Geometric representation theory and crystals

对称性是数学的一个基本领域。 数学家对对称性的连续集合特别感兴趣，例如球体的旋转群。 我们称这些物体为李群。 一个重要的问题是理解这些李群如何作为向量空间上的线性算子出现，这些被称为表示。 世纪末和20世纪初数学的一个伟大成就是简单李群的分类和它们的表示。 李群及其表示自20世纪30年代以来在量子力学中被大量使用，以帮助描述和分类基本粒子。 近年来，量子场论和表示论之间的相互作用又重新兴起，导致双方都取得了重要进展。在过去的20年里，数学家们在表示论中发展了几何构造。 在这些构造中，我们有一个几何对象（一个代数簇），其拓扑编码了李群的表示。 这些几何构造非常漂亮，并导致更深层次的结构，如分类和规范基。这些几何构造有两个来源，它们大致是同时发明的。 第一个来自几何朗兰兹对偶理论，它是由德林费尔德发展的，作为著名的朗兰兹对偶的几何版本，它指导了现代数论的大部分。 第一种构造使用称为仿射格拉斯曼的几何对象。 第二种结构是由Lusztig和Nakajima开发的，涉及称为“几何变体”的几何对象。 第一种构造更自然，但更难处理;第二种构造更需要动手，但更特别。这两种不同的，看似无关的几何构造的存在非常神秘，并导致许多数学家提出以下问题：这两种几何构造之间的关系是什么？2012年，韦伯斯特、威克斯、雅可比和我通过辛对偶机制提出了这个问题的答案，辛对偶是某些类别的几何对象之间的微妙关系，称为圆锥辛解析。 辛对偶在理论物理学中有一个美丽的起源，更具体地说是来自N = 4的三维超对称量子场论。 对于这样一个理论，我们可以考虑所有可能的最低能量状态，这就是所谓的真空模空间。 这个模空间有几个部分，其中一个叫做希格斯分支，另一个叫做库仑分支。 真空的这两个分支形成了一个辛对偶对，我目前的研究集中在探索这种辛对偶及其对范畴化和特殊基的影响。

Symmetry is a fundamental area of mathematics. Mathematicians are particularly interested in continuous collections of symmetries, such as the group of rotations of a sphere. We call these objects Lie groups. An important problem is to understand how these Lie groups can arise as linear operators on vector spaces, these are called representations. One of the great achievements in mathematics in the late 19th and early 20th century was the classification of simple Lie groups and their representations.Much of this interest in Lie groups is motivated by theoretical physics. Lie groups and their representations have been used heavily since the 1930s in quantum mechanics to help describe and classify elementary particles. In recent years, there has been a resurgence of interaction between quantum field theory and representation theory, leading to important advances on both sides.In the past 20 years, mathematicians have developed geometric constructions in representation theory. In these constructions, we have a geometric object (an algebraic variety) whose topology encodes a representation of a Lie group. These geometric constructions are very beautiful and lead to deeper structures, such as categorification and canonical bases.There are two sources of these geometric constructions, which were invented roughly simultaneously. The first one comes from the theory of geometric Langlands duality, which was developed by Drinfeld as a geometric version of the famous Langlands conjectures, which have guided much of modern number theory. This first construction uses geometric objects called affine Grassmannians. The second construction was developed by Lusztig and Nakajima and involves geometric objects called quiver varieties. The first construction is more natural, but harder to work with; the second construction is more hands-on, but more ad-hoc.The existence of these two different, seemingly unrelated, geometric constructions was very mysterious and lead many mathematicians to the following question:What is the relationship between these two geometric constructions? In 2012, Webster, Weekes, Yacobi and I proposed an answer to this question through the mechanism of symplectic duality, a subtle relationship between certain classes of geometric objects, called conical symplectic resolutions. Symplectic duality has a beautiful origin in theoretical physics, more specifically from N = 4, 3-dimensional supersymmetric quantum field theories. To such a theory, we can consider all possible lowest energy states, which is called the moduli space of vacua. This moduli space has a few pieces, one of which is called the Higgs branch and another of which is called the Coulomb branch. These two branches of vacua form a symplectic dual pair.My current research focuses on exploring this symplectic duality and its consequences for categorification and special bases.