RUI: Algebra and Geometry of Matroids and Polytopes

在过去的50年里，为了响应现代计算、物理学和生物学的数学需求，以及所有数学领域的计算需求，组合学领域得到了极大的发展和深化。这个组合学的研究项目是由两个相关的哲学驱动的：许多数学对象是通过研究它们背后丰富的离散结构来最好地理解的，许多这些离散结构是通过为它们建立几何模型来最好地理解的。这项拨款构成了旧金山州立大学-哥伦比亚组合学计划的学术支柱，这是一个充满活力的研究和培训合作项目，以研究型课程、研究项目和暑期学校为特色。自2007年以来，该计划已培训了250多名博士预科生。学生;超过一半的美国参与者是数学领域代表性不足的群体的成员，超过80人已经攻读了博士学位。该计划的卓越和包容实践被PI和许多其他机构广泛分享，并成为全国乃至全球其他数学和科学项目的典范。本课题主要研究组合学与几何交叉的两个研究方向：1。我们构造和研究了几个矩阵的几何模型。利用热带几何、霍奇理论和交点理论的工具，我们推导出了拟阵的新的组合性质，这些性质在纯组合上似乎是无法获得的。2. 一般来说，多面体的测量是非常困难的，但当多面体具有足够的组合结构或代数结构时，测量是可以做到的。我们构建和测量在各种数学设置中出现的多面体族，并使用这些测量来计算感兴趣的代数和几何量。在这两个研究方向中，相关对象通常是由一个向量配置（通常是根系统）构建的。多面体和拟阵提供了一个强大的工具来研究这种结构。提出的项目将进一步加深我们对组合学、离散几何、表示理论和代数几何等基本问题的理解。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。

The field of combinatorics has grown and deepened enormously in the last 50 years, in response to the mathematical needs of modern computing, physics, and biology, and the computational needs of all areas of mathematics. This research project in combinatorics is driven by two related philosophies: many mathematical objects are best understood by studying the rich discrete structures underlying them, and many of those discrete structures are best understood by building geometric models for them. This grant constitutes the academic backbone of the San Francisco State University-Colombia Combinatorics Initiative, a vibrant research and training collaboration featuring research-based courses, research projects, and summer schools. Since 2007 the initiative has trained more than 250 pre-Ph.D. students; more than half of the US participants are members of underrepresented groups in mathematics, and more than 80 have gone on to Ph.D. programs. The initiative’s practices of excellence and inclusion are shared broadly by the PI and many others, and they serve as a model for other math and science programs nationwide and beyond.This project studies two research directions at the intersection of combinatorics and geometry: 1. We construct and study several geometric models of a matroid. Using tools from tropical geometry, Hodge theory, and intersection theory, we derive novel combinatorial properties of matroids that do not seem accessible purely combinatorially. 2. Measuring polytopes is very difficult in general, but it can be done when the polytopes have sufficient combinatorial or algebraic structure. We construct and measure families of polytopes arising in various mathematical settings, and use those measurements to compute algebraic and geometric quantities of interest. In both of these research directions, relevant objects are often built from a configuration of vectors – usually a root system. Polytopes and matroids offer a powerful toolkit to study such configurations. The proposed projects will further our understanding of fundamental questions in combinatorics, discrete geometry, representation theory, and algebraic 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.