Geometric Representation Theory and W-algebras

在十九世纪早期，数学家和物理学家是不可能区分的，因为最伟大的科学家在这些领域的每一个重要问题上都有工作。那个时代最有影响力的博学者之一是艾米·诺特，他发展了启发这项研究项目的所有基本理论。她对物理学的最大贡献可能是诺特的第一定理，该定理说，如果你想了解宇宙的守恒定律，那么理解宇宙的对称性就足够了。守恒定律是物理学中最基本的定律，它给我们提供了关于物质本质和空间形状的线索，因此诺特定理开启了一波发现的浪潮，随着数学家和物理学家试图理解宇宙的对称性，这一发现已经持续了一百多年。今天，数学家和物理学家更容易区分，然而，这两个学科仍然深深地交织在一起。在现代数学语言中，对对称性的研究被称为表示理论，这个项目的目标是理解Noether的代数结构如何表示为对称性。换句话说，我的目标是理解某些重要的代数族的表示法。古希腊人相信所有的物质都可以由不可分的碎片组成--“原子”这个词的字面意思是“不可分割的”--在现代粒子物理学的语言中，众所周知，宇宙中的所有物质都可以由基本粒子组成。以完全相同的方式，我试图理解的表示也是由基本的构件构建的，称为不可约表示。我们能明确地描述这些不可约的表示吗？我们能确定它们的结构并计算它们的尺寸吗？在这个研究项目中，我们将通过将每个表示与一个重要的几何空间相关联来回答这些基本的、难以捉摸的问题，该几何空间被称为泊松变量的辛叶。这一领域中一些最重要的未回答的问题与我们所称的代数有关：这是因为基本的数字系统不像实数线那样是线性的，而是像钟面上的数字一样是圆形的。由于增加了几何和运算的复杂性，模表示理论中的问题往往更加困难。通过使用抽象代数和几何之间的接口工具，该项目将在模表示理论中一些最具挑战性的问题上取得令人兴奋的进展，表明Noether的发现浪潮仍在数学的海洋上生长。

In the early nineteenth century it was impossible to draw a distinction between mathematicians and physicists, since the greatest scientists worked on every important problem in these fields. One of the most influential polymaths of the era was Emmy Noether, who developed all of the foundational theories which inspired this research project. Her greatest contribution to physics was probably Noether's first theorem, which says that if you want to understand the conservation laws of the universe then it suffices to understand the symmetries of the universe. Conservation laws are the most fundamental laws of physics, giving us clues about the nature of matter, and the shape of space, and so Noether's theorem started a wave of discovery which has been growing and growing for over a hundred years, as mathematicians and physicist seek to understand the symmetries of the universe. Today mathematicians and physicists are much easier to distinguish, however the subjects are still deeply intertwined. In modern day mathematical language, the study of symmetries is called representation theory and the goal of this project is to understand how Noether's algebraic structures can be expressed as symmetries. To rephrase this, my objective is to understand the representations of certain important families of algebras.The ancient Greeks believed that all matter could be built up from indivisible pieces - the word "atom" literally means "indivisible" - and in the language of modern particle physics it is well-understood that all matter in the universe can be built up from the fundamental particles. In precisely the same way, the representations I seek to understand are also built from fundamental building blocks, known as irreducible representations. Can we describe these irreducible representations explicitly? Can we determine their structure and calculate their dimensions? In this research project we will answer these fundamental, elusive questions by relating each representation to an important geometric space, known as a symplectic leaf of a Poisson variety.Some of the most important unanswered questions in this field pertain to algebras which we call "modular": this is because the underlying number system is not linear, like the real number line, but is circular like the numbers on the face of a clock. Questions in modular representation theory tend to be significantly harder due to the added complexity of the geometry and the arithmetic.By working with tools on the interface between abstract algebra and geometry this project will make substantial exciting progress in some of the most challenging problems in modular representation theory, showing that Noether's wave of discovery is still growing on the ocean of mathematics.