Past

Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will be studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. The connections between SSAs and the deterministic models (based on reaction-diffusion PDEs) will be presented. I will consider chemical reactions both at a surface and in the bulk. I will show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which SSA is used. I will also present multiscale algorithms which use models with a different level of detail in different parts of the computational domain.

We consider the iterative solution of large sparse linear least squares (LS) problems. Specifically, we focus on the design and implementation of reliable stopping criteria for the widely-used algorithm LSQR of Paige and Saunders. First we perform a backward perturbation analysis of the LS problem. We show why certain projections of the residual vector are good measures of convergence, and we propose stopping criteria that use these quantities. These projections are too expensive to compute to be used directly in practice. We show how to estimate them efficiently at every iteration of the algorithm LSQR. Our proposed stopping criteria can therefore be used in practice.

This talk is based on joint work with Xiao-Wen Chang, Chris Paige, Pavel Jiranek, and Serge Gratton.

I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point.

However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof -- using an entirely different method -- which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups.

Many problems in computer science can be modelled as metric spaces, whereas for mathematicians they are more likely to appear as the opening question of a second year examination. However, recent interesting results on the geometry of finite metric spaces have led to a rethink of this position. I will describe some of the work done and some (hopefully) interesting and difficult open questions in the area.

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

Abstract: Flagella and cilia are ubiquitous in biology as a means of motility and critical for male gametes migration in reproduction, to mucociliary clearance in the lung, to the virulence of devastating parasitic pathogens such as the Trypanosomatids, to the filter feeding of the choanoflagellates, which are constitute a critical link in the global food chain. Despite this ubiquity and importance, the details of how the ciliary or flagellar waveform emerges from the underlying mechanics and how the cell, or the environs, may control the beating pattern by regulating the axoneme is far from fully understood. We demonstrate in this talk that mechanics and modelling can be utilised to interpret observations of axonemal dynamics, swimming trajectories and beat patterns for flagellated motility impacts on the science underlying numerous areas of reproductive health, disease and marine ecology. It also highlights that this is a fertile and challenging area of inter-disciplinary research for applied mathematicians and demonstrates the importance of future observational and theoretical studies in understanding the underlying mechanics of these motile cell appendages.

The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.

The signature of the path is an essential object in rough path theory which takes value in tensor algebra and it is anticipated that the expected signature of Brownian motion might characterize the rough path measure of Brownian path itself. In this presentation we study the expected signature of a Brownian path in a Bananch space E stopped at the first exit time of an arbitrary regular domain, although we will focus on the case E=R^{2}. We prove that such expected signature of Brownian motion should satisfy one particular PDE and using the PDE for the expected signature and the boundary condition we can derive each term of expected signature recursively. We expect our method to be generalized to higher dimensional case in R^{d}, where d is an integer and d >= 2.

All this is related to my ongoing research projects with Chris Douglas and Arthur Bartels, in which we investigate conformal nets from a category
theoretical
perspective.

Abstract: We construct solutions to Burgers type equations perturbed by a multiplicative

space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation. We show that our solutions are stable under smooth approximations of the driving noise. A more general class of approximations will also be discussed. This is joint work with Martin Hairer and Jan Maas.

We present theory and numerics of affine processes and several of their applications in finance. The theory is appealing due to methods from probability theory, analysis and geometry. Applications are diverse since affine processes combine analytical tractability with a high flexibility to model stylized facts like heavy tails or stochastic volatility.

We are interested in measure theory and integration theory that ¯ts into the
o-minimal context. Therefore we introduce the following de¯nition:
Given an o-minimal structure M on the ¯eld of reals and a measure ¹ de¯ned on the
Borel sets of some Rn, we call ¹ M-tame if there is an o-minimal expansion of M such
that for every parameter family of functions on Rn that is de¯nable in M the family of
integrals with respect to ¹ is de¯nable in this o-minimal expansion.
In the ¯rst part of the talk we give the de¯nitions and motivate them by existing and
many new examples. In the second one we discuss the Lebesgue measure in this context.
In the ¯nal part we obtain de¯nable versions of important theorems like the theorem of
Radon-Nikodym and the Riesz representation theorem. These results allow us to describe
tame measures explicitly.
1

This talk will focus on a family of Bayesian inference algorithms built around Gaussian processes. We firstly introduce an iterative Gaussian process for multi-sensor inference problems. Extensions to our algorithm allow us to tackle some of the decision problems faced in sensor networks, including observation scheduling. Along these lines, we also propose a general method of global optimisation, Gaussian process global optimisation (GPGO). This paradigm is extended to the Bayesian decision problem of sequential multi-scale observation selection. We show how the hyperparameters of our system can be marginalised by use of Bayesian quadrature and frame the selection of the positions of the hyperparameter samples required by Bayesian quadrature as a sequential decision problem, with the aim of minimising the uncertainty we possess about the values of the integrals we are approximating.

In various fields of application, one must solve very large scale boundary value problems using parallel solvers and supercomputers. The domain decomposition approach partitions the large computational domain into smaller computational subdomains. In order to speed up the convergence, we have several ``optimized'' algorithm that use Robin transmission conditions across the artificial interfaces (FETI-2LM). It is known that this approach alone is not sufficient: as the number of subdomains increases, the number of iterations required for convergence also increases and hence the parallel speedup is lost. A known solution for classical Schwarz methods as well as FETI algorithms is to incorporate a ``coarse grid correction'', which is able to transmit low-frequency information more quickly across the whole domain. Such algorithms are known to ``scale weakly'' to large supercomputers. A coarse grid correction is also necessary for FETI-2LM methods. In this talk, we will introduce and analyze coarse grid correction algorithms for FETI-2LM domain decomposition methods.

This talk will be an introduction to the space of Bridgeland stability conditions on a triangulated category, focussing on the case of the derived category of coherent sheaves on a curve. These spaces of stability conditions have their roots in physics, and have a mirror theoretic interpretation as moduli of complex structures on the mirror variety.

For curves of genus g > 0, we will see that any stability condition comes from the classical notion of slope stability for torsion-free sheaves. On the projective line we can study the more complicated behaviour via a derived equivalence to the derived category of modules over the Kronecker quiver.

Considering kinetic equations (Boltzmann, BGK, say...) in the small mean free path regime lead to conservation laws (the Euler system, typically) When the problem is set in a domain, boundary layers might occur due to the fact that incoming fluxes could be far from equilibrium states. We consider the problem from a numerical perspective and we propose a definition of numerical fluxes for the Euler system which is intended to account for the formation of these boundary layers.

A brief survey of the above.

As many of you are no doubt aware, a Study Group of particular significance will take place in Thuwal, Saudi Arabia, from 23rd January - 26th January 2011.

The problem statements, in their preliminary form can be found at:

http://people.maths.ox.ac.uk/bentahar/KSG/

Communication between observers in a relativistic scenario has proved to be

a setting for a fruitful dialogue between quantum field theory and quantum

information theory. A state that an inertial observer in Minkowski space

perceives to be the vacuum will appear to an accelerating observer to be a

thermal bath of radiation. We study the impact of this Davies-Fulling-Unruh

noise on communication, particularly quantum communication from an inertial

sender to an accelerating observer and private communication between two

inertial observers in the presence of an accelerating eavesdropper. In both

cases, we establish compact, tractable formulas for the associated

communication capacities assuming encodings that allow a single excitation

in one of a fixed number of modes per use of the communications channel.

We consider the Relativistic Vlasov-Maxwell system of equations which

describes the evolution of a collisionless plasma. We show that under

rather general conditions, one can test for linear instability by

checking the spectral properties of Schrodinger-type operators that

act only on the spatial variable, not the full phase space. This

extends previous results that show linear and nonlinear stability and

instability in more restrictive settings.

An old conjecture of Hardy and Littlewood posits that on average, the number of representations of a positive integer N as a sum of k, k-th powers is "very small." Recently, it has been observed that this conjecture is closely related to properties of a discrete fractional integral operator in harmonic analysis. This talk will give a basic introduction to the two key problems, describe the  correspondence between them, and show how number theoretic methods, in particular the circle method and mean values of Weyl sums, can be used to say something new in abstract harmonic analysis.

We analyse stability properties of stochastic Lagrangian Navier stokes flows on compact Riemannian manifolds.

What is a random group? What does it look like? In Gromov's few relator
and density models (with density < 1/2) a random group is a hyperbolic
group whose boundary at infinity is homeomorphic to a Menger curve.
Pansu's conformal dimension is an invariant of the boundary of a
hyperbolic group which can capture more information than just the
topology. I will discuss some new bounds on the conformal dimension of the
boundary of a small cancellation group, and apply them in the context of
random few relator groups, and random groups at densities less than 1/24.

Abstract: In this talk, we first introduce the notion of ergodic BSDE which arises naturally in the study of ergodic control. The ergodic BSDE is a class of infinite-horizon BSDEs:
Y_{t}^{x}=Y_{T}^{x}+∫_{t}^{T}[ψ(X^{x}_{σ},Z^{x}_{σ})-λ]dσ-∫_{t}^{T}Z_{σ}^{x}dB_{σ}, P-<K1.1/>, ∀0≤t≤T<∞,
<K1.1 ilk="TEXTOBJECT" > <screen-nom>hbox</screen-nom> <LaTeX>\hbox{a.s.}</LaTeX></K1.1> where X^{x} is a diffusion process. We underline that the unknowns in the above equation is the triple (Y,Z,λ), where Y,Z are adapted processes and λ is a real number. We review the existence and uniqueness result for ergodic BSDE under strict dissipative assumptions.
Then we study ergodic BSDEs under weak dissipative assumptions. On the one hand, we show the existence of solution to the ergodic BSDE by use of coupling estimates for perturbed forward stochastic differential equations. On the other hand, we show the uniqueness of solution to the associated Hamilton-Jacobi-Bellman equation by use of the recurrence for perturbed forward stochastic differential equations.
Finally, applications are given to the optimal ergodic control of stochastic differential equations to illustrate our results. We give also the connections with ergodic PDEs.

• Chris Farmer - Numerical simulation of anisotropic diffusion
• Jean-Charles Seguis - Introduction to the Fictitious Domain Method for Finite Elements Method
• Amy Smith - Multiscale Models of Cardiac Contraction and Perfusion
• Mark Curtis - Developing a novel Slender Body Theory incorporating regularised singularities

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes.

In this talk, I will present a new theory to study the phenomenon of cavitation in soft solids that, contrary to existing approaches,

simultaneously: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii)  applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and

(iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as the homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. Then, by means of a novel iterated homogenization procedure, exact solutions are constructed for such a problem. These include solutions for the change in size of the underlying cavities as a function of the applied loading conditions, from which the onset of cavitation - corresponding to the event when the initially infinitesimal cavities suddenly grow into finite sizes - can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the cavities plays the role of "time" and the applied load plays the role of "space".

An application of the theory to the case of Ne-Hookean solids containing a random isotropic distribution of vacuous defects will be presented.

We will describe some joint work with V. G. Maz’ya and I. E. Verbitsky, concerning homogeneous quasilinear differential operators. The model operator under consideration is:

\[ L(u) = - \Delta_p u - \sigma |u|^{p-2} u. \]

Here $\Delta_p$ is the p-Laplacian operator and $\sigma$ is a signed measure, or more generally a distribution. We will discuss an approach to studying the operator L under only necessary conditions on $\sigma$, along with applications to the characterisation of certain Sobolev inequalities with indefinite weight. Many of the results discussed are new in the classical case p = 2, when the operator L reduces to the time independent Schrödinger operator.

I will discuss a few problems  that involve randomness , chosen randomly  (?) from the following : (i) the probability of a coin landing on a side  (ii) optimal strategies for throwing accurately, (iii)  the statistical mechanics of a ribbon, (iv) the intermittent dynamics of a growing polymeric assembly (v) fat tails from feedback.



• Nick Hale - 'Rectangular pseudospectral differentiation matrices' or, 'Why it's not hip to be square'

Boundary conditions in pseudospectral collocation methods are imposed by removing rows of the discretised differential operator and replacing them with others to enforce the required conditions at the boundary. A new approach, based upon projecting the discrete operator onto a lower-degree subspace to create a rectangular matrix and applying the boundary condition rows to ‘square it up’, is described.
We show how this new projection-based method maintains characteristics and advantages of both traditional collocation and tau methods.

• Cameron Hall - 'Discrete-to-continuum asymptotics of functions defined as sums'

When attempting to homogenise a large number of dislocations, it becomes important to express the stress in a body due to the combined effects of many dislocations. Assuming linear elasticity, this can be written as a simple sum over all the dislocations. In this talk, a method for obtaining an asymptotic approximation to this sum by simple manipulations will be presented. This method can be generalised to approximating one-dimensional functions defined as sums, and work is ongoing to achieve the same results in higher dimensions.

• Vladimir Zubkov - 'On the tear film modeling'

A great number of works about the tear film behaviour was published. The majority of these works based on modelling with the use of the lubrication approximation. We explore the relevance of the lubrication tear film model compare to the 2D Navier-Stokes model. Our results show that the lubrication model qualitatively describe the tear film evolution everywhere except region close to an eyelid margin. We also present the tear film behaviour using Navier-Stokes model that demonstrates that here is no mixing near the MCJ when the eyelids move relative to the eyeball.

• Kostas Zygalakis - 'Numerical methods for stiff stochastic differential equations'

Multiscale differential equations arise in the modelling of many important problems in the science and engineering. Numerical methods for such problems have been extensively studied in the deterministic case. In this talk, we will discuss numerical methods for (mean-square stable) stiff stochastic differential equations. In particular we will discuss a generalization of explicit stabilized methods, known as Chebyshev methods to stochastic problems.

Free-discontinuity problems describe situations where the solution of

interest is defined by a function and a lower dimensional set consisting

of the discontinuities of the function. Hence, the derivative of the

solution is assumed to be a "small function" almost everywhere except on

sets where it concentrates as a singular measure.

This is the case, for instance, in certain digital image segmentation

problems and brittle fracture models.

In the first part of this talk we show new preliminary results on

the existence of minimizers for inverse free-discontinuity problems, by

restricting the solutions to a class of functions with piecewise Lipschitz

discontinuity set.

If we discretize such situations for numerical purposes, the inverse

free-discontinuity problem in the discrete setting can be re-formulated as

that of finding a derivative vector with small components at all but a few

entries that exceed a certain threshold. This problem is similar to those

encountered in the field of "sparse recovery", where vectors

with a small number of dominating components in absolute value are

recovered from a few given linear measurements via the minimization of

related energy functionals.

As a second result, we show that the computation of global minimizers in

the discrete setting is an NP-hard problem.

With the aim of formulating efficient computational approaches in such

a complicated situation, we address iterative thresholding algorithms that

intertwine gradient-type iterations with thresholding steps which were

designed to recover sparse solutions.

It is natural to wonder how such algorithms can be used towards solving

discrete free-discontinuity problems. This talk explores also this

connection, and, by establishing an iterative thresholding algorithm for

discrete inverse free-discontinuity problems, provides new insights on

properties of minimizing solutions thereof.