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.

The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($\nabla u + \nabla u^T)/2$) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.

Azema martingales arise naturally in the study of the chaotic representation property; they also provide classical interpretations of quantum stochastic calculus. The talk will not insist on these aspects, but only define these processes and address the problem of their classification. This raises algebraic questions concerning tensors. Everyone knows that matrices can be diagonalized in some common orthonormal basis if and only if they are symmetric and commute with each other; we shall see an analogous statement for tensors with more
than two indices. This, and other theorems in the same vein, make it possible to associate to any multidimensional Azema martingale an orthogonal decomposition of the state space into one- and two-dimensional subspaces; the behaviour of the process becomes simpler when split into its components in these sub-spaces.

We shall discuss the moduli problem for irreducible holomorphic symplectic manifolds. If these manifolds are equipped with a polarization (an ample line bundle), then they are parametrized by (coarse) moduli spaces. We shall relate these moduli spaces to arithmetic quotients of type IV domains and discuss when they are rational or not. This is joint work with V.Gritsenko and G.K.Sankaran.

 "Scattering amplitudes in gauge theories and gravity have extraordinary properties that are completely invisible in the textbook formulation of quantum field theory using Feynman diagrams. In this usual approach, space-time locality and quantum-mechanical unitarity are made manifest at the cost of introducing huge gauge redundancies in our description of physics. As a consequence, apart from the very simplest processes, Feynman diagram calculations are enormously complicated, while the final results turn out to be amazingly simple, exhibiting hidden infinite-dimensional symmetries. This strongly suggests the existence of a new formulation of quantum field theory where locality and unitarity are derived concepts, while other physical principles are made more manifest. The past few years have seen rapid advances towards uncovering this new picture, especially for the maximally supersymmetric gauge theory in four dimensions.

These developments have interwoven and exposed connections between a remarkable collection of ideas from string theory, twistor theory and integrable systems, as well as a number of new mathematical structures in algebraic geometry. In this talk I will review the current state of this subject and describe a number of ongoing directions of research."

We show how to solve optimization problems in the presence of proportional transaction costs by determining a shadow price, which is a solution to the dual problem. Put differently, this is a fictitious frictionless market evolving within the bid-ask spread, that leads to the same optimization problem as in the original market with transaction costs. In addition, we also discuss how to obtain asymptotic expansions of arbitrary order for small transaction costs. This is joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer.

The timing of developmental milestones such as egg hatch or bud break

can be important predictors of population success and survival. Many

insect species rely directly on temperature as a cue for their

developmental timing. With environments constantly under presure to

change, developmental timing has become highly adaptive in order to

maintain seasonal synchrony. However, climatic change is threatening

this synchrony.

Our model couples existing models of developmental timing to a

quatitative genetics framework which descibes the evolution of

developmental parameters. We use this approach to examine the ability of a

population to adapt to an enviroment that it is highly maladapted to.

Through a combination of numerical and analtyical approaches we explore

the dynamics of the infinite dimensional system of

integrodifference equations. The model indicates that developmental timing

is surprisingly robust in its ability to maitain synchrony even under

climatic change which works constantly to maintain maladaptivity.

The mechanics of thin elastic or viscous objects has applications in e.g. the buckling of engineering structures, the spinning of polymer fibers, or the crumpling of plates and shells. During the past decade the mathematics, mechanics and physics communities have witnessed an upsurge of interest in those issues. A general question is to how patterns are formed in thin structures. In this talk I consider two illustrative problems: the shapes of an elastic knot, and the stitching patterns laid down by a viscous thread falling on a moving belt. These intriguing phenomena can be understood by using a combination of approaches, ranging from numerical to analytical, and based on exact equations or low-dimensional models.

In this talk, we focus on the analytical properties of a decentralized auction, namely the PAUSE Auction Procedure. We prove that the revenue of the auctioneer from PAUSE is greater than or equal to the profit from the well-known VCG auction when there are only two bidders and provide lower bounds on the profit for arbitrary number of bidders. Based on these bounds and observations from auctions with few items, we propose a modification of the procedure that increases the profit. We believe that this study, which is still in progress, will be a milestone in designing better decentralized auctions since it is the first analytical study on such auctions with promising results.

Much mathematical work has gone into the creation of time-consistent nonlinear expectations. When we think of implementing these, various problems arise and destroy the beautiful consistency properties we have worked so hard to create. One of these problems is to do with horizon dependence, in particular, where a portfolio's value is considered at a time t+m, where t is the present time and m is a fixed horizon.

In this talk we shall discuss various notions of time consistency and the corresponding solution concepts. In particular, we shall focus on notions which pay attention to the space of available policies, allowing for commitment devices and non-markovian restrictions. We shall see that, for any time-consistent nonlinear expectation, there is a notion of time consistency which is satisfied by the moving horizon problem.

We'll survey some of the ways that hyperbolic groups have been studied

using analysis on their boundaries at infinity.

We analyze the question of the minimal index of a normal subgroup in a free group which does not contain a given element. Recent work by BouRabee-McReynolds and Rivin give estimates for the index. By using results on the length of shortest identities in finite simple groups we recover and improve polynomial upper and lower bounds for the order of the quotient. The bounds can be improved further if we assume that the element lies in the lower central series.

I describe a conjecture equating the two items appearing in the title.