Lifting cusp forms to Maass forms with an application to partitions

For 2 < k ∈ 1/2 &Z; , we define lifts of cuspidal Poincaré series in S_k(Γ₀(N)) to weight 2 - k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general framework "explaining" Ramanujan's mock theta functions. As an application, we show that the number of partitions of a positive integer n is the "trace" of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a Calabi-Yau threefold.