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

这项研究项目是由这样一种哲学驱动的，即数学中的许多对象和关系都是通过研究它们背后丰富的离散结构来最好地理解的。在过去的几十年里，组合数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。这个项目研究了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关键作用，并为长期存在的问题提供了意想不到的创新方法。这一研究项目构成了旧金山州立大学-哥伦比亚组合学倡议的学术支柱，该倡议是美国和哥伦比亚主要本科生机构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，该计划已经培训了200多名博士前学生，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。该计划还通过分发课程录像、课堂讲稿和研究项目，帮助培训世界各地的数学家。该项目研究不同数学领域的重要问题，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(正阵及其代数结构)和组合交换代数(Fröberg猜想)。这些学科以令人惊讶的方式联系在一起，来自一个领域的技术成为了其他领域的强大工具。许多这些问题的核心是一种载体的配置--通常是一个根系--它起着至关重要的作用。多面体、(Coxeter)拟阵和超平面排列的组合理论就是为了研究这些构型而设计的，它们提供的强大工具包是这个项目的统一线索。对正在研究的问题的解决将对组合学和离散几何产生强烈的影响，并将加深我们对代数和几何中的中心问题的理解。

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.