This proposal details a comprehensive program which uses diagrammatic methods in categorified representation theory to obtain new results in representation theory and low-dimensional topology. One research objective seeks to exploit an analogy between knot homology and categorified representation theory to produce new representation theoretic objects arising from odd Khovanov homology. Another research direction seeks to use the powerful tool of categorical skew Howe duality to solve open problems in link homology theory for the special linear Lie algebra. This work will have applications to Landau-Ginzburg models in theoretical physics via the Kapustin-Li formula. The same diagrammatics used to encode the structure of categorified quantum groups will be used to transform current research into unique educational opportunities including an undergraduate research program at USC and innovative curriculum development of an undergraduate course on 'Diagrammatic Algebra'. The PI will also tap into the existing infrastructure of USC's community involvement to recruit high school students for a mathematics based event using diagrammatic algebra as a tool for engaging high school students.
An emerging mathematical philosophy known as "categorification" has significantly altered our perception of mathematics and the way we analyze our surroundings. Imagine if an architect studied a building by only examining its shadow. In much the same way, the perspective of categorification uncovers a hidden layer in mathematical objects allowing mathematicians to see the entire structure rather than its shadow. The PI's research uncovers these hidden structures in an area of mathematics known as representation theory which is closely connected to some of the most sophisticated models in theoretical physics. Following the philosophy that fundamental structures in mathematics should be simple and intuitive the PI utilizes a diagrammatic framework to encode a great deal of complexity into an intuitive diagrammatic language that greatly simplifies computations. The PI will incorporate this diagrammatic philosophy into transformative educational programs aimed at high school students, undergraduates, and graduate students making the tools of modern research accessible to a new generation of researchers. This award is co-funded by the Algebra and Number Theory program and the Topology and Geometric Analysis program.