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There has recently been signiﬁcant progress on several fronts of the arithmetic Langlands programme building on a better understanding of p-adic automorphic forms and developments of modularity lifting theorems for Galois representations beyond the so-called Taylor-Wiles method. Underlying this are developments on geometric and algebraic aspects of of modular and automorphic forms, in particular, breakthroughs in the theory of p-adic families of geometric modular forms, p-adic eigenvarieties, and Scholze’s theory of perfectoid spaces. Eigenvarieties are rigid analytic spaces (in the sense of Tate) with a deep arithmetic meaning, encoding part of the Langlands program, in particular, its p-adic variant recently studied by Breuil and Colmez. Kisin’s work on the Fontaine-Mazur conjecture is a prime example of work mixing Galois representation, p-adic Hodge theory and p-adic deformation techniques to prove results about eigenvarieties. A more recent highlight of applications of p-adic families is the paper by G.Boxer, F.Calegari, T. Gee and V.Pilloni in which they proved potential modularity of abelian surfaces using the p-adic variation of automorphic sheaves on Siegel schemes and higher Hida theory.
The purpose of this conference is to gather experts on these topics to present the latest advances, encourage discussions to foster collaborations between the participants and expose PhD students to these exciting new developments.