Theta functions in differential and arithmetic geometry

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My research lies in the field of arithmetic geometry, at the interface of two mathematical subfields: number theory and geometry. Number theory is the study of integers, which are essentially discrete and rigid in nature. On the other hand, geometry deals with objects that are continuous, that can be stretched and pulled and deformed in a fluid manner. Arithmetic geometry marries these two points of view, applying tools and intuitions from the world of geometry to gain greater insight into number theoretic phenomena, and vice versa.*** I'm particularly interested in Shimura varieties, which are geometric objects that have fascinated mathematicians for decades, in part because they seem to carry deep information about number theory, and are an ideal proving ground for the tools and techniques of arithmetic geometry. Indeed, many of the major recent successes in number theory, including the spectacular resolution of Fermat's last theorem, can be viewed in these terms.****** In some cases, there is a nesting phenomenon whereby one Shimura variety contains many sub-Shimura varieties called special cycles. In recent years, evidence has emerged that special cycles possess very subtle and mysterious symmetries, which can be expressed precisely in terms of a mathematical property known as modularity, and which mirror, in a sense, the behaviour of the classical theta functions that have been studied for well over 150 years. However, despite a wealth of beautiful mathematics inspiring deep conjectures around this phenomenon, at present a complete conceptual account is quite out of reach.****** The research described in this proposal is aimed towards closing this gap. In particular, I hope to make significant strides on the geometric aspects of modularity questions, in part by leveraging recent joint work with Stephan Ehlen that develops certain conceptual tools in this context. At the same time, there are interesting, and interrelated, problems in the arithmetic setting that I intend to study, assisted by a team of three graduate students. This work would provide compelling evidence for the conjectural picture described above. As a whole, the outcome of the proposed research will advance the state of the art in this area, and point the way towards a systematic understanding of this fascinating circle of ideas.