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. 

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.

On a symplectic manifold M, the group of Hamiltonian diffeomorphisms Ham(M) is a natural object of study. When M models the phase space of a mechanical system, Ham(M) describes all possible motions. In 1990, Hofer introduced a bi-invariant metric on Ham(M). Intuitively, the Hofer distance between a Hamiltonian diffeomorphism and the identity can be thought of as the minimal energy necessary to generate the diffeomorphism. We will outline the definition and some of the properties of this metric. We will explain how spectral invariants coming from Floer theory can used to study the Hofer metric. Using spectral invariants, we will show that Ham(M) admits quasi-isometric embeddings of ℝⁿ or ℝ^∞ in various settings.

In this talk I want to explain the basics of operad theory. These are homotopical tools which, when acting on objects such as topological spaces, allow us to talk about homotopically-coherent algebraic structures on these. After briefly talking about why one might care about such objects, I want to also give a large range of examples. Finally, if time allows, I want to explain how the language of trees offers itself as an alternative way for studying operads, via the formalism of dendroidal sets.

We discuss how to approximate norms of classes in the singular homology of a product manifold using the singular homology of the factors. In particular, we focus on cross products and how the behaviour of their norms depends on the coefficients. The motivation and examples are related to simplicial volume. This is based on ongoing joint work with C. Löh.

Abstract: Atari founder Nolan Bushnell once said that "every good game is easy to learn but hard to master". As an instance of this principle, it is very easy to define Artin groups, which arise naturally in representation theory and topology, but most basic questions about this class of groups have been open for over a century. We don't even have a unified solution to the isomorphism problem, which asks to determine when two Artin groups are isomorphic!

 In this talk we characterise which Artin groups admit a splitting over Z (that is, an action on a tree with infinite cyclic edge stabilisers), and we use this to shed more light on the isomorphism problem. In the process, we also construct a JSJ splitting for any Artin group, which "refines" every splitting over Z. If time allows, we shall also mention some applications to the study of automorphism groups of Artin groups, and further study directions. 

This talk is based on joint work with Oli Jones and Giovanni Sartori.

Kaplansky's group ring conjectures date back to at least the 1940's, and very little progress had been made towards to the general case for decades. Recently, Gardam offered a counterexample to the unit conjecture. In my talk I will survey the techniques used for proving groups satisfy the unit and zero divisor conjectures via a property known as diffuseness, which can be shown using both geometric and algebraic techniques. Time permitting, I will present some of my work on finding interesting examples of non-diffuse groups.

The study of finitely generated and locally compact groups as geometric objects has been a highly productive area for some time. In this talk I will discuss what group structures (or free transitive actions) can look like on metric spaces that are typically very far from being locally finite or locally compact: real trees. These are continuous analogues of simplicial trees which play an important role in geometric group theory, and the group structures on them can be thought of as analogues of free groups. This can be made precise using work of Berestovskii—Plaut, which shows that any group with the metric structure of a length space is the (metric and group) quotient of a group with the structure of a real tree. I will describe some methods of constructing group structures on real trees and their products, and discuss some of the surprising behaviours that show up.

I will describe how Green's functions of natural differential operators behave as analytic substitutes for distance functions on Riemannian manifolds with positive Ricci curvature, extending results of Colding and Colding-Minicozzi in the non-negatively curved case. 

For a natural comparison operator, I will show that its Green's function satisfies a sharp gradient estimate, with equality characterizing the round sphere. I will also show how this estimate is related to a family of monotonic quantities associated with its (sub)level sets, which may be regarded as elliptic analogues of normalised areas and volumes. These formulae also recover the Colding-Minicozzi theory in the limit as the curvature lower bound tends to zero.

I will conclude with some geometric applications, including a new proof of Bishop's volume comparison theorem in dimension four.