Knot Homology and Moduli of Sheaves

纽结是一个圆在三维空间中的嵌入。纽结不变量是分配给嵌入圆的量，这些量在嵌入的连续变化下不会变化。从量子场论的工具产生新的，代数的，称为结同调的结不变量的结构。在这个项目中，PI将给出结同源的严格数学构造，以及计算这些结不变量的有效算法。特别是，PI将解释纽结同调的模空间的几何是由多项式方程定义，并使用解析几何的方法来研究这些模空间，从而揭示纽结同调的新性质。在本研究中，PI将针对本科生和博士生开展研究培训活动。更准确地说，在本项目中，PI将研究层的模量和结的同源性之间的关系。特别是，PI将研究一个猜想，涉及HOMFLYPT同源性的一个链接的平面曲线奇点的同源性的希尔伯特计划的点的曲线。为了计算纽结的HOMFLYPT同调，PI将进一步发展一种理论，将链接的HOMFLYPT同调解释为全局截面的空间，即平面上点的希尔伯特方案上自然定义的相干层。PI还将研究一般三重Pandharipande-Thomas模空间的后代不变量的Virasoro约束，并且将进一步发展后代不变量的Gromov-Witten/Pandharipande-Thomas对应，这样Virasoro约束就可以从Gromov维滕理论。该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值的支持和更广泛的影响审查标准。

A knot is an embedding of a circle in three dimensional space. Knot invariants are quantities that are assigned to an embedded circle that do not change under continuous change of the embedding. Tools from quantum field theory yield conjectural constructions for new, algebraic, knot invariants known as knot homologies. In this project, the PI will give rigorous mathematical constructions of knot homologies, as well as efficient algorithms for computing these knot invariants. In particular, the PI will interpret knot homology in terms of the geometry of moduli spaces that are defined by polynomial equations and use methods in analytic geometry to study these moduli spaces, thus uncovering new properties of knot homology. In connection with this research, the PI will undertake research-training activities aimed at both undergraduate and doctoral students.More precisely, in this project the PI will study relations between moduli of sheaves and knot homology. In particular, the PI will study a conjecture that relates the HOMFLYPT homology of a link of a plane curve singularity with the homology of the Hilbert scheme of points on the curve. To compute the HOMFLYPT homology of a knot, the PI will to develop further a theory that interprets the HOMFLYPT homology of a link as a space of global sections a naturally defined coherent sheaf on the Hilbert scheme of points on the plane. The PI will also study Virasoro constraints for the descendant invariants of Pandharipande-Thomas moduli spaces for a general threefold, and the Gromov-Witten/Pandharipande-Thomas correspondence for invariants with descendants will be developed further, so that Virasoro constraints can be transported from Gromov-Witten theory.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.