CAREER: Integer Point Transforms of Polytopes

多项式是代数的基本组成部分。多面体是高维空间中具有平坦边的集合。这个项目的目的是使用多面体来回答关于多元多项式的特殊族的系数的问题。例如，哪些系数是非零的？这些系数在大小上如何相互比较？如果我们画出非零系数单项式的指数向量，它们是否构成一个多面体的整数点？如果我们绘制一个系数的直方图，连续高度的比率是否形成一个递减序列？该项目还包括教育部分，其核心目标是通过向年轻女性介绍离散数学（这是计算机科学和数据科学日益重要的领域），使她们能够在STEM职业中取得成功。舒伯特多项式和格罗滕迪克多项式是分别表示旗流形上的上同调和K理论类的多元多项式。尽管在过去的三十年里为它们开发了美丽的公式，但这些多项式的系数仍然是神秘的。甚至不清楚如何判断给定的系数是否为非零！PI将研究Schubert和Grothendieck多项式系数的凸性。例如，这些多项式的牛顿多面体饱和了吗？它们的系数是沿着沿着对数凹的吗？有没有一个多面体的整数点变换专门用于舒伯特和格罗滕迪克多项式？该项目致力于了解这些多项式的系数，并着眼于揭示其性质的多面体原因。该奖项反映了NSF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 the basic building blocks of algebra. Polytopes are sets in high-dimensional space with flat sides. The aim of this project is to use polytopes to answer questions about the coefficients of special families of polynomials in many variables. For example, which coefficients are nonzero? How do the coefficients compare to each other in size? If we plot the exponent vectors of the monomials with nonzero coefficients, do they form the integer points of a polytope? If we plot a histogram of the coefficients, do the ratios of consecutive heights form a decreasing sequence? This project also has an educational component, with the core goal of enabling young women to succeed in STEM careers by introducing them to discrete mathematics, an area of growing importance for computer science and data science.Schubert and Grothendieck polynomials are multivariate polynomials representing cohomology and K-theory classes on the flag manifold, respectively. Despite the beautiful formulas developed for them over the past three decades, the coefficients of these polynomials remain mysterious. It is not even evident how to tell if a given coefficient is nonzero! The PI will investigate convexity properties of the coefficients of Schubert and Grothendieck polynomials. For instance, are the Newton polytopes of these polynomials saturated? Are their coefficients log-concave along lines? Is there a polytope whose integer point transform specializes to Schubert and Grothendieck polynomials? This project is dedicated to understanding the coefficients of these polynomials, with an eye for exposing polytopal reasons for their properties.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.