Past

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.

When modeling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation diagram.

Stochastic Simulation Algorithms (SSAs) exist which draw exact trajectories from the Chemical Master Equation (CME). However, these methods can be very computationally expensive, particularly where there is a separation of time scales of the evolution of some of the chemical species. Some of the species may react many times on a time scale for which others are highly unlikely to react at all. Simulating all of these reactions of the fast species is a waste of computational effort, and many different methods exist for reducing the system to one which only contains the slow variables.

In this talk we will introduce the conditional Gillespie algorithm, a method for sampling directly from the conditional distribution on the fast variables, given a static value for the slow variables. Using this, we will go on to describe the constrained Gillespie approach, which uses simulations of the CG algorithm to estimate the drift and diffusion terms of the effective dynamics of the slow variables.

If there is time at the end, I will briefly describe my work on another project, which involves full sampling of the posterior distributions in various problems in data assimilation using Monte Carlo Markov Chain (MCMC) methods.

Implementations of the revised simplex method for solving large scale sparse linear programming (LP) problems are highly efficient for single-core architectures. This talk will discuss the limitations of the underlying techniques in the context of modern multi-core architectures, in particular with respect to memory access. Novel techniques for implementing the dual revised simplex method will be introduced, and their use in developing a dual revised simplex solver for multi-core architectures will be described.

I will sketch a method to prolong certain classes of differential equations on manifolds using Lie algebra cohomology. The talk will be based on articles by Branson, Cap, Eastwood and Gover (arXiv:math/0402100 and ESI preprint 1483).

In [Liang, Lyons and Qian(2009): Backward Stochastic Dynamics on a Filtered Probability Space, to appear in the Annals of Probability], the authors demonstrated that BSDEs can be reformulated as functional differential equations, and as an application, they solved BSDEs on general filtered probability spaces. In this paper the authors continue the study of functional differential equations and demonstrate how such approach can be used to solve FBSDEs. By this approach the equations can be solved in one direction altogether rather than in a forward and backward way. The solutions of FBSDEs are then employed to construct the weak solutions to a class of BSDE systems (not necessarily scalar) with quadratic growth, by a nonlinear version of Girsanov's transformation. As the solving procedure is constructive, the authors not only obtain the existence and uniqueness theorem, but also really work out the solutions to such class of BSDE systems with quadratic growth. Finally an optimal portfolio problem in incomplete markets is solved based on the functional differential equation approach and the nonlinear Girsanov's transformation.

The talk is based on the joint work with Lyons and Qian:

http://arxiv4.library.cornell.edu/abs/1011.4499

Given a block, b, of a finite group, Alperin's weight conjecture predicts a miraculous equality between the number of isomorphism classes of simple b-modules and the number of G-orbits of b-weights. Radha Kessar showed that the latter can be written in terms of the fusion system of the block and Markus Linckelmann has computed it as an Euler characteristic of a certain space (provided certain conditions hold). We discuss these reformulations and give some examples.

I will speak about a geometric method, based on the classical trace map, for investigating word maps on groups PSL(2, q) and SL(2, q). In two different papers (with F. Grunewald, B. Kunyavskii, and Sh. Garion, F. Grunewald, respectively) this approach was applied to the following problems.

1. Description of Engel-like sequences of words in two variables which characterize finite

solvable groups. The original problem was reformulated in the language of verbal dynamical

systems on SL(2). This allowed us to explain the mechanism of the proofs for known

sequences and to obtain a method for producing more sequences of the same nature.

2. Investigation of the surjectivity of the word map defined by the n-th Engel word

[[[X, Y ], Y ], . . . , Y ] on the groups PSL(2, q) and SL(2, q). Proven was that for SL(2, q), this

map is surjective onto the subset SL(2, q) $\setminus$ {−id} $\subset$ SL(2, q) provided that q $\ge q_0(n)$ is

sufficiently large. If $n\le 4$ then the map was proven to be surjective for all PSL(2, q).

Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.

In a healthy human brain, cerebrospinal fluid (CSF), a water-like liquid, fills a system of cavities, known as ventricles, inside the brain and also surrounds the brain and spinal cord. Abnormalities in CSF dynamics, such as hydrocephalus, are not uncommon and can be fatal for the patient. We will consider two types of models for the so-called infusion test, during which additional fluid is injected into the CSF space at a constant rate, while measuring the pressure continuously, to get an insight into the CSF dynamics of that patient.

In compartment type models, all fluids are lumped into compartments, whose pressure and volume interactions can be modelled with compliances and resistances, equivalent to electric circuits. Since these models have no spatial variation, thus cannot give information such as stresses in the brain tissue, we also consider a model based on the theory of poroelasticity, but including strain-dependent permeability and arterial blood as a second fluid interacting with the CSF only through the porous elastic solid.

We show the existence of global-in-time weak solutions to a general class of bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.

The talk is based on joint work with John W. Barrett [Imperial College London].

Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions.  He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line.  Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions.  In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition.

Study the parabolic Hitchin fibrations for Langlands dual groups. Sketch the proof of a duality theorem of the natural symmetries on their cohomology.

There will be three problems discussed all of which are open for consideration as MSc projects.

1. Reduction of Ndof in Adaptive Signal Processing

2. Calculus of Convex Sets

3. Dynamic Response of a disc with an off centre hole(s)

In my talk I will introduce the concept of spectral discrete solitons

(SDSs): solutions of nonlinear Schroedinger type equations, which are localized on a regular grid in frequency space. In time domain such solitons correspond to periodic trains of pulses. SDSs play important role in cascaded four-wave-mixing processes (frequency comb generation) in optical fibres, where initial excitation by a two-frequency pump leads to the generation of multiple side-bands. When free space diffraction is taken into consideration, a non-trivial generalization of 1D SDSs will be discussed, in which every individual harmonic is an optical vortex with its own topological charge. Such excitations correspond to spatio-temporal helical beams.

We propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities.

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, convergence of the algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented.

All of Joyce's constructions of compact manifolds with special holonomy are in some sense generalisations of the Kummer construction of a K3 surface. We will begin by reviewing manifolds with special holonomy and the Kummer construction. We will then describe Joyce's constructions of compact manifolds with holonomy G_2 and Spin(7).

Extend the affine Weyl group action in Lecture I to double affine Hecke algebra action, and (hopefully) more examples.

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.