FRG: Collaborative Research: Algebra and Geometry Behind Link Homology

FRG：协作研究：链接同调背后的代数和几何

• 批准号: 1800498
• 负责人: Benjamin Elias
• 金额: $ 10万
• 依托单位: University of Oregon Eugene
• 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: 2018-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800498&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Algebra; Geometry

The Hecke algebra and its generalizations are central objects in modern representation theory. They naturally appear in number theory, representation theory, algebraic geometry, and even low-dimensional topology. A categorification of the Hecke algebra was used to define a new topological invariant of knots and links, known as HOMFLY-PT homology. However, it is extremely difficult to compute this invariant from the definition. The project is focused on understanding the algebraic, geometric, and combinatorial structure of link homology and categorified Hecke algebras, with the goal of unifying, deepening, and clarifying connections between these concepts.Recent progress strongly indicates a connection between the HOMFLY-PT homology and algebraic geometry of the Hilbert scheme of points on the plane, a central object in modern algebraic geometry and geometric representation theory. In this collaborative project, the investigators plan to compare and unify different approaches to the study of this connection and to develop the fundamental understanding of the relation between the category of Soergel bimodules and the Hilbert scheme. They also plan to provide an algebro-geometric construction of HOMFLY-PT homology and to understand its relation to the combinatorics of Macdonald polynomials.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.

Hecke代数及其概括是现代表示理论中的中心对象。它们自然出现在数字理论，表示理论，代数几何形状，甚至是低维拓扑结构中。 Hecke代数的分类用于定义一个新的结和链路拓扑，称为Homfly-PT同源性。但是，从定义中计算出这种不变的非常困难。该项目的重点是理解链接同源性和分类Hecke代数的代数，几何和组合结构，目的是统一，加深和澄清这些概念之间的联系。结果表明，Homfly-PT PTP同源性和同源性与Homfly-PT同源性和同源性之间的联系有很强的联系。平面上希尔伯特方案的代数几何形状，这是现代代数几何形状和几何表示理论中的中心对象。 在这个合作项目中，研究人员计划比较和统一研究这种联系的不同方法，并发展对Soergel Bimodules类别与希尔伯特计划之间关系的基本理解。他们还计划提供Homfly-PT同源性的代数几何结构，并了解其与MacDonald多项式组合的关系。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和更广泛影响的评估来支持的。审查标准。