Explore the intricate connections between eigenvarieties and concepts in locally analytic p-adic cohomology representation theory in this advanced mathematics lecture. Delve into p-adic families of finite slope automorphic eigenforms as Matthew Emerton from the University of Chicago presents his insights on the subject. Gain a deeper understanding of these complex mathematical structures and their relationships over the course of this hour-long talk delivered at the Institut des Hautes Etudes Scientifiques (IHES).