Talk 1:

Title: P-adic Hodge Theory, Introduction and Motivation
Abstract: We give an introduction to p-adic Hodge theory, including motivation both from classical non-p-adic and p-adic theory. We introduce some techniques we will need in the next talks.



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Talk 2:

Title: Classes of Representations and Rings of Periods
Abstract: We discuss Fontaine's various classes of representations and how they correspond to certain "rings of periods". These are "big" algebras that try to recapture some geometric content which is otherwise hard to translate to the p-adic world. We try to elucidate an axiomatic approach as well as some geometric intuition. We sketch proofs for the hierarchy of representations (Hodge-Tate, de Rham, semistable, crystalline) and discuss the p-adic monodromy theorem.



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Talk 3:

Title: ($\varphi,$$\Gamma$)-modules and Beyond
Abstract: We present Fontaine's equivalence of categories between *all* p-adic Galois representations and objects of semilinear algebra called ($\varphi,$$\Gamma$)-modules, and give an outline of a proof. We then discuss connections to earlier theory (so-called reciprocity laws), applications and possible generalizations.



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