Past

4D-Var is a widely used data assimilation method, particularly in the field of Numerical Weather Prediction. However, it is highly sequential: integrations of a numerical model are nested within the loops of an inner-outer minimisation algorithm. Moreover, the numerical model typically has a low spatial resolution, limiting the number of processors that can be employed in a purely spatial parallel decomposition. As computers become ever more parallel, it will be necessary to find new dimensions over which to parallelize 4D-Var. In this talk, I consider the possibility of parallelizing 4D-Var in the temporal dimension. I analyse different formulations of weak-constraint 4D-Var from the point of view of parallelization in time. Some formulations are shown to be inherently sequential, whereas another can be made parallel but is numerically ill-conditioned. Finally, I present a saddlepoint formulation of 4D-Var that is both parallel in time and amenable to efficient preconditioning. Numerical results, using a simple two-level quasi-geotrophic model, will be presented.

Links between:

• storm tracks, sediment movement and an icy environment

• fluvial flash flooding to coastal erosion in the UK

Did you know that the recent Japanese, Chilean and Samoan tsunami all led to strong currents from resonance at the opposite end of the ocean?

Journey around the world, from the north Atlantic to the south Pacific, on a quest to explore and explain the maths of nature.

We discuss the question of the existence of the smallest size of a family of Banach spaces of a given density which embeds all Banach spaces of that same density. We shall consider two kinds of embeddings, isometric and isomorphic. This type of question is well studied in the context of separable spaces, for example a classical result by Banach states that C([0,1]) embeds all separable Banach spaces. However, the nonseparable case involves a lot of set theory and the answer is independent of ZFC.

Given a variety X over a number field, one is interested in the collection X(F) of rational points on X. Weil defined a variety X' (the restriction of scalars of X) defined over the rational numbers whose set of rational points is naturally equal to X(F). In this talk, I will compare the number of rational points of bounded height on X with those on X'.

Some striking, and potentially useful, effects in electrokinetics occur for

bipolar membranes: applications are in medical diagnostics amongst other areas.

The purpose of this talk is to describe the experiments, the dominant features observed

and then model the phenomena: This uncovers the physics that control this process.

Time-periodic reverse voltage bias

across a bipolar membrane is shown to exhibit transient hysteresis.

This is due to the incomplete depletion of mobile ions, at the junction

between the membranes, within two adjoining polarized layers; the layer thickness depends on

the applied voltage and the surface charge densities. Experiments

show that the hysteresis consists of an Ohmic linear rise in the

total current with respect to the voltage, followed by a

decay of the current. A limiting current is established for a long

period when all the mobile ions are depleted from the polarized layer.

If the resulting high field within the two polarized layers is

sufficiently large, water dissociation occurs to produce proton and

hydroxyl travelling wave fronts which contribute to another large jump

in the current. We use numerical simulation and asymptotic analysis

to interpret the experimental results and

to estimate the amplitude of the transient hysteresis and the

water-dissociation current.

We will go through the GIT construction of the moduli space of quiver representations. Concentrating on examples (probably the cases of Hilbert schemes of points of $\mathbb{C}^{2}$ and $\mathbb{C}^{3}$) we will try to give an idea of why this methods became relevant in modern (algebraic) geometry.

No prerequisites required, experts would probably get bored.

By classical results of Thomason, the Grothendieck group of a

triangulated category classifies the triangulated subcategories. More

precisely, there is a bijective correspondence between the set of

triangulated subcategories and the set of subgroups of the Grothendieck

group. In this talk, we extend Thomason's results to "higher"

triangulated categories, namely the recently introduced n-angulated

categories. This is joint work with Marius Thaule.

We will discuss the numerical simulation of acoustic wave propagation with localized inhomogeneities. To do this we will apply a standard finite element method (FEM) in space and explicit time-stepping in time on a finite spatial domain containing the inhomogeneities. The equations in the exterior computational domain will be dealt with a time-domain boundary integral formulation discretized by the Galerkin boundary element method (BEM) in space and convolution quadrature in time.

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We will give the analysis of the proposed method, starting with the proof of a positivity preservation property of convolution quadrature as a consequence of a variant of the Herglotz theorem. Combining this result with standard energy analysis of leap-frog discretization of the interior equations will give us both stability and convergence of the method. Numerical results will also be given.

Speaker: Tigran Atoyan\\

Title: A revised approach to hedging and pricing\\

Abstract:\\

After a brief review of the classical option pricing framework, we present a motivating example on the evaluation of hedging P&L using a simplistic strategy which does very well in practice. We then present preliminary results about a relatively unknown approach called business time hedging. Some applications of the latter approach to pricing certain derivative products as well as future research directions in this topic are discussed.\\

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Speaker: Sean Ledger\\

Title: Stochastic Evolution Equations in Portfolio Credit Modelling\\

Abstract:\\

I shall present an infinite-dimension structural model for a large portfolio of credit risky assets. As the number of assets approaches infinity we obtain a limiting system with a density process. I shall outline the properties of this density process and how one can use the SPDE satisfied by this process to estimate the loss function of the portfolio. Extensions to the model shall be onsidered, including contagion effects and Lévy noise. Finally I shall present some of the numerical testing for these models.\\

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Speaker: Peter Spoida\\

Title: Robust Pricing and Hedging of the Barrier Option with a Finite Number of Intermediate Law Constraints\\

Abstract:\\

We propose a robust superhedging strategy for simple barrier options, consisting of a portfolio of calls with different maturities and a self-financing trading strategy. The superhedging strategy is derived from a pathwise inequality. We illustrate how a stochastic control ansatz can provide a good guess for finding such strategies. By constructing a worst-case model, we demonstrate that this superhedge is the cheapest possible. Our construction generalizes the Skorokhod embedding obtained by Brown, Hobson and Rogers (2001). The talk is based on joint work with Pierre Henry-Labordere, Jan Obloj and Nizar Touzi.

We establish the exact boundary controllability of nodal profile for general first order quasi linear hyperbolic systems in 1-D. And we apply the result in a tree-like network with general nonlinear boundary conditions and interface conditions. The basic principles of choosing the controls and getting the controllability are given.

Wave-particle duality is a quantum behaviour usually assumed to have no possible counterpart in classical physics. We revisited this question when we found that a droplet bouncing on a vibrated bath could become self-propelled by its coupling to the surface waves it excites. A dynamical wave-particle association is thus formed.Through several experiments we addressed the same general question. How can a localized and discrete droplet have a common dynamics with a continuous and spatially extended wave? Surprisingly several quantum-like behaviors emerge; a form of uncertainty and a form of quantization are observed. I will show that both properties are related to the "path memory" contained in the wave field. The relation of this experiment with the pilot-wave models proposed by de Broglie and by Bohm for quantum mechanics will be discussed.

I will explain a construction of a group acting on a rooted tree, related to the Grigorchuk group. Those groups have exponential growth, at least under certain circumstances. I will also show how it can be seen that any two elements fulfil a non-trivial relation, implying the absence of non-cyclic free subgroups.

I will give a brief report on some the topics discussed at the workshop "Golod-Shafarevich groups and rank gradient" that took place this August in Vienna. I will focus on results involving rank gradient.

There is a well-acknowledged analogy between mapping class
groups and lattices in higher rank groups. I will discuss to which
extent does Margulis's superrigidity hold for mapping class groups:
examples, very partial results and questions.

Counting nodal curves in linear systems $|L|$ on smooth projective surfaces $S$ is a problem with a long history. The G\"ottsche conjecture, now proved by several people, states that these counts are universal and only depend on $c_1(L)^2$, $c_1(L)\cdot c_1(S)$, $c_1(S)^2$ and $c_2(S)$. We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecture. This can be seen as version of the MNOP conjecture for the canonical bundle $K_S$. This is joint work with R. P. Thomas.

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. By contrast to the previous literature, we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis and, under

minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience adjusted market depth.

When ice is raised to a temperature above its usual melting temperature
of 273 K, small cylindrical discs of water form within the bulk of the
ice. Subsequent internal melting of the ice causes these liquid discs to
grow radially outwards. However, many experimentalists have observed
that the circular interface of these discs is unstable and eventually
the liquid discs turn into beautiful shapes that resemble flowers or
snowflakes. As a result of their shape, these liquid figures are often
called liquid snowflakes. In this talk I'll discuss a simple
mathematical model of liquid snowflake formation and I'll show how a
combination of analytical and numerical methods can yield much insight
into the dynamics which govern their growth.