Olivier Benoist (CNRS – École Normale Supérieure – PSL)    Rationality over arbitrary fields and intermediate Jacobians
Abstract : An algebraic variety over a field k is said to be k-rational if it is birational to a projective space over k. Deciding whether a given variety is k-rational is an important and often difficult problem in birational geometry. After a general introduction to this topic, with an emphasis on the case when k is not algebraically closed, we will focus on the Clemens-Griffiths method of disproving k-rationality using intermediate Jacobians. This series of lectures will be partly based on joint work with Olivier Wittenberg.

Gergely Bérczi (Aarhus University)    Geometric invariant theory and moment maps for non-reductive group actions.
Abstract : Geometric Invariant Theory (GIT) was developed by Mumford in the 1960s to provide a systematic description of quotients of algebraic varieties by reductive algebraic groups in algebraic geometry. A key topological and computational aspect of this theory is that, due to the work of Ness and Kirwan in the 1980s, these GIT quotients are canonically isomorphic to symplectic quotients of the zero level set of moment maps.
In this mini-course, I will explain how this picture extends to a broad class of non-reductive groups. I will start with the construction, showing that all key features and computational advances of reductive GIT, such as the Hilbert-Mumford criteria, extend naturally. Then, we will introduce a canonical non-reductive moment map and explain that the constructed GIT quotient is isomorphic to the corresponding symplectic quotient. This leads to an effective intersection theory on non-reductive quotients with a Jeffrey-Kirwan type integral formula. If time permits, we will mention some recent applications in enumerative geometry.

Cédric Bonnafé (CNRS – Université de Montpellier)    K3 surfaces and complex reflection groups (Joint work with A. Sarti)
Abstract : Several exceptional varieties can be built using invariant theory of finite complex reflection groups (surfaces with many singular points, with many lines or conics, particular symplectic singularities…). In these lectures, we focus on the construction of K3 surfaces with big Picard number. We revisit and extend earlier works of Barth and Sarti, by making a more systematic use of the theory of complex reflection groups (invariants, Springer theory). This course can be seen as a systematic interplay between group theory and algebraic geometry, with the aim to construct explicitly exceptional objects in « classical » algebraic geometry.

Yohan Brunebarbe (CNRS – Université de Bordeaux)    O-minimality and applications in Hodge theory
Abstract : The parameter spaces for integral polarized Hodge structure (i.e., the arithmetic quotients of period domains) are complex analytic spaces that are only rarely algebraic. However, thanks to works of Bakker-Klingler-Tsimerman and Bakker-Brunebarbe-Klingler-Tsimerman, it turns out that the constructions of period spaces and period maps in variational Hodge theory take naturally place in the intermediate category of so-called Ran,exp-definable complex analytic spaces, which enjoy both some of the local flexibility of analytic varieties and some of the global rigidity of algebraic varieties.

The goal of this minicourse will be to give an introduction to this new point of view together with some of its applications to the geometry of period maps.

Ana-Maria Castravet (Université de Versailles St-Quentin-en-Yvelines)    Birational geometry of moduli spaces of curves
Abstract : T.B.A.