Braids, Resolvent Degree and Hilbert's 13th Problem

多项式方程无处不在。它们被用来描述、解释和预测任何物体在外力（重力、电等）作用下的运动；它们被用来模拟金融、化学和生物系统；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、或许也是最基本的问题是如何理解它们的解；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令人难以置信的力量来揭示这个问题的新曙光。理解多项式的一个主要主题是确定参数R(n)的最小数量，使一般n次多项式的解可以约简。在16 -19世纪，大量的工作致力于给出R(n)的上界。希尔伯特的第13个问题（以及相关的猜想）假设了R(n)的一些特定的下界，但直到现在还没有证明非平凡的下界。“解决度”的正式概念（由Brauer和Arnol'd-Shimura提出）使R(n)的定义明确。J. Wolfson和研究者建立了一个框架，在这个框架中，这些问题被放在一个更广泛的背景中，其中包括许多来自枚举代数几何的问题。在基辛先生的帮助下，这位研究人员利用这一新观点，结合强大的现代工具，来解决希尔伯特的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.