Past

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.

The idea of this one day meeting is to give participants the opportunity to air the 'problem you never solved'. This might be either a problem you have never had time to work on or one that has defeated you. There will be plenty of time for discussion and maybe a few problems will be solved during the day! (Alternatively, the meeting may provide John with a source of problems to work on during his retirement.)

The programme starts with coffee at 10.00 and finishes with a reception and dinner in St Anne's College. Further details at

http://www.maths.ox.ac.uk/groups/occam/forthcoming-events/open-mathemat…

$\AA^1$-homotopy theory is the homotopy theory for smooth algebraic varieties which uses the affine line as a replacement for the unit interval. The stable $\AA^1$-homotopy category is a generalisation of the topological stable homotopy category, and in particular, gives a setting where algebraic cohomology theories such as motivic cohomology, and homotopy invariant algebraic $K$-theory can be represented. We give a brief overview of some aspects of the construction and some properties of both the topological stable homotopy category and the new $\AA^1$-stable homotopy category.

Can one of the most important Italian Renaissance frescoes reduced in hundreds of thousand fragments by a bombing during the Second World War be re-composed after more than 60 years from its damage? Can we reconstruct the missing parts and can we say something about their original color?

In this talk we would like to exemplify, hopefully effectively by taking advantage of the seduction of art, how mathematics today can be applied in real-life problems which were considered unsolvable only few years ago.

We investigate ground state configurations of atomic pair potential systems in two dimensions as the number of particles tends to infinity. Assuming crystallization (which has been proved for some cases such as the Radin potential, and is believed to hold more generally), we show that after suitable rescaling, the ground states converge to a unique macroscopic Wulff shape. Moreover, we derive a scaling law for the size of microscopic non-uniqueness which indicates larger fluctuations about the Wulff shape than intuitively expected.

Joint work with Yuen Au-Yeung and Bernd Schmidt (TU Munich),

to appear in Calc. Var. PDE

We will state a theorem of Shouhei Ma (2008) relating the Cusps of the Kaehler moduli space to the set of Fourier--Mukai partners of a K3 surface. Then we explain the relationship to the Bridgeland stability manifold and comment on our work relating stability conditions "near" to a cusp to the associated Fourier--Mukai partner.



In this presentation we discuss the Heston model with stochastic interest rates driven by Hull-White or Cox-Ingersoll-Ross processes.

We present approximations in the Heston-Hull-White hybrid model, so that a characteristic function can be derived and derivative pricing can be efficiently done using the Fourier Cosine expansion technique.

This pricing method, called the COS method, is explained in some detail.

We furthermore discuss the effect of the approximations in the hybrid model on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities.

We apply the novel method of potential analysis to study climatic records. The method comprises (i) derivation of the number of climate states from time series, (ii) derivation of the potential coefficients. Dynamically

monitoring patterns of potential analysis yields indications of possible bifurcations and transitions of the system.

The method is tested on artificial data and then applied to various climatic records [1,2]. It can be applied to a wide range of stochastic systems where time series of sufficient length and temporal resolution are available and transitions or bifurcations are surmised. A recent application of the method in a model of globally coupled bistable systems [3] confirms its general applicability for studying time series in statistical physics.

[1] Livina et al, Climate of the Past, 2010.

[2] Livina et al, Climate Dynamics (submitted)

[3] Vaz Martins et al, Phys. Rev. E, 2010

Consider the valued field $\mathbb{R}((\Gamma))$ of generalised series, with real coefficients and

monomials in a totally ordered multiplicative group $\Gamma$ . In a series of papers,

we investigated how to endow this formal algebraic object with the analogous

of classical analytic structures, such as exponential and logarithmic maps,

derivation, integration and difference operators. In this talk, we shall discuss

series derivations and series logarithms on $\mathbb{R}((\Gamma))$ (that is, derivations that

commute with infinite sums and satisfy an infinite version of Leibniz rule, and

logarithms that commute with infinite products of monomials), and investigate

compatibility conditions between the logarithm and the derivation, i.e. when

the logarithmic derivative is the derivative of the logarithm.