Mathematical Conversations

In several diverse settings (variational problems and geometric flows, elliptic, parabolic, but also some dispersive PDEs) monotonicity formulas allow to get a first coarse description of singularities, which are commonly called tangent cone. Their existence usually rely on elementary compactness argument which allow to take limits of appropriate rescalings around the singularity. A major challenge is to show that the limit is unique, i.e. independent of the subsequence of rescalings. The problem is widely open even in its prototypical example, i.e. in minimal surface theory. Surprisingly, what is arguably the most beautiful result in the area, a Theorem of Leon Simon, uses a deep fact in real algebraic geometry.