RUI: Volumes in tropical geometry

RUI：热带几何中的体积

• 批准号: 2302024
• 负责人: Dustin Ross
• 金额: $ 22万
• 依托单位: San Francisco State 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: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302024&HistoricalAwards=false
• 关键词: RUI; Volumes; tropical; geometry

Throughout the last several decades, tropical geometry has emerged as an influential bridge between the disparate subjects of algebraic and discrete geometry. In essence, tropical geometry replaces geometric spaces modeled by nonlinear equations (a parabola or a sphere, for example) with geometric spaces modeled by linear equations (a line or a plane, for example). Crucially, this bridge runs in both directions, allowing one to study the rich structure of nonlinear spaces using linear and combinatorial techniques while also allowing one to import the deep geometric framework of algebraic geometry into the study of combinatorics. This project will build a new lane in this bridge that is centered around the classical concept of volume, with applications in both combinatorics and algebraic geometry. In addition to the intellectual and mathematical outcomes of this project, the principal investigator will use the line of research problems in this project as an avenue to train and support a diverse community of student researchers at his home institution of San Francisco State University, preparing them to succeed in PhD programs and research careers in the sciences. One of the most important ways in which volumes arise in algebraic geometry is through the study of divisors on algebraic varieties, which are fundamental objects for studying the defining equations of a variety. Given a divisor on a projective variety, there are at least two volume-theoretic interpretations for the degree of its top power: it is the volume of the associated compact Riemannian manifold, and it is the volume of the Newton-Okounkov body associated to the divisor. This project will develop parallels of these notions in tropical geometry by introducing volume-theoretic tools for studying divisors and intersection numbers on tropical varieties. Applications of the volume-theoretic tools introduced in this project include a new geometric understanding of recent influential results concerning log-concavity of characteristic polynomials of matroids, allowing one to generalize these log-concavity results to intersection numbers on a much larger class of tropical varieties than was accessible by previous approaches, as well as the development of new tropical methods for studying cones of divisors on tropical compactifications of algebraic varieties.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.

在过去的几十年里，热带几何已经成为代数和离散几何这两个不同学科之间有影响力的桥梁。本质上，热带几何用线性方程（如直线或平面）模拟的几何空间取代了非线性方程（如抛物线或球面）模拟的几何空间。至关重要的是，这座桥是双向的，允许人们使用线性和组合技术研究非线性空间的丰富结构，同时也允许人们将代数几何的深层几何框架引入到组合学的研究中。该项目将在这座桥上建造一条新的车道，以经典的体积概念为中心，同时应用组合学和代数几何。除了该项目的智力和数学成果外，首席研究员还将利用该项目中的研究问题作为培训和支持其所在的旧金山州立大学的多元化学生研究人员社区的途径，为他们在博士课程和科学研究事业中取得成功做好准备。在代数几何中，体积产生的最重要的途径之一是通过研究代数变量上的除数，这是研究变量方程定义的基本对象。给定射影变量上的一个除数，其上幂的程度至少有两种体积理论解释：它是相关紧致黎曼流形的体积，它是与除数相关的牛顿-奥昆科夫体的体积。本项目将通过引入研究热带品种的除数和交数的体积理论工具，在热带几何中发展这些概念的相似之处。本项目中引入的体积理论工具的应用包括对最近关于拟阵特征多项式的对数凹凸性的有影响的结果的新的几何理解，允许人们将这些对数凹凸性结果推广到比以前的方法更大的一类热带品种的交数上。以及研究代数品种热带紧化的除数锥的热带新方法的发展。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。