In the first lecture, I shall give a gentle introduction to manifolds. In particular, starting from surfaces in 3-space and an alternative way to describe them "intrinsically" (in a way that a 2-dimensional creature could understand), I shall explain how to visualise manifolds of dimension three (or even four).
When one tries to prove the existence of certain geometric structures on manifolds, or study their properties, one usually requires some structure theorem of the kind: all manifolds (of a certain dimension, or some interesting subclass) can be obtained by such and such a construction, perhaps by modifying a simple manifold we understand—and then hoping that the modification can be made compatible with the geometric structure in question.
I want to present some structure theorems for 3-dimensional manifolds—where possible in analogy with surfaces—and illustrate their usefulness by applications to the construction and classification of contact structures.
Mapping class groups
Anja Randecker
In this mini course, we will study mapping class groups, that is, the groups of homeomorphisms of a surface, up to isotopy. We will see how to generate them, how to use the curve complex as a tool, and we will also have a glimpse at the hot topic of mapping class groups of surfaces of infinite type. In particular, we will consider these "big" mapping class groups as nice new examples of Polish groups.