Geometric and category theoretic 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 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.