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RUI: Algebra and Geometry of Matroids and Polytopes

RUI：拟阵和多面体的代数和几何

基本信息

批准号：
2154279
负责人：
Federico Ardila
金额：
$ 30万
依托单位：
San Francisco State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154279&HistoricalAwards=false
关键词：
RUI Algebra Geometry Matroids Polytopes

项目摘要

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.

在过去的50年里，组合数学领域得到了极大的发展和深化，以满足现代计算，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。这个组合学的研究项目是由两个相关的哲学驱动的：许多数学对象通过研究它们背后丰富的离散结构来最好地理解，而许多离散结构通过为它们建立几何模型来最好地理解。这笔赠款构成了旧金山弗朗西斯科州立大学-哥伦比亚组合动力学计划的学术骨干，该计划是一项充满活力的研究和培训合作，以研究为基础的课程，研究项目和暑期学校为特色。自2007年以来，该计划已培养了250多名博士预科生。超过一半的美国参与者是数学代表性不足的群体的成员，超过80人获得了博士学位。程序.该项目的卓越和包容性的实践被PI和许多其他人广泛分享，他们作为全国范围内和其他地区的其他数学和科学项目的典范。该项目研究组合数学和几何交叉的两个研究方向：1.我们构造并研究了拟阵的几种几何模型。使用工具，从热带几何，霍奇理论和交叉理论，我们得到新的组合性质拟阵，似乎不访问纯粹的组合。2.测量多面体通常是非常困难的，但当多面体具有足够的组合或代数结构时，它可以做到。我们构造和测量在各种数学环境中产生的多面体族，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，相关的对象通常是从向量的配置中构建的-通常是根系。多面体和拟阵提供了一个强大的工具包来研究这样的配置。该奖项反映了NSF的法定使命，并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，被认为值得支持。