With recent advances in $p$-adic Hodge theory a new set of major challenging problems has emerged, which the collaboration will aim to address; in particular, we will study $p$-adic motives and their moduli spaces, and address the role of coefficients (including characterizing the coefficients coming from geometry). We also anticipate many applications across a wide range of mathematics. The collaboration will be structured around the following five directions which provide a roadmap for the future of the field.
- Number theory and the Langlands program. This includes projects on the local Langlands program, modularity lifting, and Shimura varieties. A key aspect of this is the geometry of moduli spaces of Galois representations.
- Analytic geometry. Motivated by the $p$-adic local Langlands correspondence the collaboration team will explore cohomology and geometry of $p$-adic analytic varieties, building on earlier work in $p$-adic Hodge theory and the newly developed solid mathematics.
- Foundational questions in $p$-adic cohomology. The aim of this part is to develop further aspects of cohomology theories in the $p$-adic setting including the $p$-adic Riemann-Hilbert functor, theories with coefficients, and flat cohomology.
- Commutative algebra. Perfectoid techniques have generated new momentum in the study of commutative algebra of mixed characteristic rings and this part of the collaboration is devoted to applying the powerful new techniques to problems in this area.
- Prismatic homotopy theory. The development of prismatic cohomology has shed great new light on $p$-adic cohomology theories. Motivated by algebraic topology, collaboration members will explore further currently hidden structures, variant cohomology theories, and their arithmetic applications.