Explore a lecture on the affine GAGA theorem and its applications in Hodge theory. Delve into a concise proof of the Peterzil-Starchenko theorem using Stoll's affine GAGA theorem and volume estimates for definable sets in o-minimal structures. Gain insights into o-minimal methods, including the proof of the volume estimate. Discover how the Peterzil-Starchenko theorem has been applied in recent developments in Hodge theory, as presented by Patrick Brossnan from the University of Maryland.