Perfectoid Spaces, Diamonds, and the Langlands Program

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The Langlands program is like a grand unified theory of numbers. It a suite of conjectures which encompasses classical patterns in numbers (called "reciprocity laws") discovered by Euler and Gauss, as well as modern results like Fermat's Last Theorem. In brief, the Langlands program unites two sorts of symmetries: one coming from continuous entities (think of all the ways a sphere might be rotated in space), and the other from roots of algebraic equations (think of the "plus or minus" in the quadratic formula). The PI proposes research on the Langlands program, specifically the part dealing with the p-adic numbers, which are the strange cousins of the real numbers. (The letter p here stands for a prime number. The real numbers form a connected continuum, whereas the p-adic numbers are totally disconnected, like infinite fractal dust.) The Langlands program as it applies to the real numbers was worked out by Langlands himself, while the p-adic story remains somewhat mysterious. The PI intends to contribute to this portion of the Langlands program by studying the geometry of some fascinating new structures discovered in the last several years, namely perfectoid spaces and diamonds. These structures were invented by Peter Scholze, who received the Fields Medal in 2018 for their discovery. The project also supports work of the PI's graduate student, Maria Fernandez, on related topics.Since their introduction around 2012, perfectoid spaces have had some unexpected applications. One of these is Laurent Fargues' program to geometrize the Langlands program over the p-adic numbers. That is, he has brought it in line with the parallel program of geometric Langlands, which seems rather more tractable. Inspired by Fargues' program, the PI has proved (in joint work with Tasho Kaletha and David Hansen) a form of Kottwitz' conjecture on the cohomology of Rapoport-Zink spaces, using a version of the Lefschetz fixed-point formula which can apply to perfectoid spaces. This can be recognized as a geometric manifestion of Langlands functoriality between a p-adic group and one of its inner twists. The PI will expand these methods to apply to other sorts of functorialities. There is a further project concerning the "smoothness" of perfectoid spaces (a beautiful concept dating to 2016), and another on the modularity of elliptic curves over function fields, which is joint with the PI's graduate student.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.