Project Grant 2201112

This National Science Foundation Project Grant of $180,000 supports research into geometric methods in the p-adic Langlands program under the Mathematical and Physical Sciences federal grant program (CFDA 47.049). Specifically, the University of Utah will apply ideas from calculus to the study of perfectoid spaces and diamonds in order to uncover new structural properties of the Langlands correspondence and help understand basic questions about integers and prime numbers. The awardee will introduce an analytic structure on diamonds and apply this theory to study representation-theoretic aspects of p-adic automorphic forms as they arise in the Langlands correspondence, with an emphasis on understanding the analytic structure of p-adic spaces analogous to universal covers of complex locally symmetric spaces. The research aims to elucidate the general shape of this analytic theory through concrete examples while connecting recent ideas in p-adic automorphic forms, the p-adic geometry of Shimura varieties, and the p-adic Langlands correspondence. The award period is from August 15, 2022 to July 31, 2025.