Geometric and category theoretic methods in representation theory

表示论中的几何和范畴论方法

基本信息

批准号：
RGPIN-2017-03854
负责人：
Savage, Alistair
金额：
$ 1.75万
依托单位：
University of Ottawa
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710594
关键词：
Geometric category theoretic methods representation

项目摘要

The field of representation theory can be thought of as the study of symmetry in mathematics and physics. Certain groups of symmetries, known as Lie groups, play a vital role in the study of the physical laws that govern our universe. Therefore, developing a better understanding of these mathematical objects results in a deeper and more sophisticated knowledge of the world around us. I use techniques in geometry and abstract algebra to study representation theory. Geometric representation theory is a field that uses concepts coming from geometry to study algebraic objects, in addition to using algebraic techniques to study geometry. This rich interplay between the two fields has proven to be extremely beneficial to both. I also study a phenomenon known as a categorification. Categorification is a relatively new and very exciting field of mathematics. Its philosophy is that many mathematical structures that appear to be fundamental are, in fact, mere shadows of a higher and richer mathematical reality. I aim to uncover this hidden higher structure, thus revealing a deeper, more unified mathematical framework. Over the next 5 years, I plan to focus on the categorification of the Heisenberg algebra, which plays a vital role in theoretical physics and Lie theory. In addition to developing the mathematical theory of Heisenberg categorification, I will also work on its applications to theoretical physics and other areas of mathematics. In addition, I also plan to continue my study of a certain type of Lie algebra known as an equivariant map algebra. The study of these objects is a fast developing field with connections to many areas of mathematics. I expect the outcomes of this research to have significant impact in the fields of representation theory, geometry/topology, combinatorics, and mathematical physics.

表示论领域可以被认为是数学和物理学中对称性的研究。某些对称群（称为李群）在研究支配我们宇宙的物理定律中发挥着至关重要的作用。因此，更好地理解这些数学对象可以使我们对周围的世界有更深入、更复杂的了解。我使用几何和抽象代数技术来研究表示论。几何表示论是一个除了使用代数技术来研究几何之外，还使用几何概念来研究代数对象的领域。事实证明，这两个领域之间丰富的相互作用对双方都极为有利。我还研究了一种称为分类的现象。分类是一个相对较新且非常令人兴奋的数学领域。它的哲学是，许多看似基本的数学结构实际上只是更高、更丰富的数学现实的影子。我的目标是揭示这个隐藏的更高结构，从而揭示一个更深层次、更统一的数学框架。在接下来的5年里，我计划重点研究海森堡代数的分类，它在理论物理和李理论中起着至关重要的作用。除了发展海森堡分类的数学理论外，我还将致力于其在理论物理和其他数学领域的应用。此外，我还计划继续研究某种类型的李代数，即等变图代数。对这些物体的研究是一个快速发展的领域，与许多数学领域都有联系。我预计这项研究的成果将对表示论、几何/拓扑、组合学和数学物理领域产生重大影响。