Structures in Khovanov-Rozansky homology

结是三维空间中的闭合环，链接是几个这样的环的联合，可能彼此链接。除了直接的数学应用，纽结理论在量子系统的物理学和长纽结分子的化学和生物学性质的研究中也有意义。纽结理论的核心问题是分类问题（一个纽结可以在不撕裂它的线的情况下转化为另一个纽结吗？它能解开看起来像一个普通的圆圈吗？）以及研究结或连接的几何性质。这两个问题都可以在链接不变量的帮助下得到部分回答：链接不变量是一组在连续拉伸下不会改变的数字。如果两个链接的不变量不同，则它们是不同的。该项目的重点是发现和研究链接不变量的模式和对称性。该项目将为学生和博士后提供研究培训机会。更详细地说，该项目的重点是Khovanov-Rozansky链接同源性，它推广了着名的HOMFLY多项式。这样一个理论的一个链接相关联的三重分级向量空间进行了许多有趣的操作。该项目将建立和统一各种现有的操作（如拉斯穆森的微分，重言式类和维特代数行动），研究这些之间的交换关系，并定义新的。这样一个大代数的作用有望解开链同调中的一些模式和对称性。另一个动机来自几何模型的链接同源性：这些包括层希尔伯特计划的点在平面上，编织品种，希尔伯特计划奇异曲线和仿射施普林格纤维。在许多情况下，几何表示理论预测了大代数在同调中的作用，研究人员将这些转化为链接同调中的显式作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

A knot is a closed loop in three-dimensional space, a link is a union of several such loops, possibly linked with each other. Besides the immediate mathematical applications, knot theory has implications in physics of quantum systems and in the study of chemical and biological properties of long knotted molecules. The central questions in knot theory are the classification problem (Can a knot be transformed to another knot without tearing its strands? Can it be untangled to look like an ordinary circle?) and the study of the geometric properties of knots or links. Both of these questions can be partially answered with the help of a link invariant: a collection of numbers that does not change under continuous stretching of a link. Two links are different if their invariants are different. The project is focused on uncovering and studying the patterns and symmetries of link invariants. The project will provide research training opportunities for students and post-docs.In more detail, the project is focused on Khovanov-Rozansky link homology which generalizes the celebrated HOMFLY polynomial. To a link such a theory associates a triply graded vector space which carries an action of many interesting operators. The project will build upon and unify a variety of existing operations (such as Rasmussen's differentials, tautological classes and Witt algebra action), to study the commutation relations between these and to define new ones. The action of such a large algebra is expected to unravel some patterns and symmetries in link homology. Another motivation comes from geometric models for link homology: these include sheaves on Hilbert schemes of points on the plane, braid varieties, Hilbert schemes on singular curves and affine Springer fibers. In many cases, geometric representation theory then predicts an action of large algebras in homology, and the investigator will translate these into explicit actions in link homology.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.