The field of homotopy theory is the study of mathematical invariants that are insensitive to deformations. It is applicable whenever one is interested in studying qualitative aspects of a system, or whenever there might be imprecision in the specification of the state of a system. In recent years the methods of homotopy theory have found use in fields as diverse as condensed matter physics and the foundations of mathematics. This project aims to bolster these relationships with new tools from algebraic topology, and to apply them to other areas of mathematics and science. There are applications of this work to condensed matter physics, classical algebraic geometry, and, in the long term, to education.
The scope of this project involves several interrelated areas of study. One of these, on algebraic vector bundles, depicts a new interface between complex analysis and algebraic topology, and is intended to get at the obstruction to topological vector bundles having algebraic structures. Another organizes tools developed for the study of differential manifolds into higher categories, in service of providing a general topological expression for the evaluation of "Gaussian integrals" in topological quantum field theories. The two principal investigators will work jointly on a project designed to establish a striking, conjectured, structural result for "groups of units" occurring in algebraic topology. This result has implications for the construction of topological quantum field theories and offers a potentially new approach to understanding the "homotopy groups of spheres." Three further projects involve new generalizations of ideas in homotopy theory, in the contexts of algebraic K-theory, "transchromatic homotopy theory" and in a relationship between Goodwillie calculus and the construction of "smash" products in the stabilizatons of infinity categories.
