Real-Analytic Orthogonal Modular Forms as Generating Series

Research Project, 2020 – 2023

How can we organize information and data in mathematical research? One popular and successful way is to introduce parameters and study several objects, possibly infinitely many, simultaneously. A particular instance of this idea are generating functions that are build from numbers, which we call their (expansion) coefficients. The more precisely we can describe a generating function by, say, analytic methods, the more we learn about its coefficients. One of the most narrow classes of functions are modular forms. In the past decades, mathematicians have found several important generating functions that are modular forms of the easiest kind. These findings impact till today as diverse fields as theoretical and mathematical physics, combinatorics, geometry, and, of cause, number theory. They have produced progress on several key questions in these areas, and have stimulated further research. In recent years generating series that are more complicated modular forms have been discovered. The goal of this project is to make accessible to other researchers a completely new kind of modular generating series that has appeared in physics but for which we lack a general understanding. In particular, as one outcome of this project researchers from other fields will have available candidate series that they can match their results against.