Braids, Resolvent Degree and Hilbert's 13th Problem

多项式方程式无处不在。它们被用来描述、解释和预测任何物体受力(引力、电学等)的运动；它们被用来对金融、化学和生物系统进行建模；它们是我们每天使用的计算机算法的一部分。关于多项式最古老也可能是最基本的问题是理解它们的解；特别是多项式的根(即解)如何依赖于它的系数。这个项目的目的是利用现代数学的不可思议的力量来阐明这个问题。理解多项式的主要主题之一是确定一般n次多项式的解可以简化到的最小参数个数R(N)。在16-19世纪，大量的工作致力于给出R(N)的上界。Hilbert的第13个问题(以及相关猜想)假定了R(N)的一些特定的下界，但到目前为止还没有给出非平凡的下界。由于Brauer和Arnol‘d-Shimura提出的“预解度”的形式概念，使得R(N)的定义是明确的。沃尔夫森和这位研究人员建立了一个框架，将这些问题放在更广泛的背景下，其中包括例如许多枚举代数几何中的问题。与M.Kisin一起，研究人员正在使用这种新的观点，以及强大的现代工具，来解决希尔伯特的问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.