Braids, Resolvent Degree and Hilbert's 13th Problem

辫子、解决度和希尔伯特第十三问题

• 批准号: 1811772
• 负责人: Benson Farb
• 金额: $ 55.5万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811772&HistoricalAwards=false
• 关键词: Braids; Resolvent; Degree; Hilbert; 13th

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.

多项式方程无处不在。 它们用于描述，解释和预测任何受力（重力，电气等）的物体的运动；它们用于建模金融，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。 关于多项式的最古老，也许最根本的问题是了解他们的解决方案。特别是，多项式的根（即溶液）如何取决于其系数。该项目的目的是利用现代数学的令人难以置信的力量来对这个问题进行新的启示。理解多项式的主要主题之一是确定可以减少一般n多项式解决方案的参数r（n）的最小次数。 在16-19世纪，大量工作专门用于在R（n）上给予上限。希尔伯特（Hilbert）的第13个问题（及相关猜想）在R（n）上提出了一些特定的下限，但直到现在尚未显示出非平凡的下限。 正式的“回避程度”（由于Brauer和Arnol'd-shimura）的正式概念使R（n）明确的定义。 J. Wolfson和研究人员建立了一个框架，其中这些问题被置于更广泛的背景下，其中包括列举代数几何形状的许多问题。 借助Kisin M. Kisin，研究人员正在利用这种新的观点以及强大的现代工具来攻击希尔伯特的问题。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。