Algebraic Geometry of Knot Homology

结是三维空间中的封闭曲线。更一般地说，一个连杆是几条这样的曲线的集合。如果一个链接可以通过连续变形和/或拉伸（不撕裂）转换为另一个链接，则两个链接是等效的。连杆不变量是指在连杆变形时不改变的量。这种不变量可以用来区分连杆和研究结点的几何性质，近几十年来已经发展了许多强大的连杆不变量。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，并在广泛的领域中有应用，包括统计学，控制理论和计算机科学。这个项目的重点是在两个不同的数学框架中出现的链路不变量和代数几何之间令人惊讶的相互作用。预计该研究将为低维拓扑学、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，它与一个小球体的交点是一个链接。同样，对于复杂曲面奇点，其连杆是三流形。研究者将研究这些曲线和曲面的代数性质之间的相互作用，以及它们的连杆的heegaard - flower同调。第二个框架不那么直接。研究者和合作者推测连杆的Khovanov-Rozansky同调可以通过研究点的Hilbert格式（四维空间中点的位形空间）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得到了很好的研究，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，研究者将发展结和希尔伯特方案之间的联系，目标是计算各种类型的链路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.