Title: A Primer on Diagrammatic Categorification
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Abstract: Categorification is the art of finding new structure.  
Typically, you take an invariant or algebra that you know and love, and then realize that it is no more than the shadow of a truly deeper thing, its categorification. On the surface level, elements that were equal in the algebra lift to objects that are isomorphic, but the new structure comes from the fact that these isomorphisms themselves satisfy equalities. The recent program of "diagrammatic categorification" has made this procedure even more like art: morphisms in the categorification can be drawn as pretty pictures on a planar canvas.
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We will explain the notion of categorification and what it is good for; demonstrate a diagrammatic categorification of the Hecke algebra; and discuss why it is reasonable to believe that the most beautiful categorifications can be drawn in this way, allowing computations with very high-tech ideas to be performed by simple manipulations of planar diagrams.