Quantitative Topology and Embedding Theory

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The field of topology possesses powerful tools for constructing or obstructing the existence of geometric spaces and continuous maps between them. Here, "continuous" describes a mapping of one space to another that can be thought of as bending and compressing, but without breaking. For example, a torus (the surface of a doughnut) can be compressed without tearing onto the surface of a sphere in a manner that does not deform continuously to a constant map to a single point. However, any continuous map in the opposite direction, taking the sphere to the torus, may be deformed without tearing to a mapping onto a single point. Classic tools of topology do not address questions of size or diameter in such constructions, and this project seeks to study those questions by bringing other ideas of topology and geometry to bear. Broader impacts of these projects include mentoring of grant-supported graduate students.These projects emphasizes notions of distortion within metric topology, and will bring to bear techniques of differential graded algebras, geometric packing constructions, cobordism, and embedding theory. Some of these issues, including embedding, have entered the study of high-dimensional data through studies of dimension reduction and persistent homology. Algebraic and geometric aspects of topology are both involved in these investigations, and among the known issues arising in quantitative geometric topology are questions of complexity and decidability in the logical senses.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.