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学科。对称性的研究与代数的一个分支“群论”有关。例如，一个物理或几何对象的对称性集合形成了数学家所说的“群”。研究群如何作用于其他数学对象，如集合和向量空间，被称为“表示理论”。 我的研究涉及两个相对较新的方向表示理论称为“几何表示理论”和“几何表示”。 几何表示论使用几何，而不是代数，研究表示论的高级主题。新的观点，结果允许一个使用几何技术来审查代表性理论的主题在一个新的光，并在某些情况下证明代数结果，数学家一直无法证明使用纯粹的代数方法。此外，它还允许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。 范畴化涉及发现隐藏在众所周知的数学概念之下的数学结构。 它往往会导致对所涉及的主题有更深入的理解。 我的目标之一是进一步发展几何表示理论和范畴化领域。我希望继续扩展代数结果的几何解释，并研究表示论的不同几何方法之间的联系。我还计划使用几何表示理论和分类的工具来解决数学和数学物理中各种未解决的问题。

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.