Geometric and category theoretic methods in representation theory

表示论中的几何和范畴论方法

• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679755
• 关键词: Geometric; category; theoretic; methods; representation; Geometric; category; theoretic; methods; representation

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.

表示理论领域可以被认为是数学和物理学中对称性的研究。 某些对称群体（称为谎言群体）在研究我们宇宙的物理定律中起着至关重要的作用。 因此，对这些数学对象进行更好的理解会导致对我们周围世界的更深入，更复杂的知识。 几何表示理论是一个使用来自几何形状的概念来研究代数对象的领域，此外还使用代数技术来研究几何形状。 事实证明，这两个领域之间的这种丰富的相互作用对两者都非常有益。******我还研究了一种被称为分类的现象。 分类是一个相对较新且非常令人兴奋的数学领域。 它的理念是，许多似乎是基本的数学结构实际上仅仅是更高，更丰富的数学现实的阴影。 我的目标是揭示这种隐藏的更高结构，从而揭示了一个更深，更统一的数学框架。******在接下来的5年中，我计划专注于海森堡代数的分类，该分类在理论物理和谎言理论中起着至关重要的作用。 除了开发海森堡分类的数学理论外，我还将在理论物理学和其他数学领域的应用中进行应用。 此外，我还计划继续研究某种类型的Lie代数，称为Eopivariant地图代数。 对这些物体的研究是一个快速发展的领域，与许多数学领域的联系。