## Adic Spaces I

We will recall the definition and some properties of Huber rings and we will define and study spaces of continuous valuations.

## Adic Spaces II: Examples

We will explain in detail some examples of affinoid adic spaces including the unit ball (for rank 1 valuations). If time permits we will talk on completion and rational subsets.

## Arithmetic Okounkov bodies I

We will review the construction of Newton-Okounkov bodies by Kaveh-Khovanskii and Lazarsfeld-Mustata, within a general framework of semigroups and cones associated to filtrations.

## On the exceptional zeros of p-adic L-functions of Hilbert modular forms

The use of modular symbols to attach p-adic Lfunctions to Hecke eigenforms goes back to the work of Manin et al in the 70s.

## Arithmetic Okounkov bodies II

We will address the construction of arithmetic Newton-Okounkov bodies, and more generally functions on Newton-Okounkov bodies arising from filtrations, following ideas of Boucksom and collaborators.

## Galois representations and torsion classes

I will describe joint work in progress with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne on potential modularity for elliptic curves over imaginary quadratic fields.

## Fermat’s Last Theorem over some totally real number fields

In the past few years, numerous progress have been achieved about the Fermat equation over number fields. We have in particular a criterion, due to Freitas and Siksek, allowing to establish the asymptotic Fermat’s Last Theorem over certain totally real number fields.

## Surjectivity of Galois representations of quadratic Q-curves

One initial tool in Wiles’ proof of Fermat’s last theorem is using the Galois representation on the p-torsion of an elliptic curve and proving that this representation is irreducible except for p very small (with an absolute bound).

## Which integers are sums of seven cubes?

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454.

## On the exceptional zeros of p-adic L-functions of Hilbert modular forms

The use of modular symbols to attach p-adic L-functions to Hecke eigenforms goes back to the work of Manin et al in the 70s.