Geometric and category theoretic methods in representation theory

代数是数学中最古老的领域之一。它涵盖了从简单的代数方程和多项式到线性和抽象代数的广泛主题。对对称性的研究与代数的一个分支“群论”有关。例如，一组物理或几何物体的对称性形成了数学家所说的“群”。研究群体如何作用于其他数学对象，如集合和向量空间，被称为“表示论”。 我的研究涉及到表征理论中的两个相对较新的方向，即几何表征理论和范畴化。几何表示理论使用几何，而不仅仅是代数，来研究表示理论中的高级主题。结果的新观点允许人们使用几何技术以新的视角来检验表示理论主题，并在某些情况下证明数学家一直无法用纯代数方法证明的代数结果。此外，它还允许人们使用代数来研究数学家和理论物理学家感兴趣的各种几何空间。分类包括发现隐藏在众所周知的数学概念下的数学结构。它通常会导致对所涉及的主题有更深的理解。 我的目标之一是进一步发展几何表示理论和分类领域。我希望继续扩展对代数结果的几何解释，并研究表示理论的不同几何方法之间的联系，其中有几种。我还计划使用几何表示理论和归类的工具来解决数学和数学物理中各种悬而未决的问题。

Algebra is one of the oldest areas in mathematics. It encompasses a wide range of subjects from simple algebraic equations and polynomials to linear and abstract algebra. The study of symmetries is related to a branch of algebra called 'group theory'. For example, the set of symmetries of a physical or geometric object form what mathematicians call a 'group'. The study of how groups act on other mathematical objects, such as sets and vector spaces, is called 'representation theory'. My research involves two relatively new directions in representation theory called 'geometric representation theory' and 'categorification'. Geometric representation theory uses geometry, rather than algebra alone, to study advanced topics in representation theory. The novel viewpoint that results allows one to use geometric techniques to examine representation theoretic topics in a new light and in some cases to prove algebraic results that mathematicians have been unable to prove using purely algebraic methods. Additionally, it also allows one to use algebra to study various geometric spaces that are of interest to mathematicians and theoretical physicists. Categorification involves the discovery of hidden mathematical structure underlying well-known mathematical concepts. It often results in a much deeper understanding of the topics involved. One of my aims is to further develop the fields of geometric representation theory and categorification. I hope to both continue to extend geometric interpretations of algebraic results as well as examine the connection between different geometric approaches to representation theory, of which there are several. I also plan to use the tools of geometric representation theory and categorification to tackle various unsolved problems in mathematics and mathematical physics.