Diagrammatic and geometric techniques in representation theory

我的研究是基于表示论，代数几何，低维拓扑和数学物理之间的接口。它围绕着研究一类特殊的非交换代数的基础理论和应用。这些代数产生于变形量子化的概念：一个交换代数可以变形为一个有趣的非交换代数，人们可以通过考虑其经典极限的几何来研究该代数的表示论。这种方法是量子力学研究的基础（海森堡测不准原理表达了交换性的失败），但在李代数的研究中也有着丰富的历史。 我考虑的相关代数簇类是“辛奇点”。每个辛奇点都有一个相关的非交换代数，我们称之为“通用包络代数”，使用变形量子化构造。这些包括李代数的通用包络代数作为特例;我研究的一个基本原则是研究如何修改关于李代数的陈述以在一般情况下成立。 数学物理学似乎是这些代数的一个来源：某些特殊的量子场论给了我们许多这些代数的例子。我在这个补助金中提出的计划是了解这些奇点的几何形状，它们的变形的表示理论，以及相关的量子场论如何相互关联，并可以应用于其他领域，如组合学和拓扑学。 在过去的十年里，我最重要的工作是致力于这些变体出现在物理学中的“对偶对”中。这使整个领域有了新的认识，并且了解了这些变体的性质在对偶性下是如何联系在一起的，这刺激了辛奇点的许多不同方面的研究。表示论和对偶簇的几何之间的联系是微妙的，但这种对偶性可以被看作是数学中许多对偶性的“几何化”和“范畴化”，例如Schur-Weyl对偶性、秩级对偶性和Gale对偶性。 展望未来，我面临的最大问题是更好地理解这些见解如何应用于数学物理，以及相反，如何将物理学的思想应用于该提案的数学问题。

My research is based on the interface between representation theory, algebraic geometry, low dimensional topology and mathematical physics. It centres around both the foundational theory and applications of studying a special class of non-commutative algebras. These algebras arise from the notion of deformation quantization: a commutative algebra can deform to an interesting non-commutative one, and one can study the representation theory of that algebra by considering the geometry of its classical limit. This approach is fundamental in the study of quantum mechanics (the Heisenberg uncertainty principle expresses the failure of commutativity), but also has a fruitful history in the study of Lie algebras. The relevant class of algebraic varieties I consider are "symplectic singularities." Every symplectic singularity has an associated non-commutative algebra, which we call its "universal enveloping algebra," constructed using deformation quantization. These include the universal enveloping algebras of Lie algebras as a special case; one basic principle of my research is to study how statements about Lie algebras can be modified to hold in the general case. Mathematical physics appears as a source of these algebras: certain special quantum field theories give us many examples of these algebras. The program I propose in this grant is to understand how the geometry of these singularities, the representation theory of their deformations, and the associated quantum field theories relate to each other, and can be applied in other areas such as combinatorics and topology. My most important work over the past decade has been dedicated to the idea that these varieties appear in "dual pairs" that arise in physics. This cast the whole field in a new light, and understanding how properties of these varieties are related under duality has stimulated study of many different aspects of symplectic singularities. The connection between the representation theory and geometry of dual varieties is subtle, but this duality can be seen as a "geometrification'' and "categorification" of many dualities in mathematics, such as Schur-Weyl, rank-level and Gale duality. Moving forward, the biggest question facing me is to understand better how these insights can be applied in mathematical physics, and conversely, how the ideas of physics can brought to bear on the mathematical questions of the proposal.