Research in Geometric and Quantitative Topology

University Of Chicago, Chicago IL

Investigators

Abstract

Topology is usually thought of as a qualitative form of geometry. This flexibility is important for the role it plays in other areas in being able to deal with somewhat noisy or imprecise data, and reason rigorously about it. For many purposes, however, one wants to know how large or complex topological constructions are, and how stable solutions are to perturbations. This requires, technically, that one studies not just nonlinear functions from one space to another, but also spaces that bound measurements (such as the size of derivatives) on the functions. This mixed analytic topological study will be at the core of the project, with applications to geometric complexity, to spaces with singularities, which benefit from such study, but also require additional geometric and algebraic tools. Application to problems of numerical computation with natural resource bounds is anticipated In more detail, this project will study the complexity measured in terms of Lipschitz constants and bi-Lipschitz constants of homotopies, embeddings, immersions and (volumes for) cobordisms in smooth and PL settings. In some sense this should have the same relation to usual geometric topology as theoretical computer science has to logic. Indeed, using undecidability results, one can prove lower complexity bounds (as in Nabutovsky's ICM talk), but much homotopy theory, especially stable homotopy theory are decidable, but not effectively so (following Brown). The new information can be thought of as providing geometric information about function spaces of Lipschitz maps, showing that they have some bounds on diameter that are quite striking in comparison to the much larger estimates (that underly approximation theory, and some learning theory) for their volumes (also known as entropy, or covering numbers). Besides the intrinsic interest in these questions, they also bear on variational problems and perhaps on computation (as in, for example, the Blum-Shub-Smale model). Earlier variants of these quantitative concerns have already arisen in the study of square integrable cohomology and controlled (and bounded) topology. These theories will continue to be studied, hopefully revealing more subtle refinements of the Borel/Baum-Connes conjectures in special cases (such as virtually solvable groups) and applications to understanding group actions on aspherical manifolds. Interestingly, although there is currently no known quantitative bound at all on the number of embeddings of one manifold in another in codimension at least three, finiteness is known (for any Bi-Lipschitz bound), there is a natural strategy combining controlled topology with new information about function spaces to give, conjecturally, sharp estimates for these finite numbers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.