Past

The new model is that the universal small scale structure of high Reynolds number turbulence is determined by the dynamics of thin evolving shear layers, with thickness of the order of the Taylor micro scale,within which there are the familiar elongated vortices .Local quasi-linear dynamics shows how the shear layers act as barriers to external eddies and a filter for the transfer of energy to their interiors. The model is consistent with direct numerical simulations by Ishihara and Kaneda analysed in terms of conditional statistics relative to the layers and also with recent 4D measurements of lab turbulence by Wirth and Nickels. The model explains how the transport of energy into the layers leads to the observed inertial range spectrum and to the generation of intense structures, on the scale of the Kolmogorov micro-scale.

But the modelling also explains the important discrepancies between data and the Kolmogorov-Richardson cascade concept ,eg larger amplitudes of the smallest scale motions and of the higher moments ,and why the latter are generally less isotropic than lower order moments, eg in thermal convection. Ref JCRHunt , I Eames, P Davidson,J.Westerweel, J Fernando, S Voropayev, M Braza J Hyd Env Res 2010

Let C be a smooth plane cubic curve over the rationals. The Mordell--Weil Theorem can be restated as follows: there is a finite subset B of rational points such that all rational points can be obtained from this subset by successive tangent and secant constructions. It is conjectured that a minimal such B can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This talk is concerned with the corresponding problem for cubic surfaces.


We extend for the first time the linear discretization theory of Schaback, developed for meshfree methods, to nonlinear operator equations, relying heavily on methods of Böhmer, Vol I. There is no restriction to elliptic problems or to symmetric numerical methods like Galerkin techniques.

Trial spaces can be arbitrary, but have to approximate the solution well, and testing can be weak or strong. We present Galerkin techniques as an example. On the downside, stability is not easy to prove for special applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. These results remain valid for the general case of fully nonlinear elliptic differential equations of second order. Some numerical examples are added for illustration.

The theory of C*-algebras provides a good realisation of noncommutative topology. There is a dictionary relating commutative C*-algebras with locally compact spaces, which can be used to import topological concepts into the C*-world. This philosophy fails in the case of homotopy, where a more sophisticated definition has to be given, leading to the notion of asymptotic morphisms.

As a by-product one obtains a generalisation of Borsuk's shape theory and a universal boundary map for cohomology theories of C*-algebras.

In this talk we will survey some aspects of social choice theory: in particular, various impossibility theorems about voting systems and strategies. We begin with the famous Arrow's impossibility theorem -- proving the non-existence of a 'fair' voting system -- before moving on to later developments, such as the Gibbard–Satterthwaite theorem, which states that all 'reasonable' voting systems are subject to tactical voting.

Given time, we will study extensions of impossibility theorems to micro-economic situations, and common strategies in game theory given the non-existence of optimal solutions.

Recently Frenkel and Hernandez introduced a kind of "Langlands duality" for characters of semisimple Lie algebras. We will discuss a representation-theoretic interpretation of their duality using quantum analogues of exceptional isogenies. Time permitting we will also discuss a branching rule and relations to Littelmann paths.

This talk introduces the topic of random walks on a finitely generated group and asks what properties of such a group can be detected through knowledge of such walks.

Both parts will deal with spherical objects in the bounded derived

category of coherent sheaves on K3 surfaces. In the first talk I will

focus on cycle theoretic aspects. For this we think of the Grothendieck

group of the derived category as the Chow group of the K3 surface (which

over the complex numbers is infinite-dimensional due to a result of

Mumford). The Bloch-Beilinson conjecture predicts that over number

fields the Chow group is small and I will show that this is equivalent to

the derived category being generated by spherical objects (which

I do not know how to prove). In the second talk I will turn to stability

conditions and show that a stability condition is determined by its

behavior with respect to the discrete collections of spherical objects.

The problem of checking existence for an Euler tour of a graph is trivial (are all vertex degrees even?). The problem of counting (or even approximate counting) Euler tours seems to be very difficult. I will describe two simple classes of graphs where the problem can be

solved exactly in polynomial time. And also talk about the many many classes of graphs where no positive results are known.

We introduce and study ergodic multidimensional diffusion processes interacting through their ranks; these interactions lead to invariant measures which are in broad agreement with stability properties of large equity markets over long time-periods.

The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset.

Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.

The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of intersection local times.

The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics, such as the Sherrington-Kirkpatrick model for spin-glasses.

"We present a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points.

For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations. We give some concrete examples, and in particular we study the case of mean variance optimal portfolios with wealth dependent risk aversion"

Both parts will deal with spherical objects in the bounded derived

category of coherent sheaves on K3 surfaces. In the first talk I will

focus on cycle theoretic aspects. For this we think of the Grothendieck

group of the derived category as the Chow group of the K3 surface (which

over the complex numbers is infinite-dimensional due to a result of

Mumford). The Bloch-Beilinson conjecture predicts that over number

fields the Chow group is small and I will show that this is equivalent to

the derived category being generated by spherical objects (which

I do not know how to prove). In the second talk I will turn to stability

conditions and show that a stability condition is determined by its

behavior with respect to the discrete collections of spherical objects.

The fundamental models for lipid bilayers are curvature based and neglect the internal structure of the lipid layers. In this talk, we explore models with an additional order parameter which describes the orientation of the lipid molecules in the membrane and compare their predictions based on numerical simulations. This is joint work with Soeren Bartels (Bonn) and Ricardo Nochetto (College Park).

(Note that the talk will be in L2 and not the usual SR1)

Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain. For instance, a smooth cubic surface in complex projective three-dimensional space contains exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational) parameterization of the points of the cubic by the points in the complex projective plane.

After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x^5+y^5+z^5+u^5+v^5=0. The space of lines on X is well-known. I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.

This talk is based on joint work with R. Heath-Brown.

We review the problem of determining the high frequency asymptotics of the spectrum of the Laplacian and its relationship to the geometry of a domain. We then establish these asymptotics for some continuum random trees as well as the scaling limit of the critical random graph.

For a superprocess in a random environment in one dimensional space, a nonlinear stochastic partial differential equation is derived for its density by Dawson-Vaillancourt-Wang (2000). The joint continuity was left as an open problem. In this talk, we will give an affirmative answer to this problem.

• Simon Cotter presents:       “Chemical Fokker-Planck equation and multiscale modelling of (bio)chemical systems”
• Lian Duan presents:            “History matching problems using Bspline Parameterization”
• Chris Prior presents:          “Helices, tubes and the Fourier Transform”

We present an O(N logN) algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing.

Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.

Presentations by:

09.30 am Bernhard Langwallner Continuum limits of atomistic energies and new computational models of fracture

09.50 am Yasemin Sengul Well-posedness of dynamics

10.10 am Kostas Koumatos X-interfaces and nonclassical austenite-martensite interfaces

10.30 am Tim Squires Models for breast cancer and heart tissue

Abstract: Quantales are ordered algebras which can be thought of as pointfree noncommutative topologies. In recent years, their connections have been studied with fundamental notions in noncommutative geometry such as groupoids and C*-algebras. In particular, the setting of quantales corresponding to étale groupoids has been very well understood: a bijective correspondence has been defined between localic étale groupoids and inverse quantale frames. We present an equivalent but independent way of defining this correspondence for topological étale groupoids and we extend this correspondence to a non-étale setting.

Wound coils or rolls accumulate essentially flat strip compactly without folding or cutting and typically, strip is wound and unwound a number of times before its end use. The variety of material that is wound into coils or rolls is very extensive and includes magnetic tape, paper, cellophane, plastics, fabric and metals such as aluminium and steel.

Stresses wound into a coil provide its structural integrity via the frictional forces between the wraps. For a coil with inadequate inter-wrap pressure, the wraps may slip or telescope (causing surface scuffing) or the coil may slump and collapse. On the other hand, large internal stresses can cause increased creep and stress relaxation, collapse at the bore, stress wrinkling and rupture of the material in the coil.

Given the range of applications, it is not surprising that the literature on calculating stresses in wound coils is large and has a long history, which goes back at least to the wire winding of gun barrels. However the basic approach of the resulting accretion models, where the residual stress is recalculated each time a layer is added, has remained essentially the same. In this talk, we take a radically different approach in analysing the winding stresses in coils. Instead of the traditional method, we seek to deduce a winding policy that will achieve a target distribution of residual stresses within a coil. In this way, optimising the coiling tension profile is much more straight-forward, by

* Specifying the residue stresses required to avoid operational problems, tight-bore collapses, and other issues such as scuffing, then

* Determining the winding tension profile to produce the required residue stresses.

Strong explosive volcanic eruptions could inject in the lower stratosphere million tons of SO2, which being converted to sulfate aerosols, affect radiative balance of the planet for a few years. During this period the volcanic radiative forcing dominates other forcings producing distinct detectable climate responses. Therefore volcanic impacts provide invaluable natural test of climate nonlinearities and feedback mechanisms. In this talk I will overview volcanic impacts on tropospheric and strsatospheric temperature, ozone, high-latitude circulation, stratosphere-troposphere dynamic interaction, and focus on the long-term volcanic effect on ocean heat content and sea level.

This workshop is half-seminar, half-workshop. \\ \\ HSBC have an on-going problem and they submitted a proposal for an MSc in Applied Stats project on this topic. Unfortunately, the project was submitted too late for this cohort of students. Eurico will talk about "the first approach at the problem" but please be aware that it is an open problem which requires further work. Eurico's abstract is as follows. \\ \\

This article examines modelling yield curves through chaotic dynamical systems whose dynamics can be unfolded using non-linear embeddings in higher dimensions. We then refine recent techniques used in the state space reconstruction of spatially extended time series in order to forecast the dynamics of yield curves.

We use daily LIBOR GBP data (January 2007-June 2008) in order to perform forecasts over a 1-month horizon. Our method seems to outperform random walk and other benchmark models on the basis of mean square forecast error criteria.

In this talk I will discuss "motivic" Donaldson-Thomas invariants, following the now not-so-recent paper of Kontsevich and Soibelman on this subject. I will, in particular, present some understanding of the mysterious notion of "orientation data," and present some recent work. I will of course do my best to make this talk "accessible," though if you don't know what a scheme or a category is it will probably make you cry.

The second talk will present conjectural motivic generalizations

of ADHM sheaf invariants as well as their wallcrossing formulas.

It will be shown that these conjectures yield recursive formulas

for Poincare and Hodge polynomials of moduli spaces of Hitchin

pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.

The first talk will present a construction of equivariant

virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.

This talk will begin with an introduction to calibrations and calibrated submanifolds. Calibrated geometry generalizes Wirtinger's inequality in Kahler geometry by considering k-forms which are analogous to the Kahler form. A famous one-line proof shows that calibrated submanifolds are volume minimizing in their homology class. Our examples of manifolds with a calibration will come from complex geometry and from manifolds with special holonomy.

We will then discuss the deformation theory of the calibrated submanifolds in each of our examples and see how they differ from the theory of complex submanifolds of Kahler manifolds.

The presence and flow of fluid inside a crack within a solid causes deformation of the solid which in turn influences the flow of the fluid.

This coupled fluid-solid problem will be discussed in the context of dyke propagation and hydrofracture. The background material will be discussed in detail and some applications to specific geometries presented.

"We investigate a construction of a Skorokhod embedding due to Root (1969), which has been the subject of recent interest for applications in Mathematical Finance (Dupire, Carr & Lee), where the construction has applications for model-free pricing and hedging of variance derivatives. In this context, there are two related questions: firstly of the construction of the stopping time, which is related to a free boundary problem, and in this direction, we expand on work of Dupire and Carr & Lee; secondly of the optimality of the construction, which is originally due to Rost (1976). In the financial context, optimality is connected to the construction of hedging strategies, and by giving a novel proof of the optimality of the Root construction, we are able to identify model-free hedging strategies for variance derivatives. Finally, we will present some evidence on the numerical performance of such hedges. (Joint work with Jiajie Wang)"

This talk will be based on the article arXiv:1006.2788.