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)作用下如何运动的东西。它们构成了确保网上银行、电子商务和电子通信安全的密码系统的基础。它们被用来为机械、化学、生物、社会和金融系统建模。给定一个多项式，我们想要找到它的解。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。这个项目的研究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。这个项目的教育部分将建立在PI与编舞雷吉·威尔逊和他的拳击与高跟鞋表演小组正在进行的合作的基础上，为中学生开发关于数学和黑人/非洲舞蹈之间的互动的创新STEAM课程。给定一个多项式，我们想找到它的解的最简单的公式，并证明不存在更简单的公式。自17世纪以来，“最简单”一直是指使用变量数最少的函数的公式。虽然对于低次多项式(直到5次)，存在一元函数的解，但除此之外，还没有这样的公式。20世纪初，大卫·希尔伯特猜想，对于次数超过5次的多项式，不存在一元公式(他对7次的具体猜想在他著名的数学问题清单上排在第13位)。称RD(N)为写出一般n次多项式所需的最小变量数。在与Benson Farb和Mark Kisin的合作中，以及在单独的工作中，PI一直致力于恢复RD(N)的研究，给出更简单的高次多项式的公式，并开发能够证明对某些n的RD(N)1的方法。PI及其合作者将研究的焦点扩大到包括解析和连续的类比，目的是将全套现代方法应用于这个问题，并为多项式的解提供新的线索。对于更广泛的影响，PI将在他与雷吉·威尔逊和Fist&Amp；Heel表演小组的长期合作的基础上，为中学生开发创新的STEAM课程，讲述黑人/非洲人舞蹈与其所运用的数学之间的互动，包括分形学、辫子、递归等。该奖项反映了NSF的法定使命，并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，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.