This project combines two principal areas of current research in homotopy theory: Goodwillie's calculus of homotopy functors and operad theory. The overall goal is to understand the universal structure possessed by the Goodwillie derivatives of a functor, from which the Taylor tower of the functor can be reconstructed. The PI and Greg Arone have proved the existence of such a structure in the form of a coalgebra over a certain comonad. In the case of spectrum-valued functors, there is a close relationship between this comonad and those associated to right modules over various operads, including the little disc operads. We now study the corresponding comonads for space-valued functors, as well as for derivatives at arbitrary base objects. We also apply our previous theory to calculations such as the Taylor tower of algebraic K-theory. The Koszul duality of operads of spectra plays a key role in this theory, and we study this duality in its own right, building on previous work of the PI and John E. Harper. The main goal here is to get an equivalence between the categories of algebras over one operad and of divided power coalgebras over the Koszul dual operad.
Topology is the study of properties of shapes and spaces in any number of dimensions. One particular goal is to develop ways to measure aspects of these spaces that are usually considered qualitative. For example, a basic problem is to give a precise description of the difference between the shapes formed by the surface of the Earth (a sphere), and the surface of a bagel (a torus). The difference is intuitively clear - the bagel has a hole - but giving a precise definition of what we mean by a 'hole' in a shape allows us to make calculations in higher dimensions, where intuition is less reliable. Some of these calculations (the so-called 'homotopy groups of spheres') turn out to be extremely complicated and have been studied extensively. This project is concerned with systematic approximations to these calculations that are analogous to the Taylor series of undergraduate calculus. Our central goal is to understand how more complex calculations can be built, in a natural way, from simpler pieces. A good understanding of the pieces should then provide us with better tools for making the hard computations. The foundational nature of our approach lends itself to applications in a range of areas of mathematics, including homological algebra and representation theory, in addition to topology.
