University of Oxford

The heart can be described as an electrically driven mechanical pump. This

pump couldn't adapt to beat-by-beat changes in circulatory demand if there

was no feedback from the mechanical environment to the electrical control

processes. Cardiac mechano-electric feedback has been studied at various

levels of functional integration, from stretch-activated ion channels,

through mechanically induced changes in cardiac cells and tissue, to

clinically relevant observations in man, where mechanical stimulation of the

heart may either disturb or reinstate cardiac rhythmicity. The seminar will

illustrate the patho-physiological relevance of cardiac mechano-electric

feedback, introduce underlying mechanisms, and show the utility of iterating

between experimental research and mathematical modelling in studying this

phenomenon.

The lattice Boltzmann equation has been used successfully used to simulate

nearly incompressible flows using an isothermal equation of state, but

much less work has been done to determine stable implementations for other

equations of state. The commonly used nine velocity lattice Boltzmann

equation supports three non-hydrodynamic or "ghost'' modes in addition to

the macroscopic density, momentum, and stress modes. The equilibrium value

of one non-hydrodynamic mode is not constrained by the continuum equations

at Navier-Stokes order in the Chapman-Enskog expansion. Instead, we show

that it must be chosen to eliminate a high wavenumber instability. For

general barotropic equations of state the resulting stable equilibria do

not coincide with a truncated expansion in Hermite polynomials, and need

not be positive or even sign-definite as one would expect from arguments

based on entropy extremisation. An alternative approach tries to suppress

the instability by enhancing the damping the non-hydrodynamic modes using

a collision operator with multiple relaxation times instead of the common

single relaxation time BGK collision operator. However, the resulting

scheme fails to converge to the correct incompressible limit if the

non-hydrodynamic relaxation times are fixed in lattice units. Instead we

show that they must scale with the Mach number in the same way as the

stress relaxation time.

An integral part of the brain is a fluid flow system that is separate from brain tissue and the cerebral blood flow system: cerebrospinal fluid (CSF) is produced near the centre of the brain, flows out and around the brain, including around the spinal cord and is absorbed primarily in a region between the brain tissue and the skull. Hydrocephalus covers a broad range of anomalous flow and pressure situations: the normal flow path can become blocked, other problems can occur which result in abnormal tissue deformation or pressure changes. This talk will describe work that treats brain tissue as a poroelastic matrix through which the CSF can move when normal flow paths are blocked, producing tissue deformation and pressure changes. We have a number of models, the simplest treating the brain and CSF flow as having spherial symmetry ranging to more complex, fully three-dimensional computations. As well as considering acute hydrocephalus, we touch on normal pressure hydrocephalus, idiopathic intracranial hypertension and simulation of an infusion test. The numerical methods used are a combination of finite difference and finite element techniques applied to an interesting set of hydro-elastic equations.

... for Torelli groups of surfaces.

We will attempt to introduce fusion systems in a way comprehensible to a Geometric Group Theorist. We will show how Bass--Serre thoery allows us to realise fusion systems inside infinite groups. If time allows we will discuss a link between the above and $\mathrm{Out}(F_n)$.

First of all, we are going to recall some basic facts and definitions about homogeneous Riemannian manifolds. Then we are going to talk about existence and non-existence of invariant Einstein metrics on compact homogeneous manifolds. In this context, we have that it is possible to associate to every homogeneous space a graph. Then, the graph theorem of Bohm, Wang and Ziller gives an existence result of invariant Einstein metrics on a compact homogeneous space, based on properties of its graph. We are going to discuss this theorem and sketch its proof.

We begin by showing the underlying ideas Bourgain used to prove that the Cayley graph of the free group of finite rank can be embedded into a Hilbert space with logarithmic distortion. Equipped with these ideas we then tackle the same problem for other metric spaces. Time permitting these will be: amalgamated products and HNN extensions over finite groups, uniformly discrete hyperbolic spaces with bounded geometry and Cayley graphs of cyclic extensions of small cancellation groups.