Louis Yudowitz: Semi-Continuity of the Morse Index for Ricci Shrinkers

When studying the compactness theory for solutions to geometric PDEs, one can encounter singular behavior via some manner of concentration phenomenon. This occurs, for instance, for sequences of harmonic maps, Einstein manifolds, minimal surfaces, and Ricci shrinkers. Provided the singular set consists of isolated point singularities, a very successful strategy to understand the possible degenerations has been to construct a "bubble tree" through an iterative blow-up procedure. Some standard consequences of this are a diffeomorphism finiteness result and an "energy identity", which shows any lost geometric/topological information can be recovered. In this talk I will go over current progress toward proving analogous results for the Morse index of a sequence of Ricci shrinking solitons which exhibit bubbling behavior. I will also discuss a potential application and possible extensions to other settings.