RUI: Exploring the Algebraic Geometry of Matroids

RUI：探索拟阵的代数几何

• 批准号: 1802263
• 负责人: Noah Giansiracusa
• 金额: $ 16.5万
• 依托单位: Swarthmore College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2019-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802263&HistoricalAwards=false
• 关键词: RUI; Exploring; Algebraic; Geometry; Matroids

There have been recent breakthroughs in parts of combinatorics related to matroid theory coming from the importation of tools and ideas from algebraic geometry, which is the study of systems of polynomial equations and their solutions. The PI will further develop the bridge between algebraic geometry and matroid theory, thereby allowing insight and progress in one subject to carry over to the other. This paves the way for well-known applications of matroids in topics such as optimization and network theory to draw from the heavily developed, and computationally powerful, machinery of algebraic geometry. The PI will also run a summer enrichment program, focusing on interactions between math and law, for disadvantaged minority students at a nearby underfunded public high school.The PI's program of bringing more algebraic geometry into matroid theory consists of three projects. (1) Matroids can be viewed as modules over the two-element Boolean semifield or as tensors in a tropical exterior algebra. By extending this framework the PI willgeometrize various combinatorial constructions and results in the matroid literature. Motivating problems focus on transversal matroids and matroid irreducibility. (2) The maximal torus action on the Grassmannian beautifully brings together topology, combinatorics, representation theory, and algebraic geometry. The PI is extending portions of this story to diagonal subtori. This opens several paths of investigation, including the possibility of extending the K-theoretic interpretation of the Tutte polynomial from matroids to their multiset generalization, discrete polymatroids. (3) The PI is applying the tropical scheme theory he co-developed to construct a functorial moduli space of matroids. This entails a careful understanding of how matroids vary algebraically in families and could lead to an eventual Schubert calculus on the Dressian.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.

最近在与矩阵理论有关的组合学部分的一部分中取得了突破，来自代数几何学的工具和思想的进口，这是对多项式方程式及其解决方案系统的研究。 PI将进一步发展代数几何学和矩形理论之间的桥梁，从而允许一个主题的洞察力和进步延续到另一个主题。 这为基质中的众所周知的应用在诸如优化和网络理论等主题中的应用铺平了道路，从而从代数几何形状的大量开发和计算功能强大的机械中汲取了借助。 PI还将在附近一所资金不足的公立高中提供夏季丰富计划，重点介绍数学与法律之间的互动。 （1）可以将曲霉视为两元素布尔半场上的模块，也可以看作是热带外部代数中的张量。通过扩展该框架，PI将对各种组合构建体进行构成，并导致Matroid文献。激励问题的重点是横向矩形和矩阵不可约性。 （2）对硕士的最大圆环动作将拓扑，组合学，代表理论和代数几何结合在一起。 PI将这个故事的一部分扩展到对角线siptori。这打开了几个研究途径，包括将TUTTE多项式从矩形扩展到其多元概括的离散多膜质的可能性。 （3）PI应用了他共同开发的热带方案理论来构建矩阵的功能模量空间。这需要仔细地了解原子体在家庭中如何变化的代数，并可能导致最终的舒伯特微积分，该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准来评估的支持。