Topological minimal surfaces and applications to topology and group theory

Topology is classically a qualitative study of the shape of spaces, which can help understand the shape of the universe, robotics, and paths of vehicles. It is helpful to probe the space in question using low-dimensional objects such as surfaces. Relevant to this idea, this project aims to advance a more modern theme in topology: Studying quantitative and optimization problems regarding surfaces in spaces to obtain deeper insights about the ambient space. In terms of broader impacts, the PI will continue co-organizing a regional workshop and sectional meetings, mentor graduate students, and supervise the directed reading program at Purdue University. In more technical terms, the goal of this research is to better understand the minimal complexity of surface maps into various spaces under different constraints, where the complexity is measured using negative Euler characteristics. One setting is about the existence of optimizers for the stable commutator length (scl) and the Gromov norm in negatively curved 2-complexes, such as the presentation complex of hyperbolic one-relator groups. This is related to Gromov’s surface subgroup problem and unfolds into two new attempts: Reducing the problem to understanding immersed surfaces, or studying the optimization problem from a dual point of view. Another setting is about sharp estimates of a geometric-degree analog of scl, which is well-connected to various fundamental open problems in topology and group theory such as the cabling conjecture, the Kervaire-Laudenbach conjectures, and the Wiegold problem. 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.