The focus of this FRG project is on the Calculus of Functors, a systematic method of studying fundamental geometric objects, particularly spaces of functions of various sorts (e.g. embeddings), through focusing on whole processes (functors) which construct the geometric objects. It allows for systematic stratification of objects in a hierarchical way that reveals invariants that satisfy `polynomial' local-to-global behavior. Pioneered by Tom Goodwillie and Michael Weiss in the late 80's and early 90's, it is only more recently that the broad power of these methods has been becoming clear. Principal Investigators Arone, Ching, Dwyer, Kuhn, Lesh, and Turchin have all been involved in important discoveries in this area, which range from new results about the periodic homotopy of spheres, to giving new models for spaces of knots. Operad Theory is another algebraic machine that has been developed to study systems of operations satisfying specified algebraic properties (associativity, commutativity, etc.) up to some sort of controlled deformation. The current work of the PIs has led to the emerging perspective that Functor Calculus has deep connections with the more studied Theory of Operads, and that one might be able to use equivariant homotopy methods to measure how much simpler the latter is than the the former. The range of application is also growing with the placement of Calculus within the broader context of Homotopical Algebra.
In Topology, one is studying geometric objects ranging from manifolds (higher dimensional versions of curves and surfaces) and knots, in the case of Geometric Topology, to spaces of continuous functions and structured rings up to deformation, in the case of Algebraic Topology. One studies such things by means of algebraic invariants. Such invariants need to be computable, which in practice means that if a `global' object is built out of `local' pieces, there is some process that allows one to attempt to calculate the global invariant from the local invariants. The purposes of this project are to (a) investigate the Calculus of Functors method of organizing and constructing such invariants, (b) to connect this to Operad Theory, the very important theory of algebraic operations, and (c) to bring these methods to a broad spectrum of mathematicians through workshops and a conference. The methods studied in this project should give new insights into many mathematical topics of ongoing and wide interest, ranging from topological complexity of algorithms to representation theory to topological field theory.
Algebraic topology is concerned in general with developing algebraic strategies for answering geometric questions about complex higher-dimensional shapes. This project follows that pattern. In one instance, the researchers found a way to synthesize collections of higher-dimensional knots by using what is essentially an algebraic model. It turns out that in a global sense each knot is determined by a correspondence betwee one algebraic object and another. The ideas behind this synthesis are fairly general, and there is hope that they will apply to other type of problems. In another case, the researchers used collections of symmetries to tease apart a family of geometric threads that weave together to give the circle. Since the circle is very simple, it was clear in advance that these threads, which are individially rather intricate, must in some way cancel one another out. The challenge was to determine how the cancellation actually takes place. This is a problem which is interesting both for its own sake, and because it is one step in the process of learning more about such threaded constructions in general. This project is part of a long-term effort by many mathematicians to reveal the deep mathematical structures behind higher-dimensional topology, and to make it possible to do practical calculations within these structures.
