Groups in Geometry and Topology: Discrete, Continuous, and Higher

In this project, the PI will develop new approaches to elucidate and apply the notion of symmetry, a unifying principle within modern mathematics that has important applications in theoretical physics, and cryptographic systems. The research will be organized around two research streams: one is to understand the structure of "higher" continuous symmetries which first emerged in investigations in theoretical physics in the early 1990s, and the second is to explore discrete symmetries arising in algebraic equations, which play an essential role throughout science and engineering. By studying the symmetries of these equations, the PI will find deeper ways to solve them, and to measure the ease or difficulty of finding their solutions. Alongside this work, the PI will contribute to the mathematical training of STEM learners from middle school on up. The two streams used by the PI can be encapsulated as "higher Lie theory" and "resolvent problems". Both build upon fundamental inquiries at the interface of group theory, geometry and topology, and homotopical algebra. For the first major stream, the PI will establish analogues of Lie’s Second and Third Theorems for finite type L-infinity-algebras and finite dimensional Lie infinity-groups. For the second major stream, the PI will continue his ongoing investigation of resolvent problems and Hilbert’s 13th problem in three interrelated ways: a) initiating the study of resolvent degree of finitely generated groups, with a focus on braid groups and arithmetic lattices; b) developing new methods for generating solvable points on varieties, and use this to give p-adic analogues of Klein’s modular solutions of polynomials of low degree (with a focus on degree 8 and below); and most speculatively, c) seeking to find invariants capable of detecting nontrivial lower bounds on resolvent degree. 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.