Polytopes and Matroids in Algebra and Geometry

在过去的几十年里，组合学作为一个领域得到了极大的发展和成熟，以响应现代计算的数学需求和所有数学领域的计算需求。尤其是近年来，组合学和几何学通过它们直接联系的方式，以及通过相互学习--S的哲学和技术--已经极大地互惠互利。该项目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。这种跨学科的视角为解决长期存在的问题提供了意想不到的有效方法。这一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、垂直和地理上综合的研究项目，以及一年两次的哥伦比亚组合学院，学生们在参与真正的国际合作的同时，为组合数学做出重大的科学贡献。自2007年以来，这一计划已经培训了150多名美国博士前学生，其中一半以上是数学专业代表性不足的群体的成员，其中50多人已经攻读博士学位。该倡议还通过分发课程视频、课堂讲稿和研究项目为世界各地的数学家服务。该项目研究了三个跨学科和相互关联的研究方向：1.通过研究拟阵的几何结构，我们得到了纯组合结果，这些结果在没有几何观点的情况下似乎很难处理，我们在热带几何和组合霍奇理论中建立了基础结果。2.我们通过承认多面体是权利的一部分来衡量感兴趣的多面体。多面体的家族，并找到一个公式来衡量这个家族中的所有多面体。这些结果在环面几何和表示论中有重要意义。3.研究了各类多面体的Hopf-代数结构。由此产生的对象在许多组合和代数应用中很有用，并且它们提出了独立感兴趣的几何问题。在这三个项目中，每一个项目的核心都是一种载体的配置--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构型而设计的，它们提供的强大工具包是这个项目的统一主线。提出的问题的解决方案将在组合学和离散几何中产生强烈的影响，并将促进我们对代数和几何中的中心问题的理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.