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

职业：结不变量、滑轮模空间和表示论

• 批准号: 1352398
• 负责人: Alexei Oblomkov
• 金额: $ 42.12万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352398&HistoricalAwards=false
• 关键词: CAREER; Knot; invariants; moduli; spaces

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.

本课题的主题是小维变量内点集合的位形空间的几何，更一般地说，是这些变量上的轴的模空间。主要目标是揭示新的和进一步探索以前已知的模空间和其他数学领域的对象之间的联系，特别是表示理论和低维拓扑。PI将致力于证明关于平面奇异曲线上点的Hilbert格式的拓扑不变量和曲线奇异点的连杆的HOMFLY结同调的数学猜想（Hilb/HOMFLY公式）。该猜想还揭示了环面结的同调的意想不到的对称性：从推测上讲，它们形成了a型有理Cherednik代数的不可约表示。PI将探讨与非a型的对称反射代数和有理Cherednik代数的表示理论相关的广义Hilb/HOMFLY猜想。最后，PI描述了拟齐次奇点的紧化jacobian的上同调环。PI（与Zhiwei Yum共同）推测了曲线的紧化雅可比矩阵的上同环与曲线的有理映射的模空间的结构环之间的关系：Gromov-Witten/Donaldson-Thomas关系的局部变化。该项目的教育部分为马萨诸塞大学REU项目提供了一个新的模式。结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。复杂的形状在生物学（如蛋白质、DNA）、理论物理学（弦）和其他自然科学领域中自然存在。因此，开发新的不变量和计算方法来理解复杂形状的整体结构是一个具有许多潜在应用的重要数学问题。PI努力理解已经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数学其他领域的意想不到的应用。该项目还将通过一个由教师和研究生共同指导的暑期研究项目，让本科生参与最前沿的研究。PI的目的是通过在夏季研究项目中为当地两所女子学院的学生保留特定的名额，从代表性不足的群体中吸引更多的学生从事数学研究。PI将在一年内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。这个新的夏季研究项目结构将增加多样性，加强学术界的垂直整合，并改善现在和未来不同世代研究人员之间的交流和思想流动。