Given a number field K, it is natural to ask whether it has a finite extension with ideal class number one. This question can be translated into a fundamental question in class field theory, namely the class field tower problem. In this talk, we are going to discuss this problem as well as its solution due to Golod and Shafarevich using methods of group cohomology.
 

Modularity is a function on graphs which is used in algorithms for community detection. For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The (max) modularity $q^\ast(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0 \leq q^\ast(G) \leq 1$.

We analyse when community structure of an underlying graph can be determined from an observed subset of the graph. In a natural model where we suppose edges in an underlying graph $G$ appear with some probability in our observed graph $G'$ we describe how high a sampling probability we need to infer the community structure of the underlying graph.

Joint work with Colin McDiarmid.

Gromov hyperbolicity is a property to metric spaces that generalises the notion of negative curvature for manifolds.
After an introduction about these spaces, we will explain the construction of horocyclic products related to lamplighter groups, Baumslag solitar groups and the Sol geometry.
We will describe the shape of geodesics in them, and present rigidity results on their quasi-isometries due to Farb, Mosher, Eskin, Fisher and Whyte.

Right-angled Artin groups (RAAGs) were first introduced in the 70s by Baudisch and further developed in the 80s by Droms.
They have attracted much attention in Geometric Group Theory. One of the many reasons is that it has been shown that all hyperbolic 3-manifold groups are virtually finitely presented subgroups of RAAGs.
In the first part of the talk, I will discuss some of their interesting properties. I will explain some of their relations with manifold groups and their importance in finiteness conditions for groups.
In the second part, I will focus on my PhD project concerning subgroups of direct products of RAAGs.

The automorphism group of a d-regular tree is a topological group with many interesting features. A nice thing about this group is that while some of its features are highly non-trivial (e.g., the existence of infinitely many pairwise non-conjugate simple subgroups), often the ideas involved in the proofs are fairly intuitive and geometric. 
I will present a proof for the fact that the outer automorphism group of (Aut(T)) is trivial. This is original joint work with Gil Goffer, but as is often the case in this area, was already proven by Bass-Lubotzky 20 years ago. I will mainly use this talk to hint at how algebra, topology and geometry all play a role when working with Aut(T).
 

Whitehead published two papers in 1936 on free groups. Both concerned decision problems for equivalence of (sets of) elements under automorphisms. The first focused on primitive elements (those that appear in some basis), the second looked at arbitrary sets of elements. While both of the resulting algorithms are combinatorial, Whitehead's proofs that these algorithms actually work involve some nice manipulation of surfaces in 3-manifolds. We will have a look at how this works for primitive elements. I'll outline some generalizations due to Culler-Vogtmann, Gertsen, and Stallings, and if we have time talk about how it fits in with some of my current work.

If asked, we all say we aim to to good research and make sensible decisions. In mathematics, the choice of criteria to optimise is often explicit, and we know there is no complete ordering in more than one dimension.

Statisticians involved in multi-disciplinary research need to reflect on how their understanding of uncertainty and statistical methods can contribute to reliable and reproducible research. The ISI Declaration of Professional Ethics provides a framework for statisticians.  Judging what is "normal" and what is "best" requires an appreciation of the assumptions and guidelines of other disciplines.

I will briefly discuss the requirements for design and analysis in medical research, and relate this to debates on reproducible research and p-values in social science research. Issues arising from informed and uninformed consent will be outlined.

Examples might include medical research in developing countries, toxic tort or wrongful birth claims, big data and use of routine administrative or commercial data.

Sustainability is a highly complex topic, containing interwoven economic, ecological, and social aspects.  Simply defining the concept of sustainability is a challenge in itself.  Assessing the impact of sustainability efforts and generating effective policy requires analyzing the interactions and challenges presented by these different aspects. To address this challenge, it is necessary to develop methods that bridge fields and take into account phenomena that range from physical analysis of climate to network analysis of societal phenomena. In this talk, I will give an insight into areas of mathematical research that try to account for these inter-dependencies. The aim of this talk is to provide a critical discussion of the challenges in a joint discussion and emphasize the importance of multi-disciplinary approaches.