Polytopes and Matroids in Algebra and Geometry

在过去的几十年中，组合学作为一个领域已经发展和成熟，以满足现代计算的数学需求和所有数学领域的计算需求。特别是，组合学和几何学在最近几年中相互受益匪浅，无论是通过它们直接连接的方式，还是通过相互学习？的哲学和技术。本计画研究代数与几何对象的离散结构，以及拟阵与多面体的代数与几何结构。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。该研究计划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骨干。通过基于研究的课程，垂直和地理整合的研究项目，以及一年两次的Encuentro Coconano de Combinatoria，学生参与真正的国际合作，同时为组合学做出重大科学贡献。自2007年以来，该计划已培养了150多名博士生。美国学生，其中一半以上是数学代表性不足的群体的成员，其中50多人获得了博士学位。程序.该计划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提供服务。该项目研究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，我们得到了纯粹的组合结果，似乎棘手的几何角度，我们建立热带几何和组合霍奇理论的基础结果。2.我们衡量多面体的利益认识到，他们是一部分？右边吗一个多面体的家庭，并找到一个公式的措施，所有的多面体在该家庭。结果在复曲面几何和表示理论的后果。3.我们探讨了各种家庭的多面体的霍普夫代数结构。由此产生的对象是有用的，在许多组合和代数应用，他们提出了独立的兴趣几何问题。这三个项目的核心都是载体的配置-通常是根系-发挥着至关重要的作用。多面体和（Coxeter）拟阵的组合理论旨在研究这种配置，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数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NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.