Combinatorial Methods in Algebraic Geometry

这个研究项目涉及代数方面的一系列令人惊讶的最新成果，由于几个作者，有关主题的代数几何，代数理论的结，拓扑结构，困难的组合（计数）问题，数论和数学物理。涉及如此广泛的主题的猜想通常对数学家特别有吸引力，并且可以导致特别强大的结果。举一个例子，数学家经常发现在参数化数学对象的空间上“计数点”是有用的。下面提到的所谓的HLV拓扑不仅预言了一个公式，用于对通常称为“特征标簇”的某些空间这样做，而且还以一种与它们的拓扑有关的方式推广了它们。这个家族中的其他一些图表将密切相关的空间与一些非常引人注目的公式联系起来，这些公式涉及图表，称为停车函数，这些公式是手工测试的基础。虽然这个项目是基于代数方法，但对这个主题的全面数学理解有望揭示许多深层次开放问题背后的几何学，其中一些问题源于物理学。研究人员还计划让本科生和研究生研究人员参与该项目。这项活动将集中在组合方法，需要最低限度的学生先决条件，并创建代数软件进行计算实验，一个特别有效的方法，为学生研究人员不熟悉这些主题。通用计算机软件的开发是该项目的另一个影响，预计将对计算领域的研究人员有用。该项目研究的一些主题是仿射Springer纤维的上同调，Khovanov-Rozanksy结不变量，Hausel，Letellier和Rodriguez-Villegas（HLV）的一些著名的理论，Alday，Gaiotto，和Tachikawa（AGT），以及洗牌猜想的证明的相关组合扩展，例如nabla正性猜想。一方面，几乎所有这些主题都与复平面上的希尔伯特点系上的层有着共同的关系。另一方面，它们通过黎曼曲面的存在而连接起来，该黎曼曲面的意义隐藏在希尔伯特格式一侧，除非通过公式。例如，这个黎曼曲面在施普林格纤维的例子中可以是穿孔圆盘C^*，在HLV构型的例子中可以是定义特征标簇的亏格g的穿孔曲面，在AGT的例子中可以是共形场论发生的二维曲面。这个项目的目标是在这些命题的数学证明方面取得进展，发现新的命题，并最终理解一般的数学图景。该方法的一个主要方面是从显式组合公式中进行外推，当它们可用时，例如出现在洗牌猜想中的排序，在麦克唐纳理论中通常被称为“纳布拉公式”。理解这种联系对数论、代数几何和组合学都有相当大的意义。第二个方面是创建复杂的计算机软件来测试新的几何图形，以及生成数据来预测与几何图形的一般关系。该奖项反映了NSF的法定使命，并被认为值得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来支持。

This research project concerns algebraic aspects of a range of surprising recent conjectures due to several authors, relating topics in algebraic geometry, the algebraic theory of knots, topology, difficult combinatorial (counting) problems, number theory, and mathematical physics. Conjectures that relate such a broad range of topics are often especially compelling to mathematicians, and can lead to particularly powerful results. To give one example, mathematicians often find it useful to "count points" on spaces that parametrize mathematical objects. So-called HLV conjectures mentioned below predict not only a formula for doing this for certain spaces commonly called "character varieties," but also generalize them in a way that is connected to their topology. Other conjectures in this family connect closely related spaces with some extremely compelling formulas involving diagrams, called parking functions, that are elementary to test by hand. While this project is based on algebraic methods, a full mathematical understanding of this topic is expected to reveal the geometry behind many deep open problems, some of which have roots in physics. The investigator also plans to involve undergraduate and graduate researchers in the project. This activity will focus on combinatorial methods that require minimal student prerequisites, and on the creation of algebra software for conducting computational experiments, an especially effective approach for student researchers unfamiliar with these topics. The development of general computer software is another impact of this project, which is expected to be useful to researchers in computational fields.Some of the topics this project examines are the cohomology of the affine Springer fiber, Khovanov-Rozanksy knot invariants, some famous conjectures of Hausel, Letellier, and Rodriguez-Villegas (HLV), conjectures relating four-dimensional gauge theory to conformal theory due to Alday, Gaiotto, and Tachikawa (AGT), and related combinatorial extensions of the proof of the shuffle conjecture, such as the nabla-positivity conjecture. On one side of these conjectures, nearly all these topics have in common (conjectured) relationships with sheaves on the Hilbert scheme of points in the complex plane. On the other side, they are connected by the presence of a Riemann surface whose significance is hidden on the Hilbert scheme side, except through formulas. For instance, this Riemann surface would be the punctured disc C^* in the example of the Springer fiber, the punctured surface of genus g defining the character variety in the case of the HLV conjectures, or the two-dimensional surface on which the conformal field theory takes place in the case of AGT. The goal of this project is to make progress towards mathematical proofs of these conjectures, discover new ones, and ultimately understand the general mathematical picture. A major aspect of the approach is to extrapolate from explicit combinatorial formulas when they are available, such as the sort that appear in the shuffle conjecture, often called "nabla formulas" in Macdonald theory. Understanding this connection is of considerable interest to number theory, algebraic geometry, and combinatorics. A second aspect is the creation of sophisticated computer software for testing new conjectures, as well as for generating data to make predictions about the general relationship with geometry.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.