Explore the intricacies of Lusternik-Schnirelmann theory as applied to 4-manifolds in this hour-long lecture. Delve into various interpretations of the Lusternik-Schnirelman category, focusing on covers and fillings of 4-manifolds by different sets. Gain insights into Gay-Kirby trisections, a specific type of decomposition of 4-manifolds into 1-handlebodies. Enhance your understanding of advanced topics in algebraic topology and manifold theory through this in-depth presentation.