Past

comonotonicity joint work with Carlier and Galichon Abstact This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a \emph{comonotone dominance principle}, due to Landsberger and

Meilijson that efficiency is

characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and it is addressed by using techniques from convex duality and optimal transportation.

We consider valued fields with a distinguished isometry or contractive derivation, as valued modules over the Ore ring of difference operators. This amounts to study linear difference/differential

equations with respect to the distinguished isometry/derivation.

Under certain assumptions on the residue field, but in all characteristics, we obtain quantifier elimination in natural languages, and the absence of the independence property.

We will consider other operators of interest.

How does form emerge from cellular processes? Using cell-based mechanical models of growth, we investigated the geometry of leaf vasculature and the cellular arrangements at the shoot apex. These models incorporate turgor pressure, wall mechanical properties and cell division. In connection with experimental data, they allowed us to, on the one hand, account for characteristic geometrical property of vein junctions, and, on the other hand, speculate that growth is locally regulated.

We consider saddle point problems arising as (linearized) optimality conditions in elliptic optimal control problems. The efficient solution of such systems is a core ingredient in second-order optimization algorithms. In the spirit of Bramble and Pasciak, the preconditioned systems are symmetric and positive definite with respect to a suitable scalar product. We extend previous work by Schoeberl and Zulehner and consider problems with control and state constraints. It stands out as a particular feature of this approach that an appropriate symmetric indefinite preconditioner can be constructed from standard preconditioners for those matrices which represent the inner products, such as multigrid cycles.

Numerical examples in 2D and 3D are given which illustrate the performance of the method, and limitations and open questions are addressed.

A Hyperkähler manifold is a riemannian manifold carrying three complex structures which behave like quaternions such that the metric is Kähler with respect to each of them. This means in particular that the manifold is a symplectic manifold in many different ways. In analogy to the Marsden-Weinstein reduction on a symplectic manifold, there is also a quotient construction for group actions that preserve the Hyperkähler structure and admit a moment map. In fact most known (non-compact) examples of hyperkähler manifolds arise in this way from an appropriate group action on a quaternionic vector space.

In the first half of the talk I will give the definition of a hyperkähler manifold and explain the hyperkähler quotient construction. As an important application I will discuss the moduli space of solutions to the gauge-theoretic "Self-duality equations on a Riemann surface", the space of Higgs bundles, and explain how it can be viewed as a hyperkähler quotient in an infinite-dimensional setting.

What do historians of mathematics do? What sort of questions do they ask? What kinds of sources do they use? This series of four informal lectures will demonstrate some of the research on history of mathematics currently being done in Oxford. The subjects range from the late Renaissance mathematician Thomas Harriot (who studied at Oriel in 1577) to the varied and rapidly developing mathematics of the seventeenth century (as seen through the eyes of Savilian Professor John Wallis, and others) to the emergence of a new kind of algebra in Paris around 1830 in the work of the twenty-year old Évariste Galois.

Each lecture will last about 40 minutes, leaving time for questions and discussion. No previous knowledge is required: the lectures are open to anyone from the department or elsewhere, from undergraduates upwards.

tba

We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the Lovász-Szegedy notion of convergence of dense graph sequences. We investigate how the limit objects evolve under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limiting object from its initial state up to the stationary state using the theory of exchangeable arrays, the Pólya urn model, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits “aging”.

This talk considers the problem of sampling an independent set uniformly at random from a bipartite graph (equivalently, the problem of approximately counting independent sets in a bipartite graph). I will start by discussing some natural Markov chain approaches to this problem, and show why these lead to slow convergence. It turns out that the problem is interesting in terms of computational complexity – in fact, it turns out to be equivalent to a large number of other problems, for example, approximating the partition function of the “ferromagnetic Ising model’’ (a 2-state particle model from statistical physics) in the presence of external fields (which are essentially vertex weights). These problems are all complete with respect to approximation-preserving reductions for a logically-defined complexity class, which means that if they can be approximated efficiently, so can the entire class. In recent work, we show some connections between this class of problems and the problem of approximating the partition function of the ``ferromagnetic Potts model’’ which is a generalisation of the Ising model—our result holds for q>2 spins. (This corresponds to the approximation problem for the Tutte polynomial in the upper quadrant

above the hyperbola q=2.) That result was presented in detail at a recent talk given by Mark Jerrum at Oxford’s one-day meeting in combinatorics. So I will just give a brief description (telling you what the Potts model is and what the result is) and then conclude with some more recently discovered connections to counting graph homomorphisms and approximating the cycle index polynomial.

Graphics processing units (GPU) are well suited to decrease the

computational in-

tensity of stochastic simulation of chemical reaction systems. We

compare Gillespie’s

Direct Method and Gibson-Bruck’s Next Reaction Method on GPUs. The gain

of the

GPU implementation of these algorithms is approximately 120 times faster

than on a

CPU. Furthermore our implementation is integrated into the Systems

Biology Toolbox

for Matlab and acts as a direct replacement of its Matlab based

implementation.

Topology can be generalised in at least two directions: pointless

topology, leading ultimately to topos theory, or noncommutative

geometry. The former has the advantage that it also carries a logical

structure; the latter captures quantum settings, of which the logic is

not well understood generally. We discuss a construction making a

generalised space in the latter sense into a generalised space in the

former sense, i.e. making a noncommutative C*-algebra into a locale.

This construction is interesting from a logical point of view,

and leads to an adjunction for noncommutative C*-algebras that extends

Gelfand duality.

We investigate a dynamic model of two dimensional crystal growth

described by a forward-backward parabolic equation. The ill-posed

region of the equation describes the motion of corners on the surface.

We analyze a fourth order regularized version of this equation and

show that the dynamical behavior of the regularized corner can be

described by a traveling wave solution. The speed of the wave is found

by rigorous asymptotic analysis. The interaction between multiple

corners will also be presented together with numerical simulations.

This is joint work in progress with Fang Wan.

Hamadène and Jeanblanc provided a BSDE representation for the resolution of bi-dimensional continuous time optimal switching problems. For example, an energy producer faces the possibility to switch on or off a power plant depending on the current price of electricity and corresponding comodity. A BSDE representation via multidimensional reflected BSDEs for this type of problems in dimension larger than 2 has been derived by Hu and Tang as well as Hamadène and Zhang [2]. Keeping the same example in mind, one can imagine that the energy producer can use different electricity modes of production, and switch between them depending on the commodity prices. We propose here an alternative BSDE representation via the addition of constraints and artificial jumps. This allows in particular to reinterpret the solution of multidimensional reflected BSDEs in terms of one-dimensional constrained BSDEs with jumps. We provide and study numerical schemes for the approximation of these two type of BSDEs