Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory

Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms.These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics.Here we outline the general facets of the theory, and we give several applications to number theory: partitions and q-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular L-functions, and the work of Bruinier and Yang on Gross-Zagier formulae.What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.Modular forms are central in contemporary mathematics.Indeed, modular forms play crucial roles in algebraic number theory, algebraic topology, arithmetic geometry, combinatorics, number theory, representation theory, and mathematical physics.The recent history of the subject includes (to name a few) great successes on the Birch and Swinnerton-Dyer Conjecture, Mirror Symmetry, Monstrous Moonshine, and the proof of Fermat's Last Theorem.These celebrated works are dramatic examples of the evolution of mathematics; indeed, it would have been impossible to prophesy them fifty years ago.Instead of travelling back in time to the 1950s, our story (also see [165]) begins in 1887, in a village in India.Our mathematics, 1 which is about harmonic Maass forms, begins with the legend (see [6, 39, 40, 110, 111, 133, 165]) of the great mathematician Srinivasa Ramanujan, and the mathematics he conjured from his death bed.