Let $\overline{M_{g,n}}$ be the moduli space of stable curves of genus $g$ with $n$ marked points. It is a classical problem in algebraic geometry to determine which of these spaces are rational over $\mathbb{C}$. In this talk, based on joint work with Mathieu Florence, I will address the rationality problem for twisted forms of $\overline{M_{g,n}}$ . Twisted forms of $\overline{M_{g,n}}$ are of interest because they shed light on the arithmetic geometry of $\overline{M_{g,n}}$, and because they are coarse moduli spaces for natural moduli problems in their own right. A classical result of Yu. I. Manin and P. Swinnerton-Dyer asserts that every form of $\overline{M_{0,5}}$ is rational. (Recall that the $F$-forms $\overline{M_{0,5}}$ are precisely the del Pezzo surfaces of degree 5 defined over $F$.) Mathieu Florence and I have proved the following generalization of this result.
Let $ n\geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq$ 2.
(a) If $ n$ is odd, then every twisted $F$-form of $\overline{M_{0,n}}$ is rational over $F$.
(b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form of $\overline{M_{0,n}}$ which is unirational but not retract rational over $F$.
We also have similar results for forms of $\overline{M_{g,n}}$ , where $g \leq 5$ (for small $n$ ). In the talk, I will survey the geometric results we need about $\overline{M_{g,n}}$ , explain how our problem reduces to the Noether problem for certain twisted goups, and how this Noether problem can (sometimes) be solved.

Keywords: rationality - moduli spaces of marked curves - Galois cohomology - Noether's problem[-]

Bruhat-Tits theory applies to a semisimple group G, defined over an henselian discretly valued field K, such that G admits a Borel K-subgroup after an extension of K. The construction of the theory goes then by a deep Galois descent argument for the building and also for the parahoric group scheme. In the case of unramified extension, that descent has been achieved by Bruhat-Tits at the end of [BT2]. The tamely ramified case is due to G. Rousseau [R]. Recently, G. Prasad found a new way to investigate the descent part of the theory. This is available in the preprints [Pr1, Pr2] dealing respectively with the unramified case and the tamely ramified case. It is much shorter and the method is based more on fine geometry of the building (e.g. galleries) than algebraic groups techniques.[-]

Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the existence of two different types of geometric triality. One of them is related to the octonions, but the other one is better explained in terms of a class of nonunital composition algebras discovered by the physicist Okubo (1978) inside 3x3-matrices, and which has led to the definition of the so called symmetric composition algebras.
This talk will review the history, classification, and their connections with the phenomenon of triality, of the symmetric composition algebras.[-]

In this course we give a proof of a central result in the theory of translation surfaces known as $Masur's$ $criterion$. The aim is to introduce the audience to the subject and illustrate how one can understand ergodic properties of the geodesic flow on an individual translation surface by studying the behaviour of its $SL(2,\mathbb{R})$-orbit on moduli space. The course is divided in 4 parts. Exercise sessions (TP) are also planned.

- Basic notions about translation surfaces.
- Moduli spaces of translation surfaces.
- Dynamical aspects of translation flows.
- Proof of Masur's criterion.[-]

In this course we give a proof of a central result in the theory of translation surfaces known as $Masur's$ $criterion$. The aim is to introduce the audience to the subject and illustrate how one can understand ergodic properties of the geodesic flow on an individual translation surface by studying the behaviour of its $SL(2,\mathbb{R})$-orbit on moduli space. The course is divided in 4 parts. Exercise sessions (TP) are also planned.

- Basic notions about translation surfaces.
- Moduli spaces of translation surfaces.
- Dynamical aspects of translation flows.
- Proof of Masur's criterion.[-]

This is a report on joint work with François Digne. Quasisemisimple elements are a generalisation of semisimple elements to disconnected reductive groups (or equivalently, to algebraic automorphisms of reductive groups). In the setting of reductive groups over an algebraically closed field, we discuss the classification of quasisemisimple classes, including isolated and quasi-isolated ones. The talk will start with the basic theory of non-connected reductive groups.[-]

The goal of this lecture is to present the construction of the Bruhat-Tits buildings attached to a quasi-split (that is admitting a Borel subgroup) semisimple group G defined over an henselian discretly valued field K and also the construction of the parahoric group schemes parametrized by the points of the buildings. The building part is [BT1] and the group scheme part corresponds to the four first sections of [BT2] but could also be treated by Yu's method [Y] namely by using Raynaud's theory of group schemes [BLR].[-]

In this course we give a proof of a central result in the theory of translation surfaces known as $Masur's$ $criterion$. The aim is to introduce the audience to the subject and illustrate how one can understand ergodic properties of the geodesic flow on an individual translation surface by studying the behaviour of its $SL(2,\mathbb{R})$-orbit on moduli space. The course is divided in 4 parts. Exercise sessions (TP) are also planned.

- Basic notions about translation surfaces.
- Moduli spaces of translation surfaces.
- Dynamical aspects of translation flows.
- Proof of Masur's criterion.[-]