Problems in Geometric, Algebraic and Quantitative Topology

University Of Chicago, Chicago IL

Investigators

Abstract

Topology is a very qualitative part of mathematics; it does not distinguish a round ball from an ellipsoid or a plate or bowl. The measurements done by topologists are usually mediated by algebraic structures, as the usual notions of length or volume are meaningless in this setting. The main thrusts of this proposal are to improve our understanding of the connection between the algebraic measurements and the geometry and to study the role quantitative measurements into topology. There are anticipated applications within topology (e.g., to a deeper understand of the symmetry) and outside of it (e.g., to differential geometry and possibly data science). This project is an attempt to deal with fundamental problems in geometric topology. The classical paradigm in that field is to reduce geometric problems to ones in algebra, either via homotopy theory or via K-theory or quadratic form theory. Typically involved, at its most successful, is some version of a Whitney lemma. This proposal deals with (1) problems involving symmetry, where the relevant Whitney lemma is known to be false and (2) with the understanding of quantitative aspects of solutions to topological problems that are sidestepped in the algebraic treatments. The methods involve Goodwillie's calculus of homotopy functors, controlled topology, stratified surgery, rational homotopy theory, and index theory. Besides its impact within topology, this work could expand our understanding of the geometry of function spaces, with implications in analysis, and also, indirectly, geometric methods of data analysis.