Profinite groups will be introduced and the reasons why they arise in so many contexts will be discussed. The relationship between open subgroups and subgroups of finite index raises many natural questions. One of these will be addressed in detail, and it will become apparent why some of the others are hard.

In this talk, I will explore the asymptotics of the amount of homological torsion in a group. It is conjectured for instance, that torsion in homology grows exponentially with covolume in some arithmetic groups. I will survey recent developments in the subject and present some important open questions.

The dynamics of gases can be simulated using kinetic or fluid models. Kinetic models are valid over most of the spatial and temporal scales that are of physical relevance in many application problems; however, they are computationally expensive due to the high-dimensionality of phase space. Fluid models have a more limited range of validity, but are generally computationally more tractable than kinetic models. One critical aspect of fluid models is the question of what assumptions to make in order to close the fluid model.

The approach we consider in this work for handling the fluid closure problem is based on the micro-macro partition approach of [Bennoune, Lemou, and Mieussens, J. Comp. Pays., 2008]. In particular, we develop a wave-propagation version of their scheme that handles the microscopic portion of the distribution function via a Hermite spectral method [Grad, 1949]. This formulation has important advantages over the original micro-macro decomposition approach; most notably, no mesh ever needs to be constructed in the velocity variables and no cut-off velocities need to be introduced. Several numerical examples are shown to demonstrate the viability, efficiency, and accuracy of the proposed numerical method.

I will describe joint work with Katrin Tent, in which we consider a profinite group (that is, an inverse limit of finite groups) as a first order structure in a language in which a neighourhood basis of basic open subgroups is uniformly definable. We consider when such a structure can satisfy model-theoretic `tameness' conditions originating in Shelah's work. The main theorem is that if the family is `full’ (i.e. includes all open subgroups) then the group has NIP theory if and only if it has NTP2 theory, if and only if it has an (open) normal subgroup of finite index which is a direct product of finitely many compact p-adic analytic groups (for distinct primes p). The `NIP' condition here has other interpretations -- a structure is NIP (i.e. has NIP theory) if and only if any uniformly definable family of definable sets is a Vapnik-Chervonenkis class. I will give an overview of the model-theoretic notions involved.

The aim of this talk is to give a flavour of Arakelov geometry, presented as an evolution of the classical geometry of numbers of Minkwoski. After explaining Minkowski's main result on volumes of lattices given by fractional ideals in number fields, I will move to higher dimensions. I will focus on the following question: spaces of modular forms with integral Fourier coefficients, together with their Petersson scalar product, define euclidian lattices. What are their volumes? I will give an asymptotic answer to this question, and depending on time, explain some funny exact computations as well. I will try to make the contents as self-contained and accessible as possible!