LEAPS-MPS: Special Points, Moduli Problems, and Resolvent Filtrations

Polynomials are ubiquitous across mathematics, engineering, and the physical and social sciences. We describe the motion of bodies, predict the behavior of complex systems, and design cryptographic techniques via the language of polynomials. Understanding how polynomial solutions depend on their coefficients is a fundamental and long-standing problem in mathematics.

This project concerns a recently formalized measure of complexity --resolvent degree -- introduced to quantify the difficulty of solving polynomials. Drawing on both classical and modern techniques in number theory and algebraic geometry, this project will enrich our understanding of resolvent degree as a measure of complexity in other contexts. The broader impacts of this project will support ongoing community building amongst underrepresented student affinity groups, including developing researchers from their ranks and contributing to professional development through conference attendances.

Undergraduate research projects will include data-scientific analyses and translations of classical mathematical literature informing research in resolvent degree, and the development of curricular units for math and math history courses towards broadening perspectives in math education.

The project aims to advance the theory of resolvent degree, a broadly applicable measure of complexity motivated by classical problems in mathematics (e.g., Hilbert’s 13th Problem) and consists of three main threads. The first advances the framework of “special points” through the machinery of polar cones, towards both sharpening bounds on resolvent degree of finite groups and related arithmetic interests; the others are centered around phenomena uncovered in work of Farb–Kisin–Wolfson that are yet to be fully understood, on the topology of moduli problems and resolvent filtrations of étale fundamental groups.

We expect to develop computational techniques, improve existing theorems in resolvent degree and adjacent areas of mathematics, formalize ideas from the literature into general frameworks, and explore new regimes that will inform trajectories of future research.

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.