All talks, except the last talk on Monday, will take place in the Petit Amphi of the Département de Mathématiques d'Orsay (access).
The Poster Session will take place in the Salle du thé.

### Friday

09:00 - 09:30 Welcome and registration
09:30 - 10:50 Klingler Brotbek Várilly-Alvarado Favre Favre
11:10 - 12:30 Várilly-Alvarado Klingler Brunebarbe Klingler Brotbek
Lunch break Lunch break Lunch Lunch break Lunch
14:00 - 15:20 Brotbek SAGA** Brunebarbe
15:40 - 17:00 Brunebarbe* Várilly-Alvarado Favre
17:00 - 19:00   Poster Session***
*The last talk on Monday will take place in Salle 113-115.
**The Tuesday 30 session of the regular seminar Séminaire Arithmétique et Géométrie Algébrique (SAGA), by Bruno Klingler, will take place in Salle 117-119.
***The Poster Session will take place in the Salle du thé.

## Lecture series

• Damian BROTBEK: Jet differentials and hyperbolicity of general hypersurfaces. A smooth projective complex variety X is said to be hyperbolic if it does not contain any entire curve (a non constant holomorphic map f : ℂ → X). The hyperbolicity properties of X are closely related (at least conjecturally) to the positivity properties of the canonical bundle of X. For instance, the Green-Griffiths-Lang conjecture predicts that if X is of general type, then there exists a proper algebraic subset Z ⊂ X containing all entire curves of X. In a related direction, in 1970, Kobayashi conjectured that a general hypersurface H ⊂ ℙn of sufficiently large degree is hyperbolic. This conjecture has attracted a lot of attention over the last decades and has only been proved recently by Siu.
The goal of this mini course is to give an introduction to the notion of hyperbolicity and to present another proof of this conjecture of Kobayashi. Along the way we will present the theory of jet differentials, developed among others by Green and Griffiths, Siu, and Demailly, which provides a fruitful way to study the behavior of entire curves. The two other main ingredients of this proof are a suitable Wronskain construction, and the reduction of the problem to the study of a universal family of zero dimensional complete intersections.
• Yohan BRUNEBARBE: Hyperbolicity of moduli spaces of abelian varieties. It is well-known that the level-N modular curve X(N) has genus at least 2 exactly when N>6. In particular, it follows that there is no non-trivial family of elliptic curves over the complex affine line. In my talks, I will explain how one can use Hodge theory to prove some generalizations of these hyperbolicity statements for moduli spaces of abelian varieties of dimension g>1.
• Charles FAVRE: Degeneration of endomorphisms of projective spaces. We shall describe how one can control the dynamics of a meromorphic family of endomorphisms of the projective space parameterized by the punctured unit disk as one approaches the puncture. Such a control is crucial for instance in the description of special curves in the parameter space of one-dimensional rational maps that contain infinitely many post-critically finite maps.
• Bruno KLINGLER: Bi-algebraic geometry. When X is a complex g-dimensional abelian variety, its universal cover $\stackrel{~}{X}=$ g is an algebraic variety and many geometric and arithmetic properties of X can be understood by studying the transcendance properties of the uniformization map p: ℂg → X.
In these lectures we will try to show how a general version of this picture leads to interesting new results and conjectures related to Hodge theory.
• Anthony VÁRILLY-ALVARADO: Level structures on K3 surfaces and abelian varieties. We will describe boundedness statements for certain level structures over number fields on abelian varieties and K3 surfaces that follow from a combination of recent hyperbolicity results and quantitative arithmetic conjectures. We will also describe unconditional results in the direction of these boundedness statements.