Geometric methods in representation theory

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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.