Explore the intersection of o-minimality and Hodge theory in this comprehensive lecture by Jacob Tsimerman from the University of Toronto. Delve into the development of o-minimal geometry with nilpotents, known as "definable analytic spaces," and discover how this theory proves a definable GAGA statement. Learn about the application of this concept in proving Griffiths' conjecture on the algebraic nature of period map images. Examine the o-minimal approach in the context of variations of mixed Hodge structures and its generalization to Griffiths' conjecture. Cover key topics such as Hodge structures, period maps, definable Oka coherence, algebraization theorem for non-reduced period maps, and the crucial square-zero proposition. Gain insights into further results and open questions in this fascinating area of mathematical research.