CAREER: Branched Covers in Dimensions Three and Four

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Topology is an area of mathematics with applications to many fields, from the knotting of proteins and DNA to the structure of our universe. This project concerns the topology of three- and four-dimensional spaces, or manifolds, as well as the knotted objects they contain. To classify manifolds, one studies relationships between them; one such relationship is that of a branched cover. The PI will develop combinatorial and computational tools to study branched covers of three- and four-dimensional manifolds. Using these tools to make cutting-edge problems in the field accessible, the PI will design subprojects for undergraduate and post-baccalaureate students with a wide variety of mathematical backgrounds and professional goals. The educational component of this project expands and supports the PI’s current work in programs increasing access for women and underrepresented groups in the mathematical sciences, and addresses COVID impacts on the mathematical pipeline by providing support for students whose studies were disrupted. Crucial to this is the PI’s continued leadership role in the Center for Women in Mathematics Post-Baccalaureate Program at Smith College, as well as organization of conferences for undergraduate student researchers. These activities broaden participation in the field by preparing students for graduate study in the mathematical sciences. The project explores the geography and botany problems for branched covers of three- and four-manifolds, particularly when equipped with additional geometric structure. The geography problem asks which manifolds arise as branched covers of a given manifold, subject to constraints on the degree of the cover or complexity of the branching set; the botany problem asks for a classification of branched covering maps between a given pair of manifolds. A key strategy is the development of combinatorial and diagrammatic methods for computing invariants of knots and surfaces derived from branched covers of three- and four-manifolds, respectively. Applications will include resolution of open problems in several active areas of the field, including trisections of four-manifolds, knot concordance, and contact topology.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.