CAREER: Three-manifolds with finite volume, their geometry, representations, and complexity

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The research project focuses on three dimensional manifolds. A three-manifold is a space that near each point looks like the three-dimensional space we live in. Mathematically, such spaces can be approached from different viewpoints. One of them is topological: considering properties of the space that are preserved by continuous deformations. Another viewpoint is geometric: studying certain rigid structures associated to the space. A three-manifold can also be described by equations and by an algebraic object called a group, which allows tools from algebraic geometry. Yet another point of view is computational: many sophisticated algorithms not only help calculate invariants of three-manifolds, but also raise questions about algorithmic complexity of various mathematical problems. This project includes a study of intrinsic geometric and topological properties of 3-manifolds, as well as the rich interplay between all these approaches. Subprojects stemming from interesting special cases of harder problems are suitable for early-career mathematicians, allowing the educational program to be strongly intertwined with the research goals. The PI will continue research training and mentoring at all stages, from projects with undergraduates to working with postdoctoral researchers. Through cross-disciplinary workshops, the PI aims to strengthen relations between the above mentioned fields of research. Building on her prior mentoring experience with students from underrepresented groups through the Association for Women in Mathematics and the Garden State LS Alliance for Minority Participation programs, the PI will continue to support underrepresented communities through research involvement. Additionally, to promote gender diversity in mathematics, the PI will organize quarterly “Women in Topology” lectures at Rutgers, Newark. Within the overarching theme to study intrinsic geometric and topological properties of three-manifolds with finite (hyperbolic or simplicial) volume, the project's goals encompass long-standing open questions about submanifolds of three-manifolds. They include obtaining universal upper bounds on the number of embedded surfaces, in the spirit of Mirzakani’s work on curves, but one dimension up; work inspired by open conjectures about embedded surfaces and arcs by Menasco and Reid from 1992, Sakuma and Weeks from 1995, Finkelstein and Moriah from 2000. Among other questions of interest are problems on the interface of algebraic geometry and knot theory, and conjectures about lower bounds on complexity of well-known topological problems. The outcomes will significantly contribute to low-dimensional topology and geometry, positively impact computational topology, and deepen the connections between geometry, topology, algebraic geometry and theoretical computer science. 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.