RUI: Algebraic and Geometric Aspects of Matroids, Polytopes, and Arrangements

这个研究项目是由这样一种哲学驱动的：数学中的许多对象和关系，最好是通过研究它们背后丰富的离散结构来理解的。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成长和成熟。该项目研究代数和几何中的核心问题，其中高维几何对象的组合结构起着至关重要的作用，并为长期存在的问题提供意想不到的创新方法。该研究项目构成了旧金山州立大学-哥伦比亚组合学计划的学术支柱，该计划是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。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，该计划已培训了200多名博士预科生。这些学生中有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，帮助培训全世界的数学家。本项目研究数学各个领域的重要问题，如表示论（Kostant配分函数）、计数几何（曲线的模空间）、多面体数论（Ehrhart多项式）、Brunn-Minkowski理论（多面体的赋值）、Hopf-Lie理论（多面体上的代数结构）、热带几何（热带线性空间、模空间和Hodge理论）、全正性（正子形及其代数结构）、组合交换代数（Fröberg的猜想）。这些主题以令人惊讶的方式联系在一起，一个领域的技术成为其他领域的强大工具。这些问题的核心在于载体的配置——通常是根系统——它起着至关重要的作用。多面体、（Coxeter）拟阵和超平面排列的组合理论就是为了研究这些结构而设计的，它们提供的强大工具包是这个项目的统一线索。所研究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。

This research project is driven by the philosophy that many objects and relationships in mathematics are best understood by studying the rich discrete structures underlying them. 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. This project studies central questions in algebra and geometry where the combinatorial structure of high-dimensional geometric objects plays a crucial role, and offers unexpected, innovative approaches to long-standing problems. This research program constitutes the academic backbone of the San Francisco State University-Colombia Combinatorics Initiative, 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 the initiative has trained more than 200 pre-Ph.D. 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 helps train mathematicians worldwide through the distribution of course videos, lecture notes, and research projects.This project studies important questions in various fields of mathematics, such as representation theory (Kostant partition functions), enumerative geometry (moduli spaces of curves), polyhedral number theory (Ehrhart polynomials), Brunn-Minkowski theory (valuations of polytopes), Hopf-Lie theory (algebraic structures on polytopes), tropical geometry (tropical linear spaces, moduli spaces, and Hodge theory), total positivity (positroids and their algebraic structure), and combinatorial commutative algebra (Fröberg's conjecture). These subjects are related in surprising ways, and the techniques from one field become powerful tools in the others. At the heart of many of these questions lies a configuration of vectors -- often a root system -- that plays an essential role. The combinatorial theories of polytopes, (Coxeter) matroids, and hyperplane arrangements are designed to study these configurations, and the powerful toolkit that they offer is the unifying thread of this project. Solutions to the problems under study will have a strong impact in combinatorics and discrete geometry, and will further our understanding of central questions in algebra and geometry.