This lecture explores the geometric approach to classifying 4-manifold topology, focusing on conditions under which contractible 4-manifolds admit uniform positive scalar curvature metrics. Princeton researcher Anubhav Mukherjee presents collaborative work with Otis Chodosh and Davi Maximo that demonstrates how such metrics can provide insights into 4-manifold topology. Learn how Floer theory reveals obstructions to the existence of these metrics in certain 4-manifolds. The presentation draws inspiration from the crucial role geometry plays in classifying 3-dimensional manifold topology and extends these concepts to the more complex 4-dimensional case.