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，预测和解释物体如何在力（重力，磁性等）下移动的方法。它们构成了保护在线银行，电子商务和电子通信的加密系统的基础。它们用于建模机械，化学，生物学，社会和金融系统。给定多项式，我们希望找到其解决方案。这些解决方案如何取决于合作是数学中最古老，最基本的问题之一。该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基于PI与编舞雷吉·威尔逊（Reggie Wilson）的持续合作，以及他的拳头和脚跟表演小组，为中学生开发创新的蒸汽课程，讨论数学与黑人/非洲主义舞蹈之间的互动。给定多项式，我们希望找到最简单的解决方案公式，并证明不存在更简单的公式。自17世纪以来，“最简单”意味着使用最少数量变量的函数的公式。虽然低度多项式的1变量函数中的溶液（最高5级），但除此之外，尚无此类公式。在20世纪初，戴维·希尔伯特（David Hilbert）猜想，对于超过5的程度多项式，没有1个变量的公式（他对7学位的特定协议在他著名的数学问题列表中排名第13）。调用RD（n）编写一般程度n多项式的公式所需的最小数量。在与Benson Farb和Mark Kisin的联合合作中，以及在Solo Works中，PI一直在努力恢复对RD（N）的研究，以生成更简单的公式，以用于高级多项式，并开发能够向某些n显示RD（n）1的方法。 PI及其合作者旨在将投资的重点扩大到包含分析和连续类比的重点，旨在将全面的现代方法置于解决这个问题上，并为多项式解决方案提供了新的启示。为了获得更大的影响，PI将基于他与雷吉·威尔逊（Reggie Wilson）的长期合作以及拳头和脚跟性能小组的长期合作，为中学生开发创新的蒸汽课程，讨论黑人/非洲主义舞蹈与其部署的数学之间的相互作用，包括分形，辫子，递归以及MORE。通过评估NSF的宣传奖，并取代了nsf的基础。优点和更广泛的影响审查标准。