FRG: Collaborative Research: Algebra and Geometry Behind Link Homology

Hecke代数及其推广是现代表示论的中心对象。它们自然地出现在数论、表示论、代数几何，甚至低维拓扑中。Hecke代数的一个分类被用来定义一个新的纽结和链环的拓扑不变量，称为HOMFLY-PT同调。然而，从定义中计算这个不变量是非常困难的。该项目的重点是理解代数，几何，和组合结构的链接同调和分类Hecke代数，与统一，深化和澄清这些概念之间的连接的目标。最近的进展强烈表明之间的连接HOMFLY-PT同调和代数几何的希尔伯特计划的点在平面上，一个中心对象在现代代数几何和几何表示理论。 在这个合作项目中，研究人员计划比较和统一不同的方法来研究这种联系，并发展对Soergel双模范畴与Hilbert方案之间关系的基本理解。他们还计划提供一个代数几何结构的HOMFLY-PT同源性，并了解其关系的组合Macdonald polynomies.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.