RUI: Volumes in tropical geometry

在过去的几十年里，热带几何已经成为代数和离散几何的不同主题之间的一个有影响力的桥梁。本质上，热带几何用线性方程（例如直线或平面）模拟的几何空间取代了非线性方程（例如抛物线或球面）模拟的几何空间。至关重要的是，这座桥梁在两个方向上运行，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导入组合学的研究。该项目将在这座桥上建立一条新的车道，以经典的体积概念为中心，并在组合学和代数几何中应用。除了这个项目的智力和数学成果，主要研究者将使用该项目中的研究问题线作为一种途径，在他的家乡旧金山弗朗西斯科州立大学培养和支持学生研究人员的多元化社区，为他们在科学博士课程和研究生涯中取得成功做好准备。其中最重要的方式，其中卷出现在代数几何是通过研究因子的代数簇，这是基本对象的研究定义方程的品种。给定一个投射簇上的除数，其顶幂的次数至少有两种体积论的解释：它是相关紧致黎曼流形的体积，以及它是与除数相关的牛顿-奥昆科夫体的体积。这个项目将通过引入体积理论工具来研究热带品种的因子和交叉数，从而在热带几何中发展这些概念的相似之处。在这个项目中引入的体积理论工具的应用包括一个新的几何理解最近有影响力的结果，关于拟阵的特征多项式的对数，允许一个推广这些对数结果的交集数的热带品种比以前的方法，该奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.