Mathematics of Quantum Matter
The fruitful interaction between theoretical physics and mathematics has focused within CRC TR12 in particular on the role of symmetries in mesoscopic quantum matter. Ongoing research centered around supersymmetric s-models for disordered electrons aims to develop the methods of harmonic analysis for symmetric superspaces, the Borel and superbosonization transforms, and the mathematical physics of localization and delocalization. Conformal field theory methods and the representation theory of infinite-dimensional Lie (super-)algebras are applied to condensed matter systems as well as string theory. Further projects include the asymptotics of kernel functions and the weight statistics of representations, Berezin-Toeplitz quantization, and the expansion of ultracold atoms in optical lattices. A fascinating new direction linking mathematics and physics is the classification of topological phases of gapped quantum matter, with its diverse connections to equivariant K-theory, representation theory, spectral theory, and index theory.