Collaborative Research: Toric Geometry, Tropical Geometry, and Combinatorial Buildings

这个项目涉及代数几何和组合学的交叉研究。代数几何是研究多项式方程的解集，称为代数变分。它在许多领域都有应用，如高能物理、编码、密码学和数学生物学。理解解集的形状如何随着系数的变化而变化是该领域最古老和最核心的问题之一。这种连续变形出现在代数几何及其应用的所有分支中，被称为代数族。一个重要的例子是几何朗兰兹规划，它关注于理解曲线上的主束，这是一类非常特殊的族。束也是高能物理规范理论的主要参与者。代数族是这个项目的中心焦点。本研究旨在引入代数族分类与计算的新方法。该项目将为研究生提供研究培训机会。环面变异体是一大类变异体，其几何性质与凸格多面体的组合学密切相关。它们在当代代数几何中起着核心作用。热带几何是一个相对较新的研究领域，涉及分段线性几何的研究，其根源在于凸优化。热带几何将代数和几何中的许多问题转化为通常更易于处理的组合和凸几何问题。建筑理论是组合几何的一个领域，与拓扑学和微分几何有着深刻的联系。它旨在揭示矩阵群和相关空间中隐藏的组合几何结构。本研究的主题围绕着共同的主题，即在一个环面品种或一个环面家族上研究代数变种的家族。主要的见解是，理解环面族所需的组合学来自热带几何和建筑理论。本项目所采用的方法将导致代数几何和相关领域新技术的发展。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。

This project involves research at the intersection of algebraic geometry and combinatorics. Algebraic geometry is the study of solution sets of polynomial equations called algebraic varieties. It has applications in many fields as diverse as high energy physics, coding, cryptography, and mathematical biology. Understanding how the shape of the solution set changes as the coefficients are varied is one of the oldest and central questions in the field. Such continuous deformations, which appear in all branches of algebraic geometry and its applications, are called algebraic families. An important example is the geometric Langlands program, which is concerned with understanding principal bundles on curves, a very special class of families. Bundles are also main players in gauge theory in high energy physics. Algebraic families are the central focus of this project. The research aims to introduce new methods to classify and compute with algebraic families. The project will provide research training opportunities for graduate students.Toric varieties are a large class of varieties whose geometry is intimately connected with combinatorics of convex lattice polytopes. They play a central role in contemporary algebraic geometry. Tropical geometry, a relatively recent area of research, concerns study of piecewise linear geometry and has roots in convex optimization. Tropical geometry translates numerous questions in algebra and geometry into combinatorial and convex geometric questions that are often more tractable. The theory of buildings is an area of combinatorial geometry that has deep connections with topology and differential geometry. It aims to unravel hidden combinatorial geometric structures in matrix groups and related spaces. The topics of this research revolve around the common theme of studying families of algebraic varieties over a toric variety, or a toric family for short. The main insight is that the combinatorics needed to understand toric families comes from both tropical geometry and the theory of buildings. The approach followed in this project will lead to the development of new techniques in algebraic geometry and related fields.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.