Algebraic Geometry of Knot Homology

节点是三维空间中的一条闭合曲线。更广泛地说，链接是几条这样的曲线的集合。如果两个链接可以通过连续变形和/或拉伸(不撕裂)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的变形下不变的量。这种不变量可以用来区分链环和研究纽结的几何性质，近几十年来已经发展了许多强大的链环不变量。在另一个方向上，代数几何的数学学科研究用多项式方程描述的空间，并在包括统计、控制理论和计算机科学在内的广泛领域中得到应用。这个项目专注于在两个不同的数学框架中出现的链接不变量和代数几何之间令人惊讶的相互作用。预计这项研究将为低维拓扑、代数几何、表示理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，它与一个小球面的交点是一个纽带。同样，对于一个复杂的曲面奇点，它的链环是一个三维流形。研究人员将研究这些曲线和曲面的代数性质与它们的链环的Heegaard-Floer同调之间的相互作用。第二个框架不那么直接。研究者和合作者猜想，环的Khovanov-Rozansky同调可以通过研究点的Hilbert格式(四维空间中的点的配置空间)的代数和几何性质来计算。希尔伯特方案在代数几何、表示理论和组合学中都有很好的研究，因此这种解释将产生以前无法实现的Khovanov-Rozansky同调的显式计算。在这个项目中，研究人员将发展纽结和希尔伯特格式之间的联系，目标是计算各种类型的环的不变量，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.