Geometric and category theoretic methods in representation theory

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The field of representation theory can be thought of as the study of symmetry in mathematics and physics. Certain groups of symmetries, known as Lie groups, play a vital role in the study of the physical laws that govern our universe. Therefore, developing a better understanding of these mathematical objects results in a deeper and more sophisticated knowledge of the world around us.******I use techniques in geometry and abstract algebra to study representation theory. Geometric representation theory is a field that uses concepts coming from geometry to study algebraic objects, in addition to using algebraic techniques to study geometry. This rich interplay between the two fields has proven to be extremely beneficial to both.******I also study a phenomenon known as a categorification. Categorification is a relatively new and very exciting field of mathematics. Its philosophy is that many mathematical structures that appear to be fundamental are, in fact, mere shadows of a higher and richer mathematical reality. I aim to uncover this hidden higher structure, thus revealing a deeper, more unified mathematical framework.******Over the next 5 years, I plan to focus on the categorification of the Heisenberg algebra, which plays a vital role in theoretical physics and Lie theory. In addition to developing the mathematical theory of Heisenberg categorification, I will also work on its applications to theoretical physics and other areas of mathematics. In addition, I also plan to continue my study of a certain type of Lie algebra known as an equivariant map algebra. The study of these objects is a fast developing field with connections to many areas of mathematics.******I expect the outcomes of this research to have significant impact in the fields of representation theory, geometry/topology, combinatorics, and mathematical physics.