CAREER: Knot invariants, moduli spaces of sheaves and representation theory

这个项目的主题是小维度簇内的点的集合的配置空间的几何，更一般地，这些簇上的滑轮的模空间。主要目的是揭示新的和进一步探索已知的模空间和其他数学领域的对象之间的联系，特别是表示论和低维拓扑。PI将致力于证明关于平面奇异曲线上点的Hilbert格式的拓扑不变性和曲线奇点链环的HOMFLY纽结同调的数学猜想(Hilb/HOMFLY公式)。这个猜想还揭示了环面纽结同调的意想不到的对称性：猜想，它们形成了A型有理Cherednik代数的一个不可约表示。PI将探索广义的Hilb/HOMFLY猜想，它将辛反射代数的表示理论与A型以外的有理Cherednik代数的表示理论联系起来。最后，PI描述了拟齐次奇点的紧化雅可比的上同调环。PI(与Zhiwei Yum联合)猜想曲线的紧化Jacobian的上同调环与到曲线的有理映射的模空间的结构环之间的关系：Gromov-Witten/Donaldson-Thomas关系的局部变体。该项目的教育部分为马萨诸塞州大学REU项目提供了一个新的模式。节点不变量和拓扑不变量允许我们通过收集有关形状的局部信息来分析复杂形状的全局结构。复杂的形状自然地出现在生物学(例如蛋白质、DNA)、理论物理(弦)和自然科学的其他领域。因此，发展新的不变量和计算方法来理解复杂形状的全局结构是一个具有许多潜在应用的重要数学问题。PI努力了解已发现的不变量的隐藏对称性，开发新的不变量，并发现这些不变量在其他数学领域的意想不到的应用。PI还将通过一个暑期研究项目让本科生参与尖端研究，该项目整合了教职员工和研究生的指导。PI旨在通过在暑期研究项目中为来自当地两所女子学院的学生预留特定空间，吸引更多来自代表性不足群体的学生从事数学研究。在这一年中，国际学生协会将通过教授相关的研究生课程和阅读研讨会来培养研究生导师。这一新的暑期研究计划结构将增加多样性，加强学术界的垂直整合，并改善不同世代现在和未来研究人员之间的沟通和思想流动。

The subject of this project is the geometry of configuration spaces of collections of points inside varieties of small dimension, and more generally, the moduli spaces of sheaves on these varieties. The main objective is to reveal new and further explore previously known links between the moduli spaces and objects in other fields of mathematics, in particular Representation Theory and Lower Dimensional Topology. The PI will work toward a proof of the mathematical conjecture relating the topological invariants of the Hilbert scheme of points on plane singular curves and the HOMFLY knot homology of the links of the singularities of the curve (Hilb/HOMFLY formula). The conjecture also reveals unexpected symmetries of the homology of torus knots: conjecturally, they form an irreducible representation of the rational Cherednik algebra of type A. The PI will explore the generalized Hilb/HOMFLY conjecture that relates the representation theory of the symplectic reflection algebras and the rational Cherednik algebras of types other than A. Finally, the PI describes the cohomology ring of the compactified Jacobians of quasi-homogeneous singularities. The PI (jointly with Zhiwei Yum) conjectures a relation between the cohomology ring of the compactified Jacobian of the curve and the structure ring of the moduli space of the rational maps to the curve: a local variation of the Gromov-Witten/Donaldson-Thomas relation. The educational component of the project offers a new model for the UMass REU program. Knot invariants and topological invariants allow us to analyze the global structure of complicated shapes by collecting local information about the shape. Complicated shapes occur naturally in biology (e.g. proteins, DNA), theoretical physics (strings), and other areas of natural science. Thus developing new invariants and computational methods for understanding of the global structure of complex shapes is an important mathematical problem with many potential applications. The PI strives to understand the hidden symmetries of already discovered invariants, develop new invariants, and find unexpected applications of these invariants to other areas of mathematics. The PI will also involve undergraduate students in cutting edge research through a summer research program integrating mentorship by faculty and graduate students. The PI aims to attract more students from underrepresented groups to mathematical research by reserving specific spaces in the summer research program for students from two local women's colleges. The PI will prepare graduate student mentors during the year by teaching related graduate classes and a reading seminar. This new summer research program structure will increase diversity and strengthen vertical integration in academia and improve the communication and flow of ideas between different generations of present and future researchers.