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FRG: Collaborative Research: Matroids, Graphs, and Algebraic Geometry

FRG：协作研究：拟阵、图和代数几何

基本信息

批准号：
2053261
负责人：
Sam Payne
金额：
$ 57.82万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2053261&HistoricalAwards=false
关键词：
FRG Collaborative Research Matroids Graphs

项目摘要

Recent advances in matroid and graph theory fuse the methods of combinatorics with concepts from algebraic geometry to resolve longstanding conjectures and provide deep insights into widespread phenomena such as unimodality and log concavity of integer sequences. The influences between combinatorics and algebraic geometry flow fruitfully in both directions; combinatorial constructions such as graph complexes have recently led to resolutions of long-standing conjectures in the geometry of moduli spaces of curves. The PIs will join forces and forge timely new collaborations to address the most pressing open problems at the interface between matroids, graphs, and algebraic geometry. The project includes the participation of graduate students and postdocs.This focused research group will build on recent breakthroughs to accomplish the following goals: 1. Study matroidal generalizations of Kontsevich’s graph complex and pursue applications to the top weight cohomology of moduli spaces of abelian varieties; 2. Investigate K-theoretic analogs of the Chow ring of a matroid, with a view toward a matroidal analog of the Hecke algebra and applications to matroidal Kazhdan-Lusztig theory; 3. Prove a categorification of the Hodge-Riemann bilinear relations in the presence of a finite group action, and pursue equivariant log concavity for the characteristic polynomial of a matroid with automorphisms; 4. Use methods inspired by the hard Lefschetz theorem to attack both the Welsh conjecture on the number of isomorphism classes of matroids of given size and rank and the Harary edge reconstruction conjecture for graphs.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.

拟阵和图论的最新进展将组合学方法与代数几何的概念融合在一起，解决了长期存在的猜想，并为诸如整数序列的单峰性和对数凹性等普遍现象提供了深入的见解。组合数学和代数几何之间的影响在两个方向上都富有成果。图复形等组合结构最近解决了曲线模空间几何中长期存在的猜想。 PI 将联手并及时建立新的合作，以解决拟阵、图和代数几何之间最紧迫的开放问题。该项目包括研究生和博士后的参与。这个重点研究小组将在最近的突破的基础上实现以下目标： 1. 研究 Kontsevich 的图复形的拟阵泛化，并寻求在阿贝尔簇模空间顶权上同调上的应用； 2. 研究拟阵 Chow 环的 K 理论类似物，着眼于 Hecke 代数的拟阵类似物及其在拟阵 Kazhdan-Lusztig 理论中的应用； 3. 证明有限群作用下霍奇-黎曼双线性关系的分类，并求自同构拟阵特征多项式的等变对数凹性； 4. 使用受莱夫谢茨硬定理启发的方法来攻击关于给定大小和等级的拟阵同构类数量的威尔士猜想以及图的哈拉里边缘重建猜想。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。