Braids, Resolvent Degree and Hilbert's 13th Problem

University Of Chicago, Chicago IL

Investigators

Abstract

Polynomial equations are everywhere. They are used to describe, explain and predict the motion of any object undergoing a force (gravitational, electrical, etc); they are used to model financial, chemical and biological systems; and they are part of computer algorithms that we use every day. The oldest and perhaps most fundamental problem about polynomials are to understand their solutions; in particular, how the roots (i.e. solutions) of a polynomial depend on its coefficients. The purpose of this project is to use the incredible power of modern mathematics in order to shed new light on this question. One of the main themes in understanding polynomials is to determine the minimal number of parameters R(n) to which the solution of a general degree n polynomial can be reduced. A huge amount of work in the 16th-19th centuries was devoted to giving upper bounds on R(n). Hilbert's 13th Problem (and related conjectures) posits some specific lower bounds on R(n), but until now no nontrivial lower bound has been shown. The formal notion of "resolvent degree" (due to Brauer and Arnol'd-Shimura) makes the definition of R(n) explicit. J. Wolfson and the investigator have built a framework in which these problems are put in a much broader context, which includes for example many problems from enumerative algebraic geometry. With M. Kisin, the investigator is using this new point of view, together with powerful modern tools, to attack Hilbert's problems. 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.