Gregorio Baldi, from the Institute for Advanced Study, presents a research seminar talk on the distribution of Hodge generic points in families of algebraic varieties defined over the algebraic numbers. For a quasi-projective family f from X to S, he shows that points of S where the fiber's variation of Hodge structures is Hodge generic are analytically dense in the complex points of S. Working in the spirit of the Grothendieck period conjecture, and assuming large monodromy, he proves density of points whose periods avoid extra relations up to a given degree, and derives new cases of the Mumford-Tate conjecture outside abelian motives. When S is a curve he gives quantitative bounds. The key technical tool, developed jointly with G. Binyamini and D. Urbanik, is a new result on relations among solutions of G-operators, built on height estimates due to Bombieri and Andre. Recorded at IAS's Simonyi Hall as part of a special year learning seminar, the talk is dense, blackboard-style pure mathematics aimed at specialists in Hodge theory and arithmetic geometry.