Moduli spaces attached to singular surfaces and representation theory

附加到奇异曲面和表示理论的模空间

• 批准号: EP/R045038/1
• 负责人: Balazs Szendroi
• 金额: $ 66.13万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR045038%2F1
• 关键词: Moduli; spaces; attached; singular; surfaces

The aim of our project is to contribute to the study of two very classical constructions in mathematics: singularities of geometric spaces on the one hand, and representation theory (the theory of symmetry) on the other. One of the enduring patterns in mathematics is the so-called A-D-E classification: there are several, seemingly totally unrelated, questions in mathematics to which the answer involves a certain list of simple combinatorial patterns. One such question is very classical, in some sense going back to Euclid: find all possible finite 3-dimensional (rotational) symmetry groups. The answer is that such groups must be symmetry groups of one of the following polyhedra: a cone based on a regular n-gon (type A or cyclic); a prism based on a regular n-gon (type D or dihedral); or one of the five regular solids such as the tetrahedron, cube or dodecahedron (type E or exceptional). A classical construction translates these symmetry groups into groups of 2x2 (complex) matrices; we then obtain some singular spaces called simple (surface) singularities using these matrix groups.A seemingly totally unrelated instance of the A-D-E classification is that of simple (simply laced) Lie algebras. Lie algebras are closely related to continuous groups of symmetries. The challenge then is to understand how do these continuous groups (or algebras) of symmetries relate to simple singularities.A large part of the answer has been known for some time, and is part of what's called the McKay correspondence: given the singularity, it has a resolution, and the geometry of the resolution can be related in different ways to A-D-E patterns and continuous symmetries. Recently however, a tantalising connection has been observed in work of the PI and collaborators that suggests a relationship between the geometry of the singular space itself, and aspects of representations of Lie algebras. The objectives of our project are to study this connection in different ways: - Understand in concrete geometric ways a certain auxiliary space, the Hilbert scheme of points of the singularity;- Relate the geometry of the Hilbert scheme directly to Lie algebra symmetries;- Find new geometries attached to the singular space and study their properties;- Extend the connection to other, higher-dimensional singular spaces.Success in this project will further our understanding of singular geometric spaces and their hidden symmetries.

我们项目的目的是对数学中两个非常经典的结构的研究做出贡献：一方面是几何空间的奇点，另一方面是表示理论(对称理论)。数学中一个经久不衰的模式是所谓的A-D-E分类：数学中有几个看似完全无关的问题，其答案涉及一系列简单的组合模式。一个这样的问题非常经典，在某种意义上可以追溯到欧几里得：找到所有可能的有限三维(旋转)对称群。答案是，这样的群必须是下列多面体之一的对称群：基于正n边形(A型或循环)的圆锥体；基于正n边形(D型或二面体)的棱柱；或五种规则实体中的一种，如四面体、立方体或十二面体(E型或例外)。一种经典的构造将这些对称群转化为2x2(复)矩阵的群，然后利用这些矩阵群得到一些称为简单(曲面)奇点的奇异空间。李代数与连续的对称群密切相关。接下来的挑战是了解这些连续的对称群(或代数)如何与简单的奇点相关。答案的很大一部分已经知道一段时间了，并且是所谓的McKay对应的一部分：给定奇点，它有一个分辨率，并且分辨率的几何可以以不同的方式与A-D-E模式和连续对称相关联。然而，最近在PI和合作者的工作中观察到了一种诱人的联系，这表明奇异空间本身的几何形状与李代数表示的各个方面之间存在关系。我们项目的目标是以不同的方式研究这种联系：-以具体的几何方式理解某个辅助空间，奇点的希尔伯特方案；-将希尔伯特方案的几何直接与李代数对称联系起来；-寻找附加到奇异空间的新几何并研究它们的性质；-将这种联系扩展到其他更高维的奇异空间。这个项目的成功将加深我们对奇异几何空间及其隐藏对称性的理解。

项目成果

On the Equations Defining Some Hilbert Schemes

关于定义一些希尔伯特方案的方程

• DOI: 10.1007/s10013-021-00545-0
• 发表时间: 2022
• 期刊: Vietnam Journal of Mathematics
• 影响因子: 0.8
• 作者: Hauenstein J

Canonical spectral coordinates for the Calogero-Moser space associated with the cyclic quiver

与循环颤动相关的 Calogero-Moser 空间的规范谱坐标

• DOI: 10.1080/14029251.2020.1700634
• 发表时间: 2020
• 期刊: Journal of Nonlinear Mathematical Physics
• 影响因子: 0.7
• 作者: Gyenge Á

Traces, Schubert calculus, and Hochschild cohomology of category O

O 类的迹、舒伯特微积分和 Hochschild 上同调

• 发表时间: 2020
• 作者: Clemens Koppensteiner

Singularities and Their Interaction with Geometry and Low Dimensional Topology - In Honor of András Némethi

奇点及其与几何和低维拓扑的相互作用 - 纪念 Andrés Némethi

• DOI: 10.1007/978-3-030-61958-9_3
• 发表时间: 2021
• 作者: Gyenge Á

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

将克莱因奇点作为箭袋簇的准时希尔伯特方案

• DOI: 10.48550/arxiv.1910.13420
• 发表时间: 2019
• 期刊: arXiv e-prints
• 作者: Craw Alastair