Physical Mathematics

Physical mathematics is concerned with mathematics motivated by physics. Prime example of physical mathematics is the pioneering work of Eugene Wigner on the unitary representations of Poincare group which was motivated by his results proving that symmetries of quantum systems must be realized unitarily on their Hilbert spaces. His work opened up the huge field studying the unitary duals of noncompact Lie groups which is still an unfinished chapter of mathematics.
In a similar vein, the discovery of supersymmetry by physicists led to the development of the theory of unitary representations of Lie superalgebras.
Remarkably, though algebraically more complicated the theory of unitary representations of noncompact Lie superalgebras turned out to be simpler than those of noncompact Lie groups. Furthermore, some of the earliest results on AdS/CFT dualities were obtained, in a true Wignerian sense, within the framework of work on fitting the spectra of Kaluza-Klein supergravities into unitary supermultiplets of their underlying supersymmetry algebras.