Geometric and category theoretic methods in representation theory

表示论的领域可以被认为是数学和物理中对称性的研究。 某些对称群，称为李群，在研究支配我们宇宙的物理定律中起着至关重要的作用。 因此，对这些数学对象的更好理解会导致对我们周围世界的更深入和更复杂的知识。 我使用几何和抽象代数的技巧来学习表示论。 几何表示理论是一个使用来自几何的概念来研究代数对象的领域，除了使用代数技术来研究几何之外。 事实证明，这两个领域之间丰富的相互作用对双方都非常有利。 我还研究了一种被称为分类的现象。 范畴化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层次的结构，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，我计划专注于海森堡代数的分类，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，我还将致力于其在理论物理和其他数学领域的应用。 此外，我还计划继续我的研究某种类型的李代数称为等变映射代数。 对这些对象的研究是一个快速发展的领域，与数学的许多领域都有联系。 我希望这项研究的成果在表示论，几何/拓扑，组合学和数学物理领域产生重大影响。

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.