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.

结是在三维空间中圆圈的嵌入。结不变是分配给嵌入式圆圈的数量，在嵌入的连续变化下不会改变。量子场理论的工具产生了新的，代数，结的猜想结构，称为结同源物。在这个项目中，PI将提供结节同源性的严格数学结构，以及用于计算这些结不变的有效算法。特别是，PI将根据多项式方程定义的模量空间的几何形状来解释结的同源性，并在分析几何学中使用方法来研究这些模量空间，从而发现了结的新特性。与这项研究有关，PI将开展针对本科生和博士生的研究培训活动。更重要的是，在该项目中，PI将研究带束系的模束与打结同源性的关系。特别是，PI将研究一个猜想，该猜想将平面曲线奇异性的链接的Homflypt同源性与曲线上的Hilbert of Points的同源性联系在一起。为了计算结的homflypt同源性，PI将进一步开发一种理论，该理论将链接的Homflypt同源性解释为全球部分的空间，这是在飞机上的希尔伯特（Hilbert of Point）方案上自然定义的相干支架。 PI还将研究pandharipande-thomas moduli空间的后代不变剂的Virasoro限制，并为一般的三倍而言，Gromov-Witten/Pandharipande-Thomas通信对于具有后代的不规则的通信将进一步开发。 - 掌握理论。该奖项反映了NSF的法定使命，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。