Quantitative, Computational, and Stochastic Aspects of Topology

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Topology is the study of global properties of geometric objects which are preserved under deformation; it has recently found applications in areas such as protein folding and analysis of high-dimensional data. A particularly successful approach, which has produced innumerable results since the 1950's, is computing algebraic invariants which are then studied through algebraic means. However, in many situations this way of transforming the problem hides some inherent geometric complexity -- for example, one can deform one object to another, but only by making it very complicated somewhere in the middle. In such a case, the existence of a deformation may not be particularly meaningful from a physical, application-oriented point of view. In other cases, in contrast, one can always find a reasonably straightforward deformation, validating the use of algebraic methods for applications. The purpose of this project is to investigate these phenomena.The project will enrich our understanding of the ideas of geometric topology by proving results of three types: quantitative results, measuring the size and complexity, in various senses, of objects whose existence is known via algebra; algorithmic results, showing that certain problems can be resolved algorithmically while others cannot; and stochastic results, describing the properties of random objects. One can think of many such results as answering questions about the geometry of function and moduli spaces. Closely related questions are studied in geometric group theory, knot theory, the combinatorics of expander graphs, and the theory of topological data analysis. However, the study of high-dimensional and simply connected manifolds and complexes requires a different set of tools which have only begun to be developed. Since function spaces are found all over mathematics, this project offers connections to a number of mathematical fields ranging from theoretical computer science to fluid dynamics.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.