Metric, computational, and stochastic questions in 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, materials science, analysis of high-dimensional data, and robotics. 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. Broader impacts of this project include research training opportunities for graduate students.The past few years have seen significant progress in the area of quantitative homotopy theory: understanding the complexity, typically measured by the Lipschitz constant, of objects, such as maps and homotopies, whose existence is guaranteed by algebraic topology. This progress has been particularly pronounced for simply connected spaces. The current project will seek to harness this progress mainly in two ways. The first goal is to obtain quantitative results about topics in geometric topology such as immersions, embeddings, and surgery theory. Since these topics are usually studied via reduction to homotopy theory, this will also require a more geometric understanding of these reductions. The second goal to apply the ideas of quantitative homotopy theory to studying the geometry of nilpotent groups, an area at the intersection of geometric group theory, sub-Riemannian geometry, and analysis on metric spaces.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.