Explore the second lecture on the Yamabe problem delivered by Andrea Malchiodi as part of the Geometric Analysis and PDE program at the International Centre for Theoretical Sciences. Delve into this fundamental problem in conformal geometry, which seeks to find metrics of constant scalar curvature on Riemannian manifolds through conformal transformations. Examine the analytical techniques and differential equation methods used to approach this challenging problem, building upon concepts from the first lecture while advancing toward more sophisticated aspects of the theory. Investigate the interplay between geometric properties and elliptic partial differential equations that characterizes this important area of geometric analysis. Learn how the Yamabe problem exemplifies the broader theme of using analytical tools to solve geometric questions, demonstrating the rich connections between differential geometry and nonlinear analysis. This lecture forms part of a comprehensive workshop covering key themes in nonlinear elliptic and parabolic equations, conformal geometry, and the calculus of variations, organized by leading experts in the field and designed to deepen understanding of these fascinating mathematical subjects.