Combinatorial Methods in Algebraic Geometry

代数几何中的组合方法

• 批准号: 1802371
• 负责人: Erik Carlsson
• 金额: $ 15.01万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-10-01 至 2021-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802371&HistoricalAwards=false
• 关键词: Combinatorial; Methods; Algebraic; Geometry

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.

这项研究项目涉及由几位作者最近提出的一系列令人惊讶的猜想的代数方面的问题，相关主题包括代数几何、纽结的代数理论、拓扑学、困难的组合(计数)问题、数论和数学物理。对于数学家来说，涉及如此广泛的主题的猜想往往特别令人信服，并可能导致特别强大的结果。举个例子，数学家们经常发现，在将数学对象参数化的空间上“计数点”是很有用的。下面提到的所谓的HLV猜想不仅为某些通常称为“特征簇”的空间预测了这样做的公式，而且还以一种与它们的拓扑相关联的方式对它们进行了推广。这个家族中的其他猜想将密切相关的空间与一些极其引人注目的公式联系在一起，这些公式涉及称为停车函数的图表，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的，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原理，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参与该项目。这项活动将集中于只需要最少学生先决条件的组合方法，以及创建用于进行计算实验的代数软件，对于不熟悉这些主题的学生研究人员来说，这是一个特别有效的方法。通用计算机软件的发展是这个项目的另一个影响，预计将对计算领域的研究人员有所帮助。本项目研究的一些主题包括仿射Springer纤维的上同调，Khovanov-Rozanksy纽结不变量，Hausel，Letellier和Rodriguez-Villegas(HLV)的一些著名猜想，由于Alday，Gaiotto和Tchikawa(AGT)而导致的将四维规范理论与共形理论联系起来的猜想，以及与Shuffle猜想证明相关的组合推广，如不能正性猜想。在这些猜想的一方面，几乎所有这些主题都与复平面上点的希尔伯特格式上的层有共同的(猜想的)关系。另一方面，它们通过黎曼曲面的存在而联系在一起，除了通过公式，黎曼曲面的重要性隐藏在希尔伯特格式的一侧。例如，在Springer光纤的例子中，这个Riemann曲面将是被穿透的圆盘C^*，在HLV猜想的情况下是定义特征变化的亏格g的被穿透表面，或者在AGT的情况下是在其上发生保形场理论的二维表面。这个项目的目标是在这些猜想的数学证明方面取得进展，发现新的猜想，并最终理解一般的数学图景。这种方法的一个主要方面是从显式组合公式(如洗牌猜想中出现的那种，在麦克唐纳理论中通常被称为“纳布拉公式”)进行外推。理解这种联系对数论、代数几何和组合学都很有意义。第二个方面是创建复杂的计算机软件来测试新的猜想，以及生成数据来预测与几何的一般关系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。