Homotopies and Sweepouts of Least Complexity

William Marsh Rice University, Houston TX

Investigators

Abstract

In this project, the principal investigator seeks to understand homotopies and sweepouts of least complexity. Homotopies and sweepouts are methods of continuously moving between two objects, such as two curves in the plane. In this context, we can ask questions such as how long the intermediate curves must be, or how many times the intermediate curves must come into contact with themselves (self-intersect). Since the mid-twentieth century, numerous problems have been solved by reducing them to the study of homotopies and sweepouts; by exploring the minimal complexity of these objects, we are able to build upon and improve these results. The principal investigator will also work on several important problems in geometric analysis that are inspired by physical phenomena, such as why soap bubbles are round (the isoperimetric inequality). Many of these problems will involve collaboration with other scientists across a number of fields, as well as involvement of more junior academics such as graduate students and postdoctoral fellows. This project seeks to find homotopies and sweepouts of minimal complexity subject to constraints, where the notion of complexity is geometric and/or topological in nature. In addition, the principal investigator will explore applications of these results to a variety of problems in metric geometry and geometric analysis. One type of problem that is particularly of interest is when a homotopy or sweepout satisfying limited geometric constraints can be replaced by one possessing much stronger properties, such as when a homotopy can be replaced by an isotopy. This project will also seek to develop quantitative h-principles; by combining the techniques described above with topological h-principles used to prove existence, we will be able to prove not only existence but quantitative bounds on the geometric complexity of the objects that are produced. Lastly, this project will study stability problems for nonlocal geometric inequalities, especially those related to the Coulomb energy, and will endeavor to solve the isoperimetric problem for certain classes of manifolds with density. 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.