Geometric Representation Theory and W-algebras

在19世纪早期，数学家和物理学家之间是不可能区分开来的，因为最伟大的科学家都在研究这些领域的每一个重要问题。埃米·诺特是那个时代最有影响力的博学者之一，他提出的所有基础理论都启发了这个研究项目。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，那么了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，它为我们提供了关于物质本质和空间形状的线索，所以诺特定理开启了一波发现浪潮，这波发现浪潮在过去的一百多年里一直在不断发展，数学家和物理学家都在试图理解宇宙的对称性。今天，数学家和物理学家更容易区分，但这两个学科仍然紧密交织在一起。在现代数学语言中，对对称性的研究被称为表征理论，这个项目的目标是理解诺特的代数结构如何被表示为对称性。换句话说，我的目标是理解某些重要代数族的表示。古希腊人相信，所有的物质都可以由不可分割的碎片构成——“原子”一词的字面意思是“不可分割的”——用现代粒子物理学的语言来说，宇宙中的所有物质都可以由基本粒子构成，这是众所周知的。以完全相同的方式，我试图理解的表征也是由基本的构建块构建的，称为不可约表征。我们能明确地描述这些不可约表征吗？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，我们将回答这些基本的，难以捉摸的问题，通过将每个表示与一个重要的几何空间联系起来，称为泊松品种的辛叶。在这个领域中，一些最重要的未解问题与代数有关，我们称之为“模”：这是因为底层的数字系统不是线性的，像实数线一样，而是圆形的，像时钟表面的数字。由于几何和算术的复杂性，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，该项目将在模块化表示理论中一些最具挑战性的问题上取得实质性的令人兴奋的进展，表明诺特的发现浪潮仍在数学的海洋上增长。

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.