Explore a cutting-edge seminar on the analysis of partial differential equations, featuring renowned mathematician Alessio Figalli from ETH Zurich. Delve into the classical obstacle problem, examining the equilibrium position of an elastic membrane constrained by a fixed boundary and obstacle. Discover Figalli's recent breakthrough in solving Schaeffer's conjecture for dimensions n≤4, demonstrating that the singular set of the free boundary generically has codimension 3. Gain insights into Caffarelli's classical results on free boundary smoothness and understand the significance of this new finding in the field of PDE analysis. Moderated by Camillo De Lellis from IAS Princeton, this 72-minute talk offers a comprehensive overview of these groundbreaking results in obstacle problems and their implications for generic regularity.