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Knot Homology and Moduli of Sheaves

绳轮的结同源性和模量

基本信息

批准号：
2200798
负责人：
Alexei Oblomkov
金额：
$ 21.7万
依托单位：
University of Massachusetts Amherst
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200798&HistoricalAwards=false
关键词：
Knot Homology Moduli Sheaves

项目摘要

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.

结是在三维空间中嵌入一个圆。结不变量是分配给嵌入圆的量，它在嵌入的连续变化下不会改变。量子场论的工具给出了新的、代数的、称为结同调的结不变量的猜想构造。在这个项目中，PI将给出结同调的严格数学结构，以及计算这些结不变量的有效算法。特别是，PI将根据多项式方程定义的模空间的几何来解释结同调，并使用解析几何中的方法来研究这些模空间，从而揭示结同调的新性质。在这项研究中，PI将开展针对本科生和博士生的研究训练活动。更确切地说，在这个项目中，PI将研究轮轴模与结同源性之间的关系。特别是，PI将研究一个猜想，该猜想将平面曲线奇点的连杆的HOMFLYPT同调与曲线上点的希尔伯特格式的同调联系起来。为了计算结的HOMFLYPT同调性，PI将进一步发展一个理论，将连杆的HOMFLYPT同调性解释为平面上点的希尔伯特格式上的一个自然定义的相干束的整体截面空间。PI还将研究一般三次元Pandharipande-Thomas模空间子代不变量的Virasoro约束，并进一步发展具有子代不变量的Gromov-Witten/Pandharipande-Thomas对应关系，从而将Virasoro约束从Gromov-Witten理论中传递出来。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。