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。它们自然而然地出现在数论、表示论、代数几何，甚至低维拓扑中。利用Hecke代数的一个范畴定义了一个新的结环的拓扑不变量，称为HOMFLY-PT同调。然而，从定义中计算这个不变量是极其困难的。该项目致力于理解链环同调和范畴Hecke代数的代数、几何和组合结构，目的是统一、深化和澄清这些概念之间的联系。最近的进展有力地表明了平面上点的Hilbert方案的HOMFLY-PT同调和代数几何之间的联系，平面上点的Hilbert方案是现代代数几何和几何表示理论的中心对象。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，并发展对索尔盖尔双模范畴和希尔伯特方案之间关系的基本理解。他们还计划提供HOMFLY-PT同调的代数几何结构，并了解它与麦克唐纳多项式的组合的关系。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。

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.