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

FRG：协作研究：拟阵、图和代数几何

• 批准号: 2229915
• 负责人: June Huh
• 金额: $ 19.5万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2229915&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Matroids; Graphs

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.

Matroid和Graph理论的最新进展融合了组合学的方法与代数几何形状的概念，以解决长期存在的概念，并深入了解宽度现象，例如非偶像性和整数序列的对数凹度。组合学和代数几何形状之间的影响在两个方向上有效地流动；组合构造（例如图形复合物）最近导致了曲线模量空间几何形状的长期概念的分辨率。 PI将联合起来并及时进行新的合作，以解决Matroid，Graphs和代数几何形状之间接口上最紧迫的开放问题。该项目包括研究生和博士学位的参与。该重点研究小组将以最新的突破来实现以下目标：1。研究Kontsevich的图形复合物的Matulal概括，并采用了针对亚伯利亚变量模型空间的最高体重共同体的购买应用； 2。研究矩形的ChOW环的K理论类似物，以朝着Hecke代数的矩形类似物以及对矩阵Kazhdan-Lusztig理论的应用； 3。在存在有限的群体作用的情况下证明了霍奇 - 里曼双线关系的类别，并追求与自动形态的矩阵的特征多项式相同的对数凹陷； 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 concept for graphs.This award reflects NSF's statutory mission and has been deemed precious of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.