Self-similarity: two examples of singularity formation

Abstract: I will discuss the role that self-similarity plays in singularity formation. I will illustrate it on two examples involving novel phenomena. The first is the role of self-similar solutions of the compressible Euler equations in the proof of a finite time blow up for the energy super-critical nonlinear Schrodinger equations. The second is a new notion of “twisted self-similarity” in the context of the Einstein equations of general relativity and its role in the proof of existence of vacuum spacetimes with naked singularities.