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

拟阵和图论的最新进展融合了组合数学的方法和代数几何的概念来解决长期存在的猜想，并对整数序列的单峰性和对数凹性等普遍现象提供了深刻的见解。组合学和代数几何之间的影响在两个方向上都是卓有成效的；组合结构，如图的复形，最近导致了曲线模空间几何中长期存在的猜想的解决。PI将联合起来，建立及时的新合作，以解决拟阵、图和代数几何之间界面上最紧迫的公开问题。1.研究Kontsevich图复形的拟阵推广，并将其应用于交换变种的模空间的顶权上同调；2.研究拟阵Chow环的K-理论类比，以期得到Hecke代数的拟阵类比及其在拟阵Kazhdan-Lusztig理论中的应用；3.在有限群作用下证明Hodge-Riemann双线性关系的分类，并对具有自同构的拟阵的特征多项式追求等变量对数凹性；4.使用受硬Lefschetz定理启发的方法来攻击关于给定大小和等级的拟阵的同构类的数目的威尔士猜想和关于图的Harary边重构猜想。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。

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.