Consider a sequence of locally symmetric spaces sharing a common universal cover, a toy model being a sequence of hyperbolic surfaces covered by the hyperbolic plane. I will introduce notions of geometric and spectral convergence of the sequence to the universal cover, namely Benjamini--Schramm and Plancherel convergence, respectively. At first glance, these two notions appear completely unrelated. Upon closer inspection, they may seem to be the same. The aim of the talk is to demystify their relationship.

I will discuss the metric notion of affine maps and their appearance in the study of certain rigidity phenomena. Time permitting, I will present some of my own results within that context. 

Cops and Robbers is a two player game that has recently been generalized to Cayley graphs. The existence of winning strategies gives new group invariants for finitely generated groups. We give an overview of known results and open questions. We show how the game can be used to characterize Gromov-hyperbolicity. For this we will play some games of Cops and Robbers.

The goal of the talk is to define exotic spheres, and study certain aspects of the Kervaire-Milnor classification thereof. Finally, I will present a new result, joint with Krannich and Kupers, on whether the homotopy type of certain configuration spaces of these manifolds remember information about the exotic smooth structures. 

In a manner similar to the Artin representation for the braid group, a string link provides an automorphism of the nilpotent quotient of a free group. In this talk, we will explore the generalisation of the Artin representation for string links at the level of spaces. Perhaps surprisingly, this involves the use of an abstract homotopical tool known as Goodwillie calculus of the identity functor on spaces.

Abstract: The origins of small cancellation theory trace back to Dehn’s algorithm for solving the word problem in surface groups. The theory is a versatile tool for constructing groups with interesting properties: the classical theory provides an important source of Gromov’s hyperbolic groups, and its extension, graphical small cancellation, enables the construction of Gromov’s monster. In this talk, we will explore these developments and discuss various links between small cancellation and nonpositive curvature.

Freed introduced the mathematical notion of a special Kähler structure on a manifold $M$ and how each such structure gives rise to a Hyperkähler-structure on its cotangent bundle $T^*M$. If the special Kähler structure has integral monodromy, one also obtains a Hyperkähler-structure on certain quotients of  $T^*M$. Sun and Zhang showed a converse result, namely that 2 dimensional Gromov-Hausdorff-limits of Hyperkähler 4-folds carry a natural special Kähler structure. In this talk, I will discuss these results and also a conjecture regarding sufficient conditions for the limit structure to have integral monodromy.