LEAPS-MPS: Algorithms and Geometry in Group Theory

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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). The symmetries of any geometric object are encoded in an algebraic structure called a group. One of the primary goals of geometric group theory is to draw connections between the geometry of an object and the algebra of its symmetry group. This leads to a powerful way to link topology, algebra, combinatorics and computer science. One area of interest pursued in this project will be finding groups with interesting geometric properties that fail to be computationally “simple.” The PI will also study right-angled Coxeter groups, a class of reflection groups introduced almost 100 years ago, using techniques from probability and combinatorics to analyze their geometry and algebra. As part of this project, the PI will organize and live-stream talks by mathematicians to students at the Bard High School Early Colleges to provide opportunities to students from underrepresented groups in urban and suburban areas by connecting them with interesting work happening in the field as well learn about pathways to mathematical research on the undergraduate and post-graduate level.This project involves three research directions tied to geometric and algorithmic problems in groups. The PI will study different aspects of right-angled Coxeter groups: in one direction the Erdös-Renyi model of random graphs will be used to study negative curvature properties of random right-angled Coxeter groups; in another the topological finiteness and negative curvature properties of outer automorphism groups of right-angled Coxeter groups will be explored. The PI will also study automaticity for new examples of CAT(0) groups, and study language theoretic properties of autostackable groups.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.