The PI will continue his work in p-adic Hodge theory, with a particular focus on applications to the arithmetic of Shimura varieties. In particular, their integral models, arithmetic compactifications and the structure of their mod p points. This should eventually allow one to understand the cohomology of these varieties and the Galois representations which occur there.
p-adic Hodge theory is branch of number theory which seeks to study the local properties of Galois representations which arise from geometry. The subject has been at the heart of many of the most important developments in number theory in the last 30 years (e.g the proof of Fermat's Last theorem). The project aims to use these techniques to study the arithmetic of a class of geometric objects - Shimura varieties - which have proved particularly important in arithmetic applications.
In recent years there have been a number of astonishing breaktrhoughs in number theory, including the proof of Fermat's Last Theorem by Wiles about 20 years ago. At the heart of techniques used in these advances have been p-adic Hodge theory, Galois representations and Shimura varieties. The award The award supported research in these areas by the PI and his students. The PI completed his proof of the Langlands-Rapoport conjecture, as well as a joint paper with Eva Viehmanna dn Miaofen Chen which proved a result used in the former paper. This result describes the structure of the mod p points on a Shimura variety. The PI also completed a joint paper with Toby Gee proving cases of the Breuil-Mezard conjecture and completing the proof of the Buzzrd-Diamond-Jarvis conjecture on weights modular Galois representations. Two graduate students supported by the award graduated. Carl Erickson (Ph.D 2013) who worked on moduli spaces of Galois representaiotns. Hansheng Diao (Ph. 2014) who worked on periods of overconvergent Galois representations, and whose thesis proves a conjecture of Coleman-Mazur on the properness of the eigencurve. A number of other graduate students were supported by the award. The subjects they worked on include: algebraicity of formal subvarieties, and the Grothendieck p-curvature conjecture, the p-adic local Langlands correspodence, mod p points on Shimura varieties of parahoric level, the Hasse-Weil zeta functions of Shimura varieties.