Harmonic Maass Forms

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Harmonic weak Maass forms are the subject of much current interest. A harmonic weak Maass form of weight k is a smooth function of the upper half-plane which transforms with respect to a subgroup of the modular group, is annhilated by the hyperbolic Laplacian in weight k, and has at most linear exponential growth at cusps. Such forms decompose into holomorphic and non-holomorphic parts. Non-holomorphic parts are related to period integrals of elliptic cusp forms. The holomorphic parts, sometimes called mock modular forms, appear in a variety of mathematical contexts. In the 1970s, Zagier showed that the generating function for Hurwitz class numbers is a mock modular form. More recently,Bringmann, Ono, and Zwegers showed that Ramanujan's mock theta functions are mock modular forms. Subsequent works applied the theory of harmonic weak Maass forms to geometry, mathematical physics, and probability theory, and to topics more closely allied to number theory such as partitions and q-series, elliptic modular forms, traces of singular moduli, Borcherds products, and modular L-functions. Motivated by these connections, this proposal addresses questions concerning the holomorphic parts of harmonic weak Maass forms. In particular, the PI intends to studyproblems requiring interplay between harmonic weak Maass forms and elliptic modular forms.The PI's research interest is in number theory. The topics in this proposal originate in work of the legendary mathematician Srinivasa Ramanujan in the early twentieth century. An important part of Ramanujan's legacy concerns the large quantity of aesthetic formulas he discovered involving functions called q-series. Toward the end of his life, he discovered a particular family of q-series which he called "mock theta functions". Though these functions share many properties with modular forms, a class of functions central to modern number theory, the mock theta functions are not modular forms. Recent works of Bringmann, Ono, and Zwegers finally settled the question, as to the precise relationship between the mock theta functions and the theory of automorphic forms, a theory which subsumes the theory of modular forms. This question had been outstanding for roughly eighty years. We now know that a mock theta function is "part" or "half" of a certain automorphic form called a harmonic weak Maass form. This discovery motivates the study of a broader class of functions, called mock modular forms, which arise as "parts" of harmonic weak Maass forms. These functions have applications not only to number theory, but to other areas of mathematics such as geometry, topology, and probability, and to mathematical physics.