Algebraic Geometry of Knot Homology

结是三维空间中的闭合曲线。更一般地说，链接是几个这样的曲线的集合。如果一个链接可以通过连续变形和/或拉伸（没有撕裂）转换为另一个链接，则两个链接是等效的。链接不变量是在链接变形下不发生变化的量。这种不变量可以用来区分链接和研究纽结的几何性质，近几十年来已经开发了许多强大的链接不变量。在另一个方向上，代数几何的数学主题研究由多项式方程描述的空间，并在广泛的领域中应用，包括统计学，控制理论和计算机科学。这个项目的重点是令人惊讶的链接和代数几何的不变量之间的相互作用，出现在两个不同的数学框架。预计该研究将为低维拓扑、代数几何、表示论和组合学之间的关系提供新的视角。更详细地说，第一个框架是经典的：给定一个复杂的平面曲线奇点，它与一个小球体的相交是一个链接。类似地，对于一个复杂的曲面奇点，它的连接是一个三流形。研究人员将研究这些曲线和曲面的代数性质之间的相互作用，以及它们之间的Heegaard-Floer同调。第二个框架不那么直接。研究者和合作者已经证实，可以通过研究希尔伯特点方案（四维空间中点的配置空间）的代数和几何性质来计算链路的Khovanov-Rozansky同调。希尔伯特格式在代数几何、表示论和组合数学中得到了很好的研究，因此这种解释将产生以前无法实现的霍瓦诺夫-罗赞斯基同调的显式计算。在这个项目中，研究人员将开发节点和希尔伯特方案之间的联系，目标是计算各种链接的这些不变量并证明猜想。

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.