Explore a comprehensive lecture on higher Du Bois and higher rational singularities of hypersurfaces presented by Laurentiu Maxim from the University of Wisconsin-Madison. Delve into the homological characterization of these concepts, which were recently introduced as natural generalizations of Du Bois and rational singularities. Discover how these notions are described using the Hodge filtration on the vanishing cycle complex. Learn about the connection between these singularities and characteristic classes introduced in prior work with M. Saito and J. Schuermann. Follow the presentation as it covers classical notions, smooth cases, class transformations, hypersurfaces, vanishing cycle complexes, and new concepts. Gain insights into the proof sketch, refiltration processes, and key results in this 57-minute talk that bridges advanced mathematical concepts in algebraic geometry and singularity theory.