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.

Project Report

In Topology, one is studying geometric objects by means of algebraic invariants, with the goal of using these to classify fundamental types of geometric structure. The geometric objects to be studied range 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. 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 focus of this FRG project has been on the Calculus of Functors, a relatively new systematic method of stratifying such invariants by a hierarchy of invariants that satisfy certain "polynomial" local-to-global behavior, and its relation to Operad Theory, 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. One expects that the methods investigated will help with the study of a number of fundamental objects of ongoing interest to a wide range of mathematicians. Many parts of the project involved sophisticated use of category theory, a subject that is increasingly impacting applied subjects like computer science. The two University of Virginia PIs are well known leaders in this area. Some of Arone's work, with others in the FRG, has concentrated on the general theory: for example, a classification of "polynomial" processes (functors) that one can do to topological spaces. More specific work has given new structured models for spaces of knots and their higher dimensional analogues. Kuhn's research has included a proof of an old conjecture linking two fundamental theorems in homotopy theory, developing new tools for calculating some classic invariants of infinite loopspaces, and doing work in related areas of algebra. FRG activities have including running a workshop at the math institute in Banff in March, 2011, running a conference at the University of Virginia in June, 2012, and helping run a conference in Dubrovnik, Croatia in June, 2014.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0967649
Program Officer
Christopher W. Stark
Project Start
Project End
Budget Start
2010-06-01
Budget End
2014-05-31
Support Year
Fiscal Year
2009
Total Cost
$412,016
Indirect Cost
Name
University of Virginia
Department
Type
DUNS #
City
Charlottesville
State
VA
Country
United States
Zip Code
22904