Glasgow

The concordance group of classical knots C was introduced

over 50 years ago by Fox and Milnor. It is a much-studied and elusive

object which among other things has been a valuable testing ground for

various new topological (and smooth 4-dimensional) invariants. In

this talk I will address the problem of embedding C in a larger group

corresponding to the inclusion of knots in links.

The notion of an E-infinity ring spectrum arose about thirty years ago,

and was studied in depth by Peter May et al, then later reinterpreted

in the framework of EKMM as equivalent to that of a commutative S-algebra.

A great deal of work on the existence of E-infinity structures using

various obstruction theories has led to a considerable enlargement of

the body of known examples. Despite this, there are some gaps in our

knowledge. The question that is a major motivation for this talk is

`Does the Brown-Peterson spectrum BP for a prime p admit an E-infinity

ring structure?'. This has been an important outstanding problem for

almost four decades, despite various attempts to answer it.

I will explain what BP is and give a brief history of the above problem.

Then I will discuss a construction that gives a new E-infinity ring spectrum

which agrees with BP if the latter has an E-infinity structure. However,

I do not know how to prove this without assuming such a structure!

A mathematical model of Olufsen [1,2] has been extended to study periodic pulse propagation in both the systemic arteries and the pulmonary arterial and venous trees. The systemic and pulmonary circulations are treated as separate, bifurcating trees of compliant and tapering vessels. Each model is divided into two coupled parts: the larger and smaller vessels. Blood flow and pressure in the larger arteries and veins are predicted from a nonlinear 1D cross-sectional area-averaged model for a Newtonian fluid in an elastic tube. The initial cardiac output is obtained from magnetic resonance measurements.

The smaller blood vessels are modelled as asymmetric structured trees with specified area and asymmetry ratios between the parent and daughter arteries. For the systemic arteries, the smaller vessels are placed into a number of separate trees representing different vascular beds corresponding to major organs and limbs. Womersley's theory gives the wave equation in the frequency domain for the 1D flow in these smaller vessels, resulting in a linear system. The impedances of the smallest vessels are set to a constant and then back-calculation gives the required outflow boundary condition for the Navier--Stokes equations in the larger vessels. The flow and pressure in the large vessels are then used to calculate the flow and pressure in the small vessels. This gives the first theoretical calculations of the pressure pulse in the small `resistance' arteries which control the haemodynamic pressure drop.

I will discuss the effects, on both the forward-propagating and the reflected components of the pressure pulse waveform, of the number of generations of blood vessels, the compliance of the arterial wall, and of vascular rarefaction (the loss of small systemic arterioles) which is associated with type II diabetes. We discuss the possibilities for developing clinical indicators for the early detection of vascular disease.

References:

1. M.S. Olufsen et al., Ann Biomed Eng. 28, 1281-99 (2000)

2. M.S. Olufsen, Am J Physiol. 276, H257--68 (1999)

There is a close but underexploited analogy between the Euler characteristic

of a topological space and the cardinality of a set. I will give a quite

general definition of the "magnitude" of a mathematical structure, framed

categorically. From this single definition can be derived many

cardinality-like invariants (some old, some new): the Euler characteristic

of a manifold or orbifold, the Euler characteristic of a category, the

magnitude of a metric space, the Euler characteristic of a Koszul algebra,

and others. A conjecture states that this purely categorical definition

also produces the classical invariants of integral geometry: volume, surface

area, perimeter, .... No specialist knowledge will be assumed.

The presence and flow of fluid inside a crack within a solid causes deformation of the solid which in turn influences the flow of the fluid.

This coupled fluid-solid problem will be discussed in the context of dyke propagation and hydrofracture. The background material will be discussed in detail and some applications to specific geometries presented.