Cohomology theories, and thus period maps, attempt to study varieties via linear algebraic data like vector spaces with a filtration in the case of de Rham cohomology. The occuring linear algebraic data can however become quite elaborate, and it has become a powerful tool to geometrize it to vector bundles/quasi-coherent sheaves on associated stacks, for example through Simpson's algebraic de Rham stacks. In this course, we want to explain through examples how this geometrization works (with a focus on the p-adic case), and how it yields p-adic period maps.

References:
Fargues' ICM article
Introduction to the geometrization paper of Fargues/Scholze
Analytic stacks (Clausen-Scholze or Scholze's course on real local Langlands)
Hansen's paper on p-adic period maps.

The Hodge structure on the singular cohomology of a complex algebraic variety encodes the integrals of algebraic forms along topological cycles. In an algebraic family of algebraic varieties, the variation in this data yields a classifying map---called a period map---from the base of the family to the moduli space of Hodge structures. While period maps are intrinsically complex analytic, they have many deep applications to algebraic and arithmetic geometry. In this course we'll slowly develop the theory of variations of Hodge structures and the corresponding complex period maps. We will then describe how these structures fit into a general conjectural picture, roughly organized according to the principle "an object of transcendental origin which happens to be algebraic ought to be algebraic for a geometric reason". Finally, we will describe how recent progress has been made on understanding this picture using techniques from differential equations, o-minimality, and p-adic Hodge theory.

G-functions are power series with algebraic coefficients that solve a linear differential equation and satisfy certain growth conditions of arithmetic nature. Their significance in algebraic geometry comes from the fact that Picard-Fuchs differential equations associated with pencils of algebraic varieties have a basis of solutions consisting of G-functions with monodromy. At points where the motivic Galois group of the fibres is smaller than the generic one, there exist polynomial relations between the special values of those G-functions that do not arise from relations between the functions themselves. It is a beautiful idea of André that the diophantine properties of special values of G-functions (namely, Bombieri's principle of global relations) constraint the points at which exceptional relations can occur. This is how he was led to the André--Oort conjecture. More precisely, the method of G-functions yields bounds for the heights of such exceptional points, from which finiteness or non-density results can be deduced by means of the Pila--Zannier strategy in some favorable situations. The goal of the mini-course is to present some very recent instances of the method of G-functions due to Daw--Orr, Papas, and Urbanik. After explaining the general principles in detail, I will focus on how an interpretation of p-adic values of G-functions using p-adic Hodge theory allows Urbanik to go beyond the classic setting of families of abelian varieties.

This minicourse is an introduction to the theory of adic spaces, as defined by Huber. They provide a convenient framework for non-archimedean geometry, i.e. analytic geometry over non-archimedean fields like $\mathbb Q_p$ and $\mathbb F_p((\pi))$. After providing the general definition of adic spaces, we will discuss the two most important types: Rigid-analytic varieties and perfectoid spaces. Rigid-analytic varieties provide a p-adic analog of complex analytic varieties. Perfectoid spaces are more exotic spaces introduced by Scholze and can be seen as a form of ``universal coverings'' of rigid-analytic varieties. As such, their geometry and sheaf theory are particularly simple and they have proven to be an invaluable tool in the study of rigid-analytic varieties and beyond.

Given a smooth proper family of algebraic varieties over a base living over a p-adic field, the de Rham and etale cohomology of the fibers organize, respectively, into a vector bundle with a flat connection and into a lisse sheaf. We will discuss additional structures present on these local systems coming from p-adic Hodge theory, such as the notions of de Rham and crystalline local systems, which encode how p-adic periods of the fibers vary in a family. One of the key structure appearing in the formulation of the p-adic comparison theorems in a family is the notion of a pro-etale vector bundle -- we will also discuss recent results on p-adic Simpson correspondence describing the category of pro-etale vector bundles on smooth proper varieties over p-adic fields.