A Polytopal View of Classical Polynomials

经典多项式的多面观

• 批准号: 2348676
• 负责人: Karola Meszaros
• 金额: $ 33万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348676&HistoricalAwards=false
• 关键词: Polytopal; View; Classical; Polynomials

Knot theory is the mathematical study of knots and links. A knot is a single tangled string with the ends tied; a link consists of several knots tangled together. Knot theory has wide applications in the natural sciences, such as in the study of DNA. A basic difficult question of knot theory is how to tell if two links are different: can one be deformed to the other without untying the ends of the strings? Associating polynomials to links is one way to tackle this problem. The aim of this project is to study polynomials in knot theory and other classical branches of mathematics by associating polytopes to them. Polytopes are geometric objects in arbitrary dimensions with flat sides. The study of 3-dimensional polytopes dates back to ancient times. The project also involves mentoring of graduate students as well as outreach to middle and high school students.The support of a polynomial is the set of exponent vectors of its monomials appearing with nonzero coefficients. The Newton polytope of a polynomial is the smallest integer polytope containing its support. A polynomial has a saturated Newton polytope if every integer point in its Newton polytope is in its support. These notions extend to other bases besides the monomial basis. The goals of this project are (1) the study of saturation properties of classical multivariate polynomials with respect to various bases, such as the monomial and Schubert bases; (2) the study of the integer polytopes they give rise to; and (3) their applications to outstanding conjectures. An illustrative example of this approach is the recent progress by Hafner, Mészáros and Vidinas on Fox’s conjecture from 1962, which states that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. There are many combinatorial models for the Alexander polynomial which can be used to define combinatorial multivariate Alexander polynomials. For one such model, the support of an associated combinatorial multivariate Alexander polynomial of a special alternating link is the set of integer points in a generalized permutahedron. Such polytopal results, together with the theory of Lorentzian polynomials developed by Brändén and Huh, enabled the proof of log-concavity, and thus trapezoidal property, of the original Alexander polynomial in the case of special alternating links.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.

结理论是对结和链接的数学研究。结是一根缠在一起的绳子；链接是由多个缠结在一起的结组成的。结理论在自然科学中有着广泛的应用，例如 DNA 的研究。结理论的一个基本难题是如何判断两个链接是否不同：在不解开绳子末端的情况下是否可以将一个链接变形到另一个链接？将多项式与链接相关联是解决此问题的一种方法。该项目旨在通过将多面体与任意维度的几何对象关联来研究结理论和其他经典数学分支中的多项式。对 3 维多面体的研究可以追溯到古代。多项式的支持是其以非零系数出现的单项式的指数向量集。多项式的牛顿多胞形为。如果多项式的牛顿多面体中的每个整数点都在它的支持范围内，则该多项式具有饱和牛顿多面体。该项目的目标是（1）。经典多元多项式相对于各种基的饱和性质，例如单项式基和舒伯特基；（2）它们产生的整数多面体的研究； (3) 它们在突出猜想中的应用。这种方法的一个说明性例子是 Hafner、Mészáros 和 Vidinas 对 1962 年 Fox 猜想的最新进展，该猜想指出交替链接的亚历山大多项式的系数的绝对值。形成梯形序列 亚历山大多项式有许多组合模型，可用于定义组合多元亚历山大多项式。这种模型，特殊交替链接的相关组合多元亚历山大多项式的支持是广义置换面体中的整数点集。这种多面结果与 Brändén 和 Huh 开发的洛伦兹多项式理论一起，实现了对数证明。 -在特殊交替链接的情况下，原始亚历山大多项式的凹性和梯形性质。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。