Past

Please note that this is a short notice change from the originally advertised talk by Dr Shahrokh Shahpar (Rolls-Royce plc.)

The new talk "Multilevel Monte Carlo method" is given by Mike Giles, Oxford-Man Institute of Quantitative Finance, Mathematical Institute, University of Oxford.

Joint work with Rob Scheichl, Aretha Teckentrup (Bath) and Andrew Cliffe (Nottingham)

In this talk we will present results concerning the large scale long time behaviour of particles moving in a periodic (random) velocity field subject to molecular diffusion. The particle can be considered massless (passive tracer) or not (inertial particle). Under appropriate assumptions for the velocity field the large scale long time behavior of the particle is described by a Brownian motion with an effective diffusivity matrix K.

We then present some numerical algorithms concerning the calculation of the effective diffusivity in the limit of vanishing molecular diffusion (stochastic geometric integrators). Time permitting we will discuss the case where the driving noise is no longer white but colored and study the effects of this change to the effective diffusivity matrix.

A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators.

I will describe recent work with Charles Fefferman on a

construction of families of analytic almost-sharp fronts for SQG. These

are special solutions of SQG which have a very sharp transition in a

very thin layer. One of the main difficulties of the construction is the

fact that there is no formal limit for the family of equations. I will

show how to overcome this difficulty, linking the result to joint work

with C. Fefferman and Kevin Luli on the existence of a "spine" for

almost-sharp fronts. This is a curve, defined for every time slice by a

measure-theoretic construction, that describes the evolution of the

almost-sharp front.

A topological space $(X,\tau)$ is submaximal if $\tau$ is the maximal element of $[{\tau}_{s}]$. Submaximality was first defined and characterized by Bourbaki. Since then, some mathematicians presented several characterizations of submaximal spaces.

In this paper, we will attempt to develop the concept of submaximality and offer some new results. Furthermore, some results concerning $\alpha$-scattered space will be obtained.

A factorability structure on a group G is a specification of normal forms

of group elements as words over a fixed generating set. There is a chain

complex computing the (co)homology of G. In contrast to the well-known bar

resolution, there are much less generators in each dimension of the chain

complex. Although it is often difficult to understand the differential,

there are examples where the differential is particularly simple, allowing

computations by hand. This leads to the cohomology ring of hv-groups,

which I define at the end of the talk in terms of so called "horizontal"

and "vertical" generators.

Word maps on groups were studied extensively in the past few years, in connection to various conjectures on profinite groups, finite groups, finite simple groups, etc. I will provide background, as well as very recent works (joint with Larsen, Larsen-Tiep,

Liebeck-O'Brien-Tiep) on word maps with relations to representations (e.g. Gowers' method and character ratios), geometry and probability.

Recent applications, e.g. to subgroup growth and representation varieties, will also be described.

I will conclude with a list of problems and conjectures which are still very much open.  The talk should be accessible to a wide audience.

Counting the number of curves of degree $d$ with $n$ nodes (and no further singularities) going through $(d^2+3d)/2 - n$ points in general position in the projective plane is a problem which was already considered more than 150 years ago. More recently, people conjectured that for sufficiently large $d$ this number should be given by a polynomial of degree $2n$ in $d$. More generally, the Göttsche conjecture states that the number of $n$-nodal curves in a general $n$-dimensional linear subsystem of a sufficiently ample line bundle $L$ on a nonsingular projective surface $S$ is given by a universal polynomial of degree $n$ in the 4 topological numbers $L^2, L.K_S, (K_S)^2$ and $c_2(S)$. In a joint work with Vivek Shende and Richard Thomas, we give a short (compared to existing) proof of this conjecture.

In 1961 Hajos conjectured that if a graph contains no subdivsion of a clique of order t then its chromatic number is less than t. In 1981, Erdos and Fajtlowicz showed that the conjecture is almost always false. We show it is almost always true. This is joint work with Keevash, Mohar, and McDiarmid.

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-

point model and we give the conditions for the existence of f-minimal equivalent martingale measure. Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration when the utility is exponential. We illustrate our results considering the Black-Scholes model with change-point.

Key words and phrases: f-divergence, exponential Levy models, change-point, optimal portfolio

MSC 2010 subject classifications: 60G46, 60G48, 60G51, 91B70

Fluids that are not adequately described by a linear constitutive relation are usually referred to as   "non-Newtonian fluids". In the last 15 years we have seen a significant progress in the mathematical theory of generalized Newtonian fluids, which is an important subclass of non-Newtonian fluids. We present some recent results in the existence theory and in the error analysis for approximate solutions. We will also indicate how these techniques can be generalized to more general constitutive relations.

Over the last several decades, many mesh generation methods and a plethora of adaptive methods for solving differential equations have been developed.  In this talk, we take a general approach for describing the mesh generation problem, which can be considered as being in some sense equivalent to determining a coordinate transformation between physical space and a computational space.  Our description provides some new theoretical insights into precisely what is accomplished from mesh equidistribution (which is a standard adaptivity tool used in practice) and mesh alignment.  We show how variational mesh generation algorithms, which have historically been the most common and important ones, can generally be compared using these mesh generation principles.  Lastly, we relate these to a variety of moving mesh methods for solving time-dependent PDEs.

This is joint work with Weizhang Huang, Kansas University

We consider the problem of cavitation in nonlinear elasticity, or the formation of macroscopic cavities in elastic materials from microscopic defects, when subjected to large tension at the boundary.

The main goal is to determine the optimal locations where the body prefers the cavities to open, the preferred number of cavities, their optimal sizes, and their optimal shapes. To this aim it is necessary to analyze the elastic energy of an incompressible deformation creating multiple cavities, in a way that accounts for the interaction between the cavitation singularities. Based on the quantitative version of the isoperimetric inequality, as well as on new explicit constructions of incompressible deformations creating cavities of different shapes and sizes, we provide energy estimates showing that, for certain loading conditions, there are only the following possibilities:

• only one cavity is created, and if the loading is isotropic then it is created at the centre
• multiple cavities are created, they are spherical, and the singularities are well separated
• there are multiple cavities, but they act as a single spherical cavity, they are considerably distorted, and the distance between the cavitation singularities must be of the same order as the size of the initial defects contained in the domain.

In the latter case, the formation of thin structures between the cavities is observed, reminiscent of the initiation of ductile fracture by void coalesence.

This is joint work with Sylvia Serfaty (LJLL, Univ. Paris VI).

Postponed until May

I'll describe an approach to perturbative quantum field theory
which is philosophically similar to the deformation quantization approach
to quantum mechanics. The algebraic objects which appear in our approach --
factorization algebras -- also play an important role in some recent work
in topology (by Francis, Lurie and others).  This is joint work with Owen
Gwilliam.

We review the relation between the geometry of Kleinian singularities and Dynkin diagrams of types ADE, recalling in particular the construction of a braid group action of type A, D, or E on the derived category of coherent sheaves on the minimal resolution of a Kleinian singularity. By work of Seidel-Thomas, this action was known to be faithful in type A. We extend this faithfulness result to types ADE, which provides the missing ingredient for completing Bridgeland's description of spaces of stability conditions for certain triangulated categories associated to Kleinian singularities. Our main tool is the Garside normal form for braid group elements. This project is joint work with Hugh Thomas from the University of New Brunswick.

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.