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

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。它们构成了保护网上银行、电子商务和电子通信安全的密码系统的基础。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，我们想求它的解。这些解如何依赖于系数是数学中最古老、最基本的问题之一。本项目研究的目的是充分利用现代数学的力量，对这个问题有一个新的认识。该项目的教育部分将建立在PI与编舞Reggie Wilson和他的Fist & Heel表演小组的持续合作基础上，为中学生开发关于数学与黑人/非洲主义舞蹈之间互动的创新STEAM课程。给定一个多项式，我们要为它的解找到最简单的公式，并证明不存在更简单的公式。自17世纪以来，“最简单”的意思是使用变量最少的函数的公式。虽然一元函数的解存在于低次多项式（高达5次），但除此之外没有这样的公式。20世纪初，大卫·希尔伯特（David Hilbert）推测，对于大于5次的多项式，不存在单变量公式（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位）。称RD(n)为写出一般n次多项式公式所需的最小变量数。与本森Farb和马克Kisin和独自工作,π一直在努力重振RD (n)的研究,生产简单的多项式公式高度,和开发方法能够显示RD (n) 1 n。扩大调查的重点包括分析和连续的类似物,π和他的合作者旨在把全套的现代方法对这一问题,并揭示多项式的解决方案。对于更广泛的影响，PI将建立在他与Reggie Wilson和first &amp； Heel表演小组的长期合作基础上，为中学生开发创新的STEAM课程，介绍黑人/非洲主义舞蹈与数学之间的相互作用，包括分形、辫子、递归等。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。

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.