Algebraic Geometry of Knot Homology

结同调的代数几何

基本信息

批准号：
1700814
负责人：
Evgeny Gorskiy
金额：
$ 15万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700814&HistoricalAwards=false
关键词：
Algebraic Geometry Knot Homology

项目摘要

A knot is a closed curve in three-dimensional space. More generally, a link is a collection of several such curves. Two links are equivalent if one can be transformed to another by a continuous deformation and/or stretching (without tearing). A link invariant is a quantity that does not change under the deformations of a link. Such invariants can be used to distinguish links and to study geometric properties of knots, and many powerful link invariants have been developed in recent decades. In another direction, the mathematical subject of algebraic geometry studies spaces described by polynomial equations and has applications in a wide range of fields, including statistics, control theory, and computer science. This project is focused on surprising interactions between the invariants of links and algebraic geometry that appear in two different mathematical frameworks. It is anticipated that the research will provide new perspectives on the relationships between low-dimensional topology, algebraic geometry, representation theory, and combinatorics.In more detail, the first framework is classical: given a complex plane curve singularity, its intersection with a small sphere is a link. Similarly, for a complex surface singularity its link is a three-manifold. The investigator will the study the interactions between the algebraic properties of these curves and surfaces, and the Heegaard-Floer homology of their links. The second framework is less direct. The investigator and collaborators have conjectured that the Khovanov-Rozansky homology of links can be computed by studying the algebraic and geometric properties of the Hilbert scheme of points (the configuration space of points in four-dimensional space). The Hilbert scheme is well studied in algebraic geometry, representation theory and combinatorics, so this interpretation would yield explicit computations of Khovanov-Rozansky homology that were previously out of reach. In this project the investigator will develop the connection between knots and the Hilbert scheme, with the goals of computing these invariants for various classes of links and proving the conjecture.

结是三维空间中的封闭曲线。更一般而言，链接是几个此类曲线的集合。如果可以通过连续的变形和/或拉伸（而无需撕裂）将一个链接等效到另一个链接。链接不变的数量是在链接的变形下不变的数量。这样的不变剂可用于区分链接和研究结的几何特性，并且近几十年来已经开发了许多强大的链接不变性。在另一个方向上，由多项式方程描述的代数几何研究空间的数学主题在包括统计，控制理论和计算机科学在内的广泛领域中具有应用。该项目的重点是出现在两个不同的数学框架中的链接和代数几何形状之间的令人惊讶的相互作用。可以预见，该研究将为低维拓扑，代数几何，代表理论和组合学之间的关系提供新的观点。在更详细的情况下，第一个框架是经典的：给定一个复杂的平面曲线奇异性，它与小的相交球是一个链接。同样，对于复杂的表面奇异性，它的链接是三个manifold。研究人员将研究这些曲线和表面的代数特性与其链接的Heegaard-loer同源性之间的相互作用。第二个框架不那么直接。研究者和合作者猜想可以通过研究希尔伯特点方案的代数和几何特性来计算链接的Khovanov-Rozansky同源性（四维空间中的点的配置空间）。希尔伯特方案在代数几何，表示理论和组合方面进行了很好的研究，因此这种解释将产生以前无法触及的Khovanov-Rozansky同源性的明确计算。在这个项目中，研究人员将开发结与希尔伯特计划之间的联系，以计算这些不变性的各种链接并证明猜想的目标。