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

职业：解决度、希尔伯特第十三题和几何

基本信息

批准号：
1944862
负责人：
Jesse Wolfson
金额：
$ 45万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1944862&HistoricalAwards=false
关键词：
CAREER Resolvent Degree Hilbert 13th

项目摘要

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.

多项式在科学和工程中随处可见，我们用它们来描述、预测和解释物体在力（重力、磁力等）下如何运动，它们构成了保护网上银行、电子商务的密码系统的基础。它们用于对机械、化学、生物、社会和金融系统进行建模，我们希望找到其解决方案如何依赖于系数，这是数学中最古老和最基本的问题之一。 .这样做的目的该项目的研究旨在充分利用现代数学的力量来对这个问题产生新的理解，该项目的教育部分将建立在 PI 与编舞家 Reggie Wilson 及其 Fist & Heel Performance Group 的持续合作基础上，为该项目开发创新的 STEAM 课程。中学生关于数学和黑人/非洲舞蹈之间的相互作用给定一个多项式，我们想要找到其解决方案的最简单公式，并证明自 17 世纪以来不存在更简单的公式， “最简单”意味着使用最少数量的变量的函数的公式，虽然低次多项式（最多 5 次）存在 1 变量函数的解，但在 20 世纪初还没有这样的公式。，大卫·希尔伯特猜想，对于次数超过 5 的多项式，不存在 1 变量公式（他对 7 次的具体猜想在他著名的数学问题列表中排名第 13）。 RD(n) 编写一般 n 次多项式公式所需的最少变量数在与 Benson Farb 和 Mark Kisin 的共同工作以及单独工作中，PI 一直致力于复兴 RD(n) 的研究。 ) )，为高次多项式生成更简单的公式，并开发能够显示某些 n 的 RD(n)1 的方法。通过扩大研究范围，涵盖解析和连续类似物，PI 和他的合作者旨在引入全套现代方法来解决这个问题，并为多项式的解决方案提供新的思路。为了产生更广泛的影响，PI 将在与 Reggie Wilson 和 Fist & Heel Performance 团队的长期合作的基础上开发创新。为中学生开设的 STEAM 课程，内容涉及黑人/非洲舞蹈与其所运用的数学之间的相互作用，包括分形、辫子、递归等。该奖项反映了 NSF 的法定使命，并被认为是值得的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来提供支持。