**Speaker:** B. Bakker, UGA

**Title of talk:** o-minimal GAGA and a conjecture of Griffiths

**Abstract:** Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients $D/\Gamma$ of period domains by arithmetic groups. Except for a few very special cases, such quotients are never algebraic varieties, and this leads to many difficulties in the general theory. We explain how to partially remedy this situation by equipping $D/\Gamma$ with an o-minimal structure in which any period map is definable. We further prove a general GAGA type theorem in the definable category, and use it to prove a conjecture of Griffiths asserting that the images of period maps are quasiprojective algebraic varieties. This is joint work with Y. Brunebarbe, B. Klingler, and J.Tsimerman.