The aim of the talk is to provide a survey of the main tools, results and open problems concerning manifolds with a lower (sectional) curvature bound. It is well known that local bounds on sectional curvature can be described geometrically via local distance comparison to constant curvature spaces. For lower curvature bounds this comparison is global, as expressed in the Toponogov Comparison Theorem. This together with critical point theory for distance functions paved the way for studying manifolds with only a lower sectional curvature bound, resulting in Finiteness, Structure, and Recognition Theorems. There are several equivalent versions of Toponogov’s Comparison Theorem, some of which make sense in a general metric space. Moreover, such a metrically expressed lower curvature bound is preserved by the process of taking a Gromov-Hausdor limit. An Alexandrov space is a finite Hausdor dimensional, inner metric space with a lower curvature bound.  It turns out that, despite their general definition, Alexandrov spaces have a surprisingly rich structure and are natural objects in their own rite. Their applications and significance to Riemanian geometry stems from the fact that there are several natural geometric operations that are closed in Alexandrov geometry, but not in Riemanian geometry. These include taking Gromov-Hausdor limits, taking quotients and taking joins of positively curved spaces.  All concepts alluded to above will be explained and discussed, as will examples, some of the main results, and fundamental open problems.

Ricci curvature occurs in the Einstein equation, Ricci flow, optimal transport, and is important both in mathematics and physics. Comparison method is one of the key tools in studying the Ricci curvature. We will start with Bochner formula and derive Laplacian comparison, Bishop-Gromov volume comparison, first eigenvalue and heat kernel comparison and some application. Then we will discuss some of its generalizations to Bakry-Emery  Ricci curvature and integral Ricci curvature

To this day the only compact irreducible complete Ricci-flat Riemannian metrics arise as manifolds with special or exceptional holonomy. This pair of talks will give an introduction to special and exceptional holonomy metrics and the resulting constructions of both compact and noncompact Ricci-flat manifolds. We will indicate how ideas from Riemannian convergence theory have provided motivation for the currently known (and possibly for future) constructions of metrics with exceptional holonomy

These lectures will discuss classical and more recent research on selected topics in the area of Einstein metrics on 4-dimensional manifolds. The basic questions concern existence and uniqueness of such metrics. Broadly speaking, the first lecture will discuss uniqueness issues; more precisely the apriori structure of the (moduli) space of Einstein metrics on a given manifold. The second lecture will turn to the more difficult existence questions (on manifolds with boundary)

Ricci curvature occurs in the Einstein equation, Ricci flow, optimal transport, and is important both in mathematics and physics. Comparison method is one of the key tools in studying the Ricci curvature. We will start with Bochner formula and derive Laplacian comparison, Bishop-Gromov volume comparison, first eigenvalue and heat kernel comparison and some application. Then  we will discuss some of its generalizations to Bakry-Emery  Ricci curvature and integral Ricci curvature

In these lectures we will discuss the role that minimal submanifolds play in the study of lower curvature bounds. The first lecture will be an introduction to the theory including existence theory and the geometry of the second variation. The second lecture will be a survey of applications especially to lower scalar and isotropic curvature bounds

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can  give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, Levy-Gromov isoperimetric inequality, etc.), permitting to give further insights in the theory. The goal of the lectures is to give an introduction to the subject:
-In the first lecture I will introduce Ricci curvature lower bounds for metric measure spaces and discuss basic properties.
- The second lecture will be devoted to the study of  analytic and geometric properties of such  spaces, in particular I will present the proof of the Levy-Gromov isoperimetric inequality  (joint work with Fabio Cavalletti).