Moduli spaces birational to locally symmetric varieties of
orthogonal or unitary type -- Radu Laza, April 23, 2010

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A small number of (geometrically important cases of)
moduli spaces can be described as arithmetic quotients of
bounded symmetric domains of type IV or I_{1,n} (complex
ball). In this talk, I will survey some consequences of this
description, with a special focus on the birational geometry of
the moduli space of genus 4 curves.