# Transcendence of period maps

# Transcendence of period maps

Period domains D can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient D/\Gamma by an arithmetic group. In the very special case when D/\Gamma is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization D\rightarrow D/\Gamma is a crucial component of the modern approach to the André-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps X\rightarrow D/\Gamma which says that atypical algebraic relations between X and D are governed by Hodge loci. We will also discuss some geometric and arithmetic applications. This is joint work with J. Tsimerman.