I am a third-year student of pure mathematics at the University of Sydney, intending to enrol in Honours for 2015. While my ambition is to become a researcher, it certainly feels as though there’s much to learn before I’ll be ready to specialise into a particular field. My main interest so far has been in algebra, which is where the central ideas of my summer project have their roots. When I’m not lost in maths land, I like to read webcomics, play and watch tennis, and find new music. A few years ago, before school took priority, I also spent much of my time writing for the English Wikipedia; I hope one day to return and use my knowledge to improve Wikipedia’s coverage of maths.
Spherical harmonics and categorical representation theory
The aim of this project is to construct new example of categorical representations sl_2, the Lie algebra of 2×2 traceless matrices. While we understand very well the categorical lifting of finite dimensional representations of sl_2, the infinite dimensional analogues have not been studied extensively. We will try to construct categorical versions of a very important class of representations, called the principal series, which appear in the classical theory of spherical harmonics. We will do this by studying algebras called nilCoxeter algebras, whose representation theory allows us to categorify the operators of multiplication and differentiation on polynomial algebras.