Homotopical Methods in Arithmetic Geometry

Number theory is the study of structures which may be broadly called number systems. Number systems come in very different flavors and have a wide variety of applications; for example, the physical world is modeled on real numbers, computer systems are modeled on binary numbers, cryptographic and encoding schemes are often designed in the context of modular numbers, etc. Recently, a rich connection has formed between number theory and homotopy theory, which is an algebraic abstraction of ideas inspired by the study of shapes and topology. This project seeks to apply powerful new techniques originating in homotopy theory to resolve outstanding questions about numbers. In addition, a significant component will be devoted to the support and training of young researchers. In more specific terms, the project will develop a new theory of “derived Fourier analysis”, an expansion of Fourier analysis to derived vector spaces and motivic coefficients. This has anticipated applications in enumerative geometry, especially to modularity conjectures for arithmetic theta functions, and in the investigation of categorical period conjectures pertaining to relative Langlands duality, as formulated by Ben-Zvi – Sakellaridis – Venkatesh. Another direction will be the study of cohomology operations in p-adic geometry, using new perspectives on prismatic and syntomic cohomology due to Drinfeld and Bhatt-Lurie, with an eye towards resolving old questions about Brauer groups. Finally, methods of algebraic K-theory will be combined with the theory of Shimura varieties (extending joint work with Galatius and Venkatesh) in order to better understand the cohomology of arithmetic groups and related Galois representations. 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.