The main goal of this project is to investigate and explain modularity of q-series. Modular forms generalize classical trigonometric functions because they are periodic; however, they have more symmetries. They play a central role in many areas including algebraic topology, arithmetic geometry, combinatorics, mathematical physics, number theory, and representation theory. The situation is complicated by the fact that often modularity is broken, it is however not always a priori clear in which way. We will see several such examples of broken modularity in this project. Many areas predict modularity of important functions but fail to provide a full understanding. Any progress made towards answering such questions will imply fundamental results at the frontier of number theory and other areas. The challenge that I am taking is to push the boundaries further and to predict, prove, and understand modularity in various settings, particularly in moonshine, combinatorics, vertex algebras, and enumerative geometry.
This understanding of modularity has also wide-reaching applications to number theory. I have many years of experience in answering modularity questions and already succeeded in proving many deep conjectures and in building theories. I will achieve my goal of better understanding modularity by investigating q-series arising in particular in moonshine, combinatorics, vertex algebras, and Gromov-Witten invariants; this makes this project interdisciplinary. For this I will take predictions from these areas as guiding principle and develop new methods along the way. In the past obstructions to modularity have been a stumbling block. I will overcome this problem by a more systematic study of the occurring objects. A successful outcome of the proposed research will open new doors as I will have my newly developed machinery at hand which will apply in other areas as well.
