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.

该研究项目涉及多个作者最近引起的一系列令人惊讶的猜想的代数方面，这些作者与代数几何学的主题，结的代数理论，拓扑结构，拓扑结构，困难组合（计数）问题，数量理论和数学物理学有关。关联如此广泛主题的猜想通常对数学家尤其引人注目，并且可以带来特别有力的结果。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”点很有用。下面提到的所谓HLV猜想不仅预测了通常称为“角色品种”的某些空间的公式，而且还以与其拓扑相关的方式概括它们。该家族中的其他猜想与一些非常引人注目的公式（称为停车功能）连接了密切相关的空间，这些公式是手工测试的基础。尽管该项目基于代数方法，但对该主题的全面数学理解有望揭示许多深处开放问题背后的几何形状，其中一些问题根源在物理学中。研究人员还计划让本科和研究生研究人员参与该项目。这项活动将集中于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，这是对不熟悉这些主题的学生研究人员的特别有效方法。通用计算机软件的开发是该项目的另一个影响，预计该项目对计算领域的研究人员有用。该项目研究的一些主题是仿生的弹簧纤维，Khovanov-Rozanksy结的共同体，Khovanov-Rozanksy结的不变性，一些著名的Hausel，Letriguez-Villegas（Rodriguez-villegas contrument confient contrument contrument contrument contrument contrument）（由于Alday，Gaiotto和Tachikawa（AGT）以及相关的混乱猜想证明的组合延伸，例如Nabla Positivitive的猜想。在这些猜想的一侧，几乎所有这些主题都与复杂平面的希尔伯特方案上的滑轮有共同的（猜想）关系。在另一侧，它们是通过存在的riemann表面连接的，其意义隐藏在希尔伯特方案侧，除了通过公式。例如，在弹簧纤维的示例中，这种riemann表面将是刺穿的圆盘c^*，在HLV猜想的情况下定义了特征的刺破表面，或者在AGT的情况下发生了形式的二维表面。该项目的目的是朝着这些猜想的数学证据取得进步，发现新的猜想，并最终理解一般的数学图片。该方法的一个主要方面是在可用时从显式组合公式中推断出来，例如麦克唐纳理论中通常称为“ nabla公式”中出现的那种。了解这种联系对数字理论，代数几何和组合学具有相当大的兴趣。第二个方面是创建了用于测试新猜想的复杂计算机软件，以及生成数据以预测与几何形状的一般关系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准通过评估来进行评估的。