Structures in Khovanov-Rozansky homology

节点是三维空间中的闭合环，链接是几个这样的环的并，这些环可能相互链接。除了直接的数学应用，纽结理论在量子系统的物理学和长结分子的化学和生物性质的研究中也有意义。纽结理论的中心问题是分类问题(一个纽结能不能在不撕裂绳索的情况下转换成另一个纽结？它可以解开，看起来像一个普通的圆圈吗？)以及研究结或环的几何性质的学科。这两个问题都可以在链接不变量的帮助下部分回答：链接不变量是在链接连续拉伸下不变的数字的集合。如果两个链接的不变量不同，则它们是不同的。该项目致力于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，该项目专注于Khovanov-Rozansky链同调，它推广了著名的HOMFLY多项式。这种理论将一个三分次向量空间与一个链接联系起来，该向量空间包含许多有趣的算子的作用。该项目将建立并统一现有的各种运算(如拉斯穆森微分、重言式类和Witt代数运算)，以研究这些运算之间的对易关系并定义新的对易关系。如此大的代数的作用有望解开环同调中的一些模式和对称性。另一个动机来自于链同调的几何模型：这些模型包括平面上点的Hilbert方案、辫子簇、奇异曲线上的Hilbert方案和仿射Springer纤维。在许多情况下，几何表示理论预测了大型代数在同调中的作用，研究人员将这些行为转化为链接同调中的显式作用。这一裁决反映了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.