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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无处不在。它们是我们用来描述、预测和解释物体在力（重力、磁力等）作用下如何运动的东西。它们构成了加密系统的基础，这些加密系统保护在线银行、电子商务和电子通信。它们被用来模拟机械，化学，生物，社会和金融系统。给定一个多项式，我们想找到它的解。这些解如何依赖于系数是数学中最古老和最基本的问题之一。本项目研究的目的是利用现代数学的全部力量来发展对这个问题的新理解。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，为中学生开发创新的STEAM课程，以促进数学与黑人/非洲主义舞蹈之间的互动。给定一个多项式，我们想找到它的解的最简单公式，并证明不存在更简单的公式。自17世纪以来，“最简单”就意味着使用最少变量的函数的公式。虽然1-变量函数的解存在于低次多项式（最高5次）中，但除此之外没有这样的公式。在世纪初，大卫·希尔伯特（David Hilbert）提出，对于次数超过5次的多项式，不存在一元公式（他对7次多项式的猜想在他著名的数学问题列表中排名第13位）。把RD（n）称为写出一般n次多项式的公式所需的最小变量数。在与Benson Farb和Mark Kisin的联合工作中，以及在单独工作中，PI一直致力于恢复RD（n）的研究，为高次多项式提供更简单的公式，并开发能够证明RD（n）1的方法。通过扩大研究的重点，包括分析和连续的类似物，PI和他的合作者的目标是带来全套的现代方法来承担这个问题，并揭示新的解决方案的多项式。对于更广泛的影响，PI将建立在他与Reggie Wilson和Fist Heel Performance小组的长期合作基础上，为中学生开发创新的STEAM课程，内容涉及黑人/非洲主义舞蹈和它所使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，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。