Dissertation/Thesis Abstract

Applications of Harmonic Maass Forms
by Griffin, Michael John, Ph.D., Emory University, 2015, 306; 3708974
Abstract (Summary)

In this thesis, we prove various results in the theory of modular forms and harmonic Maass forms, representation theory, elliptic curves and differential geometry. In particular, we give a broad framework of Rogers-Ramanujan identities and algebraic values; we prove that Ramanujan's mock theta functions satisfy his original conjectured definition; and we show that certain harmonic Maass forms which arise naturally from the arithmetic of elliptic curves encode central L-values and L-derivatives involved in the Birch and Swinnerton-Dyer conjecture. We also prove a conjecture of Moore and Witten connecting the regularized u-plane integral on the complex projective plane with Donaldson invariants for the SU(2)-gauge theory. In our final two applications, we turn to moonshine phenomena. Monstrous Moonshine relates the Fourier coefficients of certain modular functions to values of the irreducible characters of the Monster group--the largest of the sporadic simple groups. We give the asymptotic distribution of these character values, answering a question of Witten with applications to mathematical physics. The Umbral Moonshine conjectures relate the the values of irreducible characters of prescribed finite groups with the Fourier coefficients of distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. We complete the proof in the remaining cases.

Indexing (document details)
Advisor: Ono, Ken
Commitee: Borthwick, David, Zureick-Brown, David M.
School: Emory University
Department: Math and Computer Science
School Location: United States -- Georgia
Source: DAI-B 76/10(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Automorphic forms, Harmonic maass forms, Number theory
Publication Number: 3708974
ISBN: 9781321844344