends with a series of conjectures about the precise nature of this relationship, and he would like to pursue these conjectures.
Arone believes that the proposal will shed new light on active areas of current research in mathematics and will lead to important new insights. The proposed research should cast an important part of geometric topology in terms familiar to algebraic topologists, and conversely bring the power of homotopy theory to geometric topology. The proposed activity involves an exciting interplay of ideas from algebraic topology, combinatorics, group cohomology and geometric topology. The PI hopes that eventually it will impact the thinking of mathematicians from fields other than topology. It should also generate exciting new ways to introduce important topics in topology to students. It is already beginning to generate a number of PhD theses in mathematics.
