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

矩阵论和图论的最新进展将组合学的方法与代数几何的概念融合在一起，以解决长期存在的猜想，并为整数序列的单峰性和对数凹性等广泛存在的现象提供深刻的见解。组合学与代数几何的相互影响是双向的；像图复合体这样的组合结构最近已经解决了曲线模空间几何中长期存在的猜想。这些pi将联合起来，及时建立新的合作关系，以解决在拟阵、图和代数几何之间的界面上最紧迫的开放问题。该项目包括研究生和博士后的参与。这个重点研究小组将以最近的突破为基础，实现以下目标：研究了Kontsevich图复合体的矩阵推广，并研究了其在阿贝尔变体模空间上权上同调中的应用；2. 研究了一类矩阵的Chow环的k -理论类似，以探讨Hecke代数的矩阵类似及其在矩阵Kazhdan-Lusztig理论中的应用3. 证明了有限群作用下Hodge-Riemann双线性关系的一种分类，并研究了一类自同构拟阵的特征多项式的等变对数凹性；4. 使用硬Lefschetz定理启发的方法来攻击关于给定大小和秩的拟阵同构类数的Welsh猜想和图的Harary边重构猜想。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。

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.