Polytopes and Matroids in Algebra and Geometry

在过去的几十年里，为了响应现代计算的数学需求和所有数学领域的计算需求，组合学作为一个领域已经发展和成熟。特别是近几年来，组合学和几何学通过它们直接联系的方式和相互学习的方式使彼此受益匪浅。S的哲学和技巧。本课题研究代数和几何对象的离散结构，以及拟阵和多面体的代数和几何结构。这种跨学科的观点为长期存在的问题提供了意想不到的有效方法。该研究项目构成了美国和哥伦比亚主要本科院校之间充满活力的研究和培训合作的学术支柱。通过以研究为基础的课程，垂直和地理整合的研究项目，以及两年一次的Encuentro Colombiano de Combinatoria，学生参与真正的国际合作，同时为组合学做出实质性的科学贡献。自2007年以来，该项目已培养了150多名博士预科生。美国学生，其中一半以上是数学领域代表性不足的群体的成员，其中50多人攻读了博士学位。该计划还通过分发课程视频、课堂讲稿和研究项目为全世界的数学家提供服务。本项目研究三个交叉交叉、相互关联的研究方向：1。通过对拟阵几何结构的研究，我们得到了在没有几何透视的情况下似乎难以解决的纯组合结果，并建立了热带几何和组合Hodge理论的基础结果。2. 我们测量感兴趣的多面体是通过识别它们是？的对吧?多面体家族，并找到一个公式来测量该家族中所有多面体。这些结果在环面几何和表示理论中具有重要意义。3. 我们探讨了各种多面体族的hopf -代数结构。所得到的对象在许多组合和代数应用中是有用的，并且它们提出了独立的几何问题。这三个项目的核心都是一个载体的配置——通常是一个根系——它起着至关重要的作用。多面体和（Coxeter）拟阵的组合理论就是为了研究这种结构而设计的，它们提供的强大工具包是这个项目的统一线索。所提出问题的解决方案将对组合学和离散几何产生重大影响，并将进一步加深我们对代数和几何中心问题的理解。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。

In the last few decades combinatorics has grown and matured immensely as a field, in response to the mathematical needs of modern computing and the computational needs of all fields of mathematics. In particular, combinatorics and geometry have benefitted each other tremendously in recent years, both by the ways in which they are connected directly, and by learning from each other?s philosophies and techniques. This project studies the discrete structure of algebraic and geometric objects, and the algebraic and geometric structure of matroids and polytopes. This interdisciplinary perspective offers unexpected and effective approaches to long-standing problems. This research program constitutes the academic backbone of a vibrant research and training collaboration among primarily undergraduate institutions in the U.S. and Colombia. Through research-based courses, vertically and geographically integrated research projects, and the biannual Encuentro Colombiano de Combinatoria, students participate in a truly international cooperation while making substantial scientific contributions to combinatorics. Since 2007 this initiative has trained more than 150 pre-Ph.D. U.S. students, more than half of whom are members of underrepresented groups in mathematics, and more than 50 of whom have gone onto Ph.D. programs. The initiative also serves mathematicians worldwide through the distribution of course videos, lecture notes, and research projects.This project studies three interdisciplinary and interrelated research directions: 1. By studying the geometric structure of matroids, we obtain purely combinatorial results that seem intractable without the geometric perspective, and we establish foundational results in tropical geometry and combinatorial Hodge theory. 2. We measure polytopes of interest by recognizing that they are part of ?the right? family of polytopes, and finding a formula for the measure of all the polytopes in that family. The results have consequences in toric geometry and representation theory. 3. We explore the Hopf-algebraic structure of various families of polytopes. The resulting objects are useful in numerous combinatorial and algebraic applications, and they raise geometric questions of independent interest. At the heart of each of these three projects lies a configuration of vectors - usually a root system - that plays an essential role. The combinatorial theories of polytopes and (Coxeter) matroids are designed to study such configurations, and the powerful toolkit that they offer is the unifying thread of this project. Solutions to the proposed problems will have a strong impact in combinatorics and discrete geometry, and will further our understanding of central questions in algebra and geometry.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.