Geometric 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学的一个叫群论的分支有关。例如，一组物理或几何对象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，称为表示理论。我的研究涉及表示理论中一个相对较新的方向。这种新的方法使用几何，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独立感兴趣的各种几何空间。我的目标之一是进一步发展几何表示理论领域。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的工具来解决数学和数学物理中各种悬而未决的问题。特别是，我正在研究统计力学中的一个模型，数学家们目前还不能令人满意地理解这个模型。我希望新的几何观点将帮助我们更好地理解这一模型。

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 a relatively new direction in representation theory. This new approach 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 independent interest to mathematicians and theoretical physicists.One of my aims is to further develop the field of geometric representation theory. 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 to tackle various unsolved problems in mathematics and mathematical physics. In particular, I am studying a certain model in statistical mechanics that mathematicians do not currently understand at a satisfactory level. I expect that the new geometric viewpoint will help us to better comprehend this model.