This project consists of a number of applications of non-archimedean (p-adic) analysis and geometry to problems in arithmetic algebraic geometry, of both theoretical and computational nature. Offshoots of the awarded research include new techniques in the theory of p-adic differential equations, which has previously found applications to computer science; for instance, one of the PI's algorithms for computing zeta functions is widely cited in the cryptography literature. The proposed activities include training of graduate students in several capacities, which promotes enhancement of the US knowledge base; increased access to enrichment activities for low-income students in New York City and Los Angeles; new training opportunities for US undergraduates seeking careers in mathematics education; development of open-source software for mathematics research; and work on interactive open-source curricular materials, including the introduction on a new course on mathematical software.
It is hardly an overstatement to assert that the theory of perfectoid spaces since 2010 has triggered a revolution in arithmetic geometry, with rapid advances coming at a previously unknown pace; however, deep improvements in the foundations of the subject are vital in order to sustain this rate of progress. Further work is also needed to fully realize the potential of perfectoid spaces to deepen our understanding of the relationship between geometric and representation-theoretic objects indicated by the Langlands correspondence; in particular, this will require deep insights in order to globalize the hitherto local constructions of p-adic Hodge theory. Separately, computational advances driven by p-adic analysis have had, and will continue to have, a transformative effect on the study of arithmetic-geometric objects and their associated L-functions, by opening up vast new territories for empirical observation. The net effect is to bring number theory back to its roots as an empirically driven subject, thus leading to a new generation of theorems based on experimental predictions (echoing the historical development of such results as quadratic reciprocity and the prime number theorem).
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.
