Explore three significant conjectures in the calculus of variations of surfaces in this conference talk by Tristan Rivière. Delve into the Yau conjecture on infinite minimal surfaces, the Willmore conjecture, and the 16π conjecture on sphere eversions in Euclidean 3-space. Discover how these conjectures have driven the development of innovative minmax techniques for producing optimal immersions. Learn about recent advancements in the field, including a novel proof of the Willmore conjecture in 3 dimensions that bypasses the use of almost minimizing varifold theory and paves the way for higher codimension applications.