Commutative algebra in algebraic geometry and algebraic combinatorics

代数几何和代数组合中的交换代数

• 批准号: 2246962
• 负责人: Patricia Klein
• 金额: $ 16.5万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246962&HistoricalAwards=false
• 关键词: Commutative; algebra; algebraic; geometry; combinatorics

Many constraints governing real-world problems, such as those arising in robotics or statistics, are described (or approximated) by solutions to polynomial equations. The set of solutions to a polynomial equation makes up a geometric object in multi-dimensional space (e.g., the set of solutions to the equation y=x^2 is a parabola in 2-space). The intersection of several of these geometric objects is made up of the points that are solutions to all of polynomial equations cutting out those geometric objects. When such an intersection consists of finitely many points, it is natural to ask how many points. Questions of this nature have been of broad interest within algebraic geometry and algebraic combinatorics since at least the time of Hermann Schubert (1848-1911), whose methodology for solving these problems forms the basis of what is now called Schubert calculus. The PI will apply techniques from commutative algebra to solve problems in modern-day Schubert calculus. Through her research program, she will train graduate students. She will also promote gender equity in the mathematical sciences by taking on leadership roles in conferences and research communities for women and non-binary mathematicians working in commutative algebra.The primary theme of this research project is the use of combinatorial models to understand affine varieties and the algebro-geometric significance of these combinatorial models. The PI will first focus on Schubert varieties in the complete flag variety and the Groebner geometry of several classes of related affine varieties. A motivating problem is to understand the Schubert structure constants. Another family of problems concerns the connection between Gorenstein liaison and Groebner degenerations. The PI will use this connection to study a range of combinatorially-natural varieties. She will also further the understanding of Gorenstein liaison itself. A link between these two focuses is the family of alternating sign matrix (ASM) varieties, which are generalizations of matrix Schubert varieties. Understanding when and why ASM varieties fail to exhibit various “niceness” properties of matrix Schubert varieties will shed light on which features of matrix Schubert varieties are essential to their good behavior. It will also facilitate the use of ASM varieties in testing an open question in Gorenstein liaison.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.

许多现实世界中的问题，如机器人或统计学中出现的问题，都是通过多项式方程的解来描述（或近似）的。 多项式方程的解的集合构成多维空间中的几何对象（例如，方程y=x^2的解的集合是2-空间中的抛物线）。 这些几何对象中的几个的交点由作为切割这些几何对象的所有多项式方程的解的点组成。 当这样一个交点由100多个点组成时，自然会问有多少个点。 至少从赫尔曼·舒伯特（Hermann Schubert，1848-1911）的时代起，这种性质的问题就在代数几何和代数组合学中引起了广泛的兴趣，他解决这些问题的方法构成了现在所谓的舒伯特微积分的基础。 PI将应用交换代数的技术来解决现代舒伯特微积分中的问题。 通过她的研究计划，她将培养研究生。 她还将通过在会议和研究社区中为女性和从事交换代数工作的非二元数学家发挥领导作用来促进数学科学中的性别平等。该研究项目的主要主题是使用组合模型来理解仿射品种和这些组合模型的代数几何意义。PI将首先关注完整旗簇中的舒伯特簇以及几类相关仿射簇的Groebner几何。一个激励性的问题是理解舒伯特结构常数。另一类问题涉及Gorenstein联络和Groebner退化之间的联系。PI将利用这种联系来研究一系列组合自然品种。她也将进一步了解Gorenstein联络本身。 这两个焦点之间的一个联系是交替符号矩阵（ASM）簇，这是矩阵舒伯特簇的推广。理解ASM簇何时以及为什么不能表现出矩阵Schubert簇的各种“niceness”性质，将揭示矩阵Schubert簇的哪些特征对其良好行为至关重要。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。