Lecture 1
Marc Kegel: Handling the Poincaré Conjecture

Poincaré conjectured in 1904 that a manifold that has the same algebraic topology as a sphere is already a sphere. Since then the various versions of this conjecture have turned out to be the leading questions in geometric and differential topology. In the last century, most of the progress in geometric topology can in some way be traced back to an attempt to prove or disprove one version of the Poincaré conjecture. Up to today, a total of seven fields medals were awarded for contributions that are connected to the Poincaré conjecture. While some of these results are obviously extremely difficult, some others are surprisingly simple (once you get the main idea). In this series of six lectures, I will survey about the parts of these results that I understand and also discuss some other results that were not awarded fields medals but are still very interesting.

Further information can be found here.

Lecture 2
Torben Kastenholz: Bounded cohomology and the Milnor-Wood Inequality

Bounded cohomology is a useful extension of classical cohomology that allows one to enrich usual cohomology with a semi-norm. As it turns out induced maps can at most decrease the norm, which allows one to restrict possible morphisms between (group-)cohomology once one knows the semi-norm on their cohomology. During this course the fundamentals of bounded cohomology will be stated and sometimes proven. 

One of the most famous applications of bounded cohomology is the modern proof of the Milnor-Wood Inequalities. These specify completely what kind of possible Euler classes can occur for flat circle bundles over surfaces. The course will explain how to prove these inequalities and discuss further generalizations.