Algebraic Geometry and Geometric Representation Theory

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Categorification is a philosophy in mathematics that suggests that much of the theory developed so far in a variety of areas is just a shadow of something deeper and more complex. In my research over the last few years I have been following this line of thinking to study several closely related fields: algebraic geometry (which is the study of polynomials), representation theory (roughly, the study of matrices) and topology (the study of surfaces, knots and other more complicated shapes).*By thinking one level higher this philosophy has found surprising applications in the fields mentioned above. A shadow gives you only a rough (and possibly wrong) depiction of the real nature of things. Discovering the actual object can reveal amazing and unexpected detail. In the same way, categorification has revealed new, remarkable, deep structures in mathematics.