Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every symplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.

During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. At last we will talk about generalisations of the curve complex, and highlight some rigidity results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".

This talk will be an introduction to L^2 homology, which is roughly "square-summable" homology. We begin by defining the L^2 homology of a G-CW complex (a CW complex with a cellular G-action), and we will discuss some applications of these invariants to group theory and topology. We will then focus on a criterion of Wise, which proves the vanishing of the 2nd L^2 Betti number in combinatorial CW-complexes with elementary methods. If time permits, we will also introduce Wise's energy criterion.
 

The asymptotic cone of a metric space X is what you see when you "look at X from infinitely far away". The asymptotic cone therefore captures much of the large scale geometry of the metric space. Furthermore, the construction often produces a smooth space from a discrete one, allowing us to apply the techniques of calculus. Notably, Gromov used asymptotic cones in his proof that finitely generated groups of polynomial growth are virtually nilpotent.

In the talk I will define asymptotic cones using the language of ultrafilters and ultralimits. We will then look at the particular cases of asymptotic cones of virtually nilpotent groups and hyperbolic metric spaces. At the end, we will prove a result of Gromov which relates the fundamental group of the asymptotic cone to the filling order of the underlying metric space.

One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored)  generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.