Moduli spaces attached to singular surfaces and representation theory

我们项目的目的是为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，另一方面是表征理论（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。其中一个问题非常经典，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：一个基于正n-gon （a型或循环）的圆锥；基于正n形（D型或二面体）的棱镜；或五种正体之一，如四面体、立方体或十二面体（E型或例外）。经典构造将这些对称群转化为2x2（复）矩阵群；然后利用这些矩阵群得到一些奇异空间，称为简单（曲面）奇异点。A- d - e分类的一个看似完全无关的实例是简单李代数。李代数与对称的连续群密切相关。接下来的挑战是理解这些对称的连续群（或代数）如何与简单的奇点联系起来。答案的很大一部分已经知道了一段时间，并且是所谓的麦凯对应的一部分：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模式和连续对称性联系起来。然而，最近在PI和合作者的工作中发现了一种诱人的联系，这种联系表明奇异空间本身的几何形状与李代数的表示方面之间存在关系。我们项目的目标是以不同的方式来研究这种联系：-以具体的几何方式理解某个辅助空间，奇点的希尔伯特格式；-将希尔伯特方案的几何直接与李代数对称联系起来；-寻找附加在奇异空间上的新几何形状，并研究它们的性质；-将连接扩展到其他更高维度的奇异空间。这个项目的成功将进一步加深我们对奇异几何空间及其隐藏的对称性的理解。

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.