I'll give an overview of a (somewhat conjectural, somewhat proven) connection between physics and representation theory. The physics part is 3-dimensional mirror symmetry, or more specifically, considering the structure of the Higgs and Coulomb branches of N=4 supersymmetric 3-dimensional field theories. The representation theory part is the Koszul duality between categories O over certain interesting pairs of noncommutative algebras. The remarkable connection is that the pairs of noncommutative algebras appearing in the second step are quantized function algebras for the varieties appearing in the first step. We still don't understand this story as well as we would like, but it's seen a lot of interesting progress in recent years, which I will do my best to explain.
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