In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.

Introduced by Bestvina and Brady in 1997, Bestvina—Brady groups form an important class of examples in geometric group theory and topology, known for exhibiting unusual finiteness properties. In this talk, I will focus on the Dehn functions of finitely presented Bestvina—Brady groups. Very roughly speaking, the Dehn function of a group measures how difficult it is to fill loops by discs in spaces associated to the group, and captures geometric information that is invariant under coarse equivalence. After reviewing known results, I will present a classification of the Dehn functions of Bestvina—Brady groups. This talk is based on joint work with Yu-Chan Chang and Matteo Migliorini.

As temperatures are increasing, so is the presence of meltwater lakes sitting on the surface of the Greenland Ice Sheet. Such lakes have the possibility of draining through cracks in the ice to the bedrock. Observed discharge rates have found that these lakes can drain at three times the flow rate of Niagara Falls. Current models of subglacial drainage systems are unable to cope with such a large and sudden volume of water. This motivates the idea of a 'subglacial blister' which propagates and slowly dissipates underneath the ice sheet. We present a basic hydrofracture model for understanding this process, before carrying out a number of extensions to observe the effects of turbulence, topography, leak-off and finite ice thickness.

In this talk I will describe how bornological spaces give a foundation for derived geometries. This works over any Banach ring allowing to define analytic and differential geometry over the integers. I will discuss applications of this approach such as the representability of certain moduli spaces and Galois actions on the cohomology of differetiable manifolds admitting a \Q-form.