- 4/10/2018: Curvature and injectivity radius estimates for Einstein -manifolds
- 9/9/2016: Ricci curvature: Some recent progress and open questions

### April 10, 2018

TITLE: Curvature and injectivity radius estimates for Einstein -manifolds

ABSTRACT: We will review some old joint work with Gang Tian ([JAMS 2005]) on the structure of possibly collapsed Einstein -manifolds with a definite bound on the Euler characteristic. Among the main results is an -regularity theorem in which the -norm of the curvature is just assumed to be sufficiently small independent of the collapsing on a ball . It leads to a bound on the curvature on the ball , the topology of which need not be that of a Euclidean ball, though the possiblities for the topology are known; essentially it is a tube around a nilmanifold. Another theorem can be referred to as “collapse implies concentration of curvature”. There is also a theorem whose conclusion can be regarded as a kind of counterpart of the Margulis Lemma, for the case of negative Einstein constant. The hope remains that these results, and the corresponding techniques of proof, could lead to a more general theory of Einstein -manifolds.

### September 9, 2016

TITLE: Ricci curvature: Some recent progress and open questions

ABSTRACT: We will survey some recent (and less recent) progress on Ricci curvature and mention a few questions

which remain open.