C6

The notion of automatic groups emerged from conversations between Bill Thurston and Jim Cannon on the nice algorithmic properties of Kleinian groups. In this introductory talk we will define automatic groups and then discuss why they are interesting. This centres on how automatic groups subsume many other classes of groups (e.g. hyperbolic groups, finitely generated Coxeter groups, and braid groups) and have good properties (e.g. finite presentability, fast solution to the word problem, and type FP_∞).

In this talk, I will introduce and analyse an elastic

surface energy recently introduced by G. Napoli and

L. Vergori to model thin films of nematic liquid crystals.

As it will be clear, the topology and the geometry of

the surface will play a fundamental role in understanding

the behavior of thin films of liquid crystals.

In particular, our results regards the existence of

minimizers, the existence of the gradient flow

of the energy and, in the case of an axisymmetric

toroidal particle, a detailed characterization of global and local minimizers.

This last item is supplemented with numerical experiments.

This is a joint work with M. Snarski (Brown) and M. Veneroni (Pavia).

Numerical models provide valuable tools for integrating understanding of riverine processes and morphology. Moreover, they have considerable potential for use in investigating river responses to environmental change and catchment management, and for aiding the interpretation of alluvial deposits and landforms. For this potential to be realised fully, such models must be capable of representing diverse river styles, and the spatial and temporal transitions between styles that can be driven by environmental forcing. However, while numerical modelling of rivers has advanced significantly over the past few decades, this has been accomplished largely by developing separate approaches to modelling different styles of river (e.g., meanders and braided networks). In addition, there has been considerable debate about what should constitute the ‘basic ingredients’ of river models, and the degree to which the environmental processes governing river evolution can be simplified in such models. This seminar aims to examine these unresolved issues, with particular reference to the simulation of large rivers and their floodplains.

Many interesting properties of groups are inherited by their subgroups examples of such are finiteness, residual finiteness and being free. People have asked whether hyperbolicity is inherited by subgroups, there are a few counterexamples in this area. I will be detailing the proof of some of these including a construction due to Rips of a finitely generated not finitely presented subgroup of a hyperbolic group and an example of a finitely presented subgroup which is not hyperbolic.

Microbial biofilms grow on most rock and stone surfaces and may play critical roles in weathering. With climate change and improving air quality in many cities in Europe biofilms are growing rapidly on many historic stone buildings and posing practical problems for heritage conservation. With many new field and lab techniques available it is now possible to identify the microbes present and start to clarify their roles. We now need help modelling microbial biofilm growth and impacts in order to provide better advice for conservators.

Hazardous geophysical mass flows, such as snow avalanches, debris-flows and pyroclastic flows, often spontaneously develop large particle rich levees that channelize the flow and enhance their run-out. Measurements of the surface velocity near an advancing flow front have been made at the United States Geological Survey (USGS) debris-flow flume, where 10m^3 of water saturated sand and gravel are allowed to flow down an 80m chute onto a run-out pad. In the run-out phase the flow front is approximately invariant in shape and advances at almost constant speed. By tracking the motion of surface tracers and using a simple kinematic model, it was possible to infer bulk motion as incoming material is sheared towards the front, over-run and shouldered to the side. At the heart of the levee formation process is a subtle segregation-mobility feedback effect. Simple models for particle segregation and the depth-averaged motion of granular avalanches are described and one of the first attempts is made to couple these two types of models together. This process proves to be non-trivial, yielding considerable complexity as well as pathologies that require additional physics to be included.

Jakobshavn Isbrae and many other fast flowing outlet glaciers of present

and past ice sheets lie in deep troughs which often have several

overdeepened sections. To make their fast flow possible their bed needs

to be slippery which in turn means high basal water pressures. I will

present a model of subglacial water flow and its application to

Jakobshavn. I find that, somewhat surprisingly, the reason for

Jakobshavn's fast flow might be the pressure dependence of the melting

point of ice. The model itself describes the unusual fluid dynamics occurring underneath the ice; it has an interesting mathematical structure that presents computational challenges.