Geometric representation theory and crystals

几何表示理论和晶体

• 批准号: RGPIN-2018-04713
• 负责人: Kamnitzer, Joel
• 金额: $ 2.99万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=655032
• 关键词: Geometric; representation; theory; crystals

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.**

对称是数学的基本领域。 数学家对连续集合对称性（例如球体的旋转组）特别感兴趣。 我们称这些对象为谎言组。 一个重要的问题是了解这些谎言组如何作为向量空间上的线性算子出现，这些谎言被称为表示。 19世纪末和20世纪初数学的伟大成就之一是对简单的谎言群体及其表达的分类。 自1930年代以来，谎言组及其表示形式已被大量用于量子力学，以帮助描述和分类基本粒子。 近年来，量子场理论与表示理论之间存在相互作用的复兴，从而导致双方的重要进展。******在过去20年中，数学家在表示理论中开发了几何结构。 在这些结构中，我们有一个几何对象（代数品种），其拓扑编码一个谎言组的表示。 这些几何结构非常美丽，并导致更深的结构，例如分类和规范基础。 第一个来自几何兰兰兹二元性的理论，德林菲尔德（Drinfeld）作为著名兰兰兹猜想的几何版本开发，该版本指导了现代数字理论的大部分。 该第一个结构使用称为Aggine Grassmanians的几何物体。 第二种结构是由Lusztig和Nakajima开发的，涉及称为Quiver品种的几何对象。 第一个结构更自然，但很难与之合作。第二个结构更动手，但更临时。******这两个不同的，看似无关的几何结构的存在非常神秘，并使许多数学家探讨了以下问题：******这两个几何结构之间的关系是什么？ ******在2012年，韦伯斯特，周刊，Yacobi和我通过符号双重性的机制提出了一个答案，这是某些类别的几何对象之间的微妙关系，称为锥形符号分辨率。 ******符号双重性在理论物理学中具有美丽的起源，更具体地说是n = 4，3维超对称量子场理论。 对于这样的理论，我们可以考虑所有可能的最低能量状态，这称为真空的模量空间。 这个模量空间有几块，其中之一称为希格斯分支，另一个称为库仑分支。 真空吸尘器的这两个分支形成了一个符号双重对。