Joint IAS/Princeton University Number Theory Seminar

In his first letter to G. H. Hardy, Ramanujan hinted at a theory of continued fractions. He offered shocking evaluations which Hardy described as: "These formulas defeated me completely...they could only be written down by a mathematician of the highest class. They must be true because no one would have the imagination to invent them." -G. H. Hardy Ramanujan had a secret device, two power series identities which were independently discovered previously by L. J. Rogers. The two Rogers-Ramanujan identities are now ubiquitous in mathematics. It turns out that these identities and Ramanujan's handful of evaluations are hints of a much larger theory. In joint work with Michael Griffin and Ole Warnaar, the author has discovered a rich multivariable framework of Rogers-Ramanujan identities, one which comes equipped with a theory of algebraic numbers. This story blends the theory of Hall-Littlewood polynomials, modular functions, and the representation theory of Kac-Moody affine Lie algebras.