Title: Ball quotients and moduli spaces

Abstract: A number of moduli problem are, via Hodge theory, closely related to ball quotients. In this situation there is often a choice of possible compactifications such as the GIT compactification´and its Kirwan blow-up or the Baily-Borel compactification and the toroidal compactificatikon. The relationship between these compactifications is subtle and often geometrically interesting. In this talk I will discuss several cases, mostly concentrating on Deligne-Mostow varieties, but also briefly touching on cubic surfaces and cubic threefolds. This discussion links several areas such as birational geometry, moduli spaces of pointed curves, modular forms and derived geometry. This talk is based on joint work with S. Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.