Past

I will consider the physical background, and general thinking behind, the recent programme aimed at applying topos theory to the foundations of physics.

• Thomas Maerz - ‘Some scalar conservation laws on some surfaces - Closest Point Method’
• Chong Luo - ‘Numerical simulation of bistable switching in liquid crystals’
• Radek Erban - ‘Half-way through my time at OCCAM: looking backwards, looking forwards’
• Hugh McNamara - ‘Challenges in locally adaptive timestepping for reservoir simulation’

Zilber constructed an exponential field B, which is conjecturally isomorphic to the complex exponential field. He did so by giving axioms in an infinitary logic, and showing there is exactly one model of those axioms. Following a suggestion of Zilber, I will give a different list of axioms satisfied by B which, under a number-theoretic conjecture known as CIT, describe its complete first-order theory

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)

In this talk we discuss the design of efficient numerical methods for solving symmetric indefinite linear systems arising from mixed approximation of elliptic PDE problems with associated constraints. Examples include linear elasticity (Navier-Lame equations), steady fluid flow (Stokes' equations) and electromagnetism (Maxwell's equations).

The novel feature of our iterative solution approach is the incorporation of error control in the natural "energy" norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a "proof of concept" MATLAB implementation of this algorithm, which we call EST_MINRES, and we illustrate its effectiveness when integrated into our Incompressible Flow Iterative Solution Software (IFISS) package (http://www.manchester.ac.uk/ifiss/).

I will explain how Chen's iterated integrals can be used to construct an $A_\infty$-version of de Rham's theorem (originally due to Gugenheim). I will then explain how to use this result to construct generalized holonomies and integrate homotopy representations in Lie theory.

Modern mathematical approaches to plasticity result in non-convex variational problems for which the standard methods of the calculus of variations are not applicable. In this contribution we consider geometrically nonlinear crystal elasto-plasticity in two dimensions with one active slip system. In order to derive information about macroscopic material behavior the relaxation of the corresponding incremental problems is studied. We focus on the question if realistic systems with an elastic energy leading to large penalization of small elastic strains can be well-approximated by models based on the assumption of rigid elasticity. The interesting finding is that there are qualitatively different answers depending on whether hardening is included or not. In presence of hardening we obtain a positive result, which is mathematically backed up by Γ-convergence, while the material shows very soft macroscopic behavior in case of no hardening. The latter is due to the vanishing relaxation for a large class of applied loads.

This is joint work with Sergio Conti and Georg Dolzmann.

An S-act over a monoid S is a representation of a monoid by tranformations of a set, analogous to the notion of a G-act over a group G being a representation of G by bijections of a set. An S-poset is the corresponding notion for an ordered monoid S.

Words are building blocks of sentences, yet the meaning of a sentence goes well beyond meanings of its words. Formalizing the process of meaning assignment is proven a challenge for computational and mathematical linguistics; with the two most successful approaches each missing on a key aspect: the 'algebraic' one misses on the meanings of words, the vector space one on the grammar.

I will present a theoretical setting where we can have both! This is based on recent advances in ordered structures by Lambek, referred to as pregroups and the categorical/diagrammatic approach used to model vector spaces by Abramsky and Coecke. Surprisingly. both of these structures form a compact category! If time permits, I will also work through a concrete example, where for the first time in the field we are able to compute and compare meanings of sentences compositionally. This is collaborative work with E. Greffenstete, C. Clark, B. Coecke, S. Pulman.

The long-open Tarski problem asked whether first-order logic can distinguish between free groups of different ranks. This was finally answered in the negative by the works of Sela and Kharlampovich-Myasnikov, which sparked renewed interest in the model theoretic properties of free groups. I will give a survey of known results and open questions on this topic.

Following the works of Sela and Kharlampovich-Myasnikov on the Tarski problem, we are interested in the first-order logic of free (and more generally hyperbolic) groups. It turns out that techniques from geometric group theory can be used to answer many questions coming from model theory on these groups. We showed with Sklinos that free groups of finite rank are homogeneous, namely that two tuples of elements which have the same first-order properties are in the same orbit under the action of the automorphism group. We also show that this is not the case for most surface groups.

I shall describe some recent results about the asymptotic behaviour of matroids.

Specifically almost all matroids are simple and have probability at least 1/2 of being connected.

Also, various quantitative results about rank, number of bases and number and size of circuits of almost all matroids are given. There are many open problems and I shall not assume any previous knowledge of matroids. This is joint work, see below.

1 D. Mayhew, M. Newman, D. Welsh and G. Whittle,

On the asymptotic properties of connected matroids, European J. Combin. to appear

2 J. Oxley, C. Semple, L. Wasrshauer and D. Welsh,

On properties of almost all matroids, (2011) submitted

How best to use the cellular Potts model? This is a boundary dynamic method for computational cell-based modelling, in which evolution of the domain is achieved through a process of free energy minimisation. Historically its roots lie in statistical mechanics, yet in modern day it has been implemented in the study of metallic grain growth, foam coarsening and most recently, biological cells. I shall present examples of its successful application to the Steinberg cell sorting experiments of the early 1960s, before examining the specific case of the colorectal crypt. This scenario highlights the somewhat problematic nuances of the CPM, and provides useful insights into the process of selecting a cell-based framework that is suited to the complex biological tissue of interest.