Differential and analytic techniques in p-adic geometry and applications to p-adic automorphic forms

This project will develop new techniques in the study of harmonics, along with applications to fundamental questions about the structure of prime numbers. More precisely, the mathematical theory of automorphic forms builds from the study of harmonics, or the fundamental tones of musical instruments, to a general theory of the vibrational modes of highly symmetric shapes in arbitrary dimensions. Because of a surprising connection between automorphic forms and prime numbers, it has also proven important to study these highly symmetric shapes with an alternative theory of geometry built up from an unusual notion of size and distance that detects divisibility by a fixed prime number. This is called p-adic geometry. The basic shapes in p-adic geometry look more like fractals such as the Cantor set than the shapes we encounter in our day-to-day life, thus much of our usual physical intuition about the real world cannot be applied in this setting. This project will improve our ability to reason intuitively in p-adic geometry by developing ideas from calculus and geometry, like derivatives and curvature, so that they can be applied also in p-adic geometry, and then use these tools to answer fundamental questions about automorphic forms and their connections to prime numbers. The mathematics of harmonics plays a fundamental role in signal processing, while questions about prime numbers are essential to the modern cryptography schemes that allow us to communicate and make purchases securely online, so that the mathematics to be developed in this project will have ties to areas of importance across the modern economy. The project also provides training opportunities for the next generation of researchers through research supervision and mentoring for undergraduate and doctoral students in mathematics. At a more technical level, this project will develop new tools for studying p-adic geometry using the language of inscribed v-sheaves, which adds a differential layer on top of the theories of diamonds and perfectoid spaces that make up the modern foundations of p-adic geometry. This theory is akin to equipping a topological space with the additional structure of a differential manifold. The project will connect these tools with other recent advances in the theory of p-adic geometry and analytic structures, and use them to study the relation between different spaces of p-adic automorphic forms and to study the geometry of moduli spaces in p-adic Hodge theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.