A central topic in Riemannian geometry is to understand the geometry of the "well-behaved" metrics on Riemannian manifolds. Those metrics in our context either have a priori controlled regularity or satisfy some geometrically motivated PDEs. The fi rst class concerns the metrics with bounded curvatures, which can be intuitively identi ed with the metrics with uniformly bounded Hessians. The second category goes to Einstein metrics. Roughly, the "Laplacian" of an Einstein metric is a constant. We aim at understanding the uniform or quantitative structure of the entire family of metrics in each of the above classes. We will explain in details how to apply such "uniformality" to formulate and solve problems in various contexts of differential geometry.


Basic background in Riemannian geometry is required (e.g. covariant derivative, geodesics, curvature tensors). It will be helpful if attendants understand a little about differential equations such as maximum principles, Sobolev inequalities and elementary elliptic estimates.


Contents:

Tentative contents of topics are, but not limited to, the following:

(1) Overview: quantitative rigidity and quantitative geometry, uniform regularity and moduli space theory

(2) Comparison geometry of Ricci curvature

(3) Introduction to Gromov-Hausdorff theory: non-collapsing and collapsing

(4) Regularity theorems of Einstein metrics

(5) Quantitative splitting and quantitative cone structures

(6) Geometry of collapsed manifolds with bounded curvatures

(7) Some recent developments in collapsed Einstein manifolds: general structure theory and exhibitions of new examples.