CAREER: Resolvent Degree, Hilbert's 13th Problem and Geometry

University Of California-Irvine, Irvine CA

Investigators

Abstract

Polynomials are everywhere in science and engineering. They are what we use to describe, predict and explain how objects move under a force (gravity, magnetism, etc.). They form the basis of the cryptographic systems which secure online banking, e-commerce, and electronic communication. They are used to model mechanical, chemical, biological, social and financial systems. Given a polynomial, we want to find its solutions. How these solutions depend on the coefficients is one of the oldest and most fundamental questions in math. The purpose of this project’s research is to use the full power of modern mathematics to develop a new understanding of this question. The education component of this project will build on the PI’s ongoing collaboration with choreographer Reggie Wilson and his Fist & Heel Performance Group to develop innovative STEAM curricula for middle school students on the interactions between math and black/Africanist dance. Given a polynomial, we want to find the simplest formula for its solutions, and to prove no simpler formula exists. Since the 17th century, “simplest” has meant a formula using functions of the least number of variables. While solutions in 1-variable functions exist for low degree polynomials (up to degree 5), no such formulas are known beyond this. At the beginning of the 20th century, David Hilbert conjectured that for polynomials of degree more than 5, no 1-variable formulas exist (his specific conjecture for degree 7 is 13th on his famous list of mathematical problems). Call RD(n) the minimal number of variables needed to write a formula of the general degree n polynomial. In joint work with Benson Farb and Mark Kisin, and also in solo work, the PI has been working to revive the study of RD(n), to produce simpler formulas for high degree polynomials, and to develop methods capable of showing that RD(n)>1 for some n. By broadening the focus of investigation to encompass analytic and continuous analogues, the PI and his collaborators aim to bring the full suite of modern methods to bear on this problem, and shed new light on solutions of polynomials. For the broader impacts, the PI will build on his long-running collaboration with Reggie Wilson and the Fist & Heel Performance group to develop innovative STEAM curricula for middle school students on the interactions between black/Africanist dance and the mathematics it deploys, including fractals, braids, recursion, and more. 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.