### Presentation

The purpose is to gather experts from two mathematical communities, arithmetic geometry and complex geometry, who study from different perspectives Shimura varieties and questions related to hyperbolicity of moduli spaces.

The geometry of quotients of bounded symmetric domains has been the subject of many works. Classical results (Mumford, Tai, Tsuyumine, etc.) and more recent ones (Bakker-Tsimerman, Brunebarbe, Cadorel, etc.) state that in most cases compactifications of such quotients have many differential forms. In particular, they are of general type. The geometry of entire curves has also been investigated (Nadel, Noguchi, Rousseau, etc.) showing that such quotients are in general hyperbolic modulo the boundary. Some recent works (Brunebarbe, Cadorel, etc.) have also established that all subvarieties of such quotients are of general type. These results can be seen as illustrations in this context of the Green-Griffiths-Lang's conjectures which make a natural bridge with arithmetic problems.

Many conjectures on the arithmetic side indeed involve the study of subvarieties of Shimura varieties. There has been a lot of work around the so-called André-Oort conjecture on special subvarieties of Shimura varieties (Klingler-Yafaev, Ullmo, Pila, Tsimerman, etc.) which was recently proved for moduli spaces of abelian varieties, by J. Tsimerman by relying in particular on the so-called Colmez conjecture in average proved by Andreatta-Goren-Howard-Pera. There is on-going work using a similar line of attack that was successful to prove André-Oort to investigate its generalization, labelled the Zilber-Pink conjecture for brevity, for example by Daw and Ren in the direction of the hyperbolic Ax-Schanuel conjecture.
The interplay between complex geometry and arithmetic geometry relies on direct applications to Shimura varieties of general theorems from complex geometry; on the suggestive analogy between the complex category and the arithmetic setting; but also on more intertwined interactions.

### Mini-course

Benjamin Bakker (Georgia) - Jacob Tsimerman (Toronto)

### Talks

Damian Brotbek (Strasbourg), Yohan Brunebarbe (Zurich), Ashay Burungale (Princeton), Benoit Cadorel (Marseille), Frederic Campana (Nancy), Christopher Daw (Oxford), Gerard Freixas (Jussieu), Carlo Gasbarri (Strasbourg), Eyal Goren (Mc Gill), Behrouz Taji (Notre-Dame), Yunqing Tang (Princeton).

### Organizers

Steven Lu, Marc-Hubert Nicole, Erwan Rousseau

### Sponsors