Past

We will discuss numerical solutions of Multi-objective Control problems governed by partial differential equations. More precisely, we will look for Nash Equilibria, which are solutions to non-cooperative differential games. First we will study the continuous case. Then, in order to compute solutions, we will combine finite difference schemes for the time discretization, finite element methods for the space discretization and a conjugate gradient algorithm (or other suitable alternative) for the iterative solution of the discrete differential game. Finally, we will apply this methodology to the solution of several test problems.

We investigate a mixed problem with Robin boundary conditions for a diffusion-reaction equation. We investigate the problem in the sublinear, linear and super linear cases, depending on the nonlinear part. We obtain relations between the parameters of the problem which are sufficient conditions for the existence of generalized solutions to the problem and, in a special case, for their uniqueness. The proof relies on a general existence theorem by Soltanov. Finally we investıgate the time-behaviour of solutions. We show that boundedness of solutions holds under some additional conditions as t is convergent to infinity. This study is joint work with Kamal Soltanov (Hacettepe University).

I'll sketch some context for future and past research around valued fields
and motivic integration, from a model theoretic viewpoint, leaving out technical details. 
The talk will be partly conjectural.

We give a brief overview of hyperbolic metric spaces and the relatively hyperbolic counterparts, with particular emphasis on the quasi-isometry class of trees. We then show that an understanding of the relative version of such spaces - quasi tree-graded spaces -  has strong consequences for mapping class groups. In particular, they are shown to embed into a finite product of (possibly infinite valence) simplicial trees. This uses and extends the work of Bestvina, Bromberg and Fujiwara.

I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|. 
I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.

Results that say local information is enough to guarantee global information provide the theoretical underpinnings of many reconstruction algorithms in evolutionary biology. Such results include Buneman's Splits-Equivalence Theorem and the Tree-Metric Theorem. The first result says that, for a collection $\mathcal C$ of binary characters, pairwise compatibility is enough to guarantee compatibility for $\mathcal C$, that is, there is a phylogenetic (evolutionary) tree that realises $\mathcal C$. The second result says that, for a distance matrix $D$, if every $4\times 4$ distance submatrix of $D$ is realisable by an edge-weighted phylogenetic tree, then $D$ itself is realisable by such a tree. In this talk, we investigate these and other results of this type. Furthermore, we explore the closely-related task of determining how much information is enough to reconstruct the correct phylogenetic tree.

I will explain recent results with Emanuele Macrì, in which we systematically study the birational geometry of moduli of sheaves on K3's via wall-crossing for

Bridgeland stability conditions. In particular, we obtain descriptions of their nef cones via the Mukai lattice of the K3, their moveable cones, their divisorial contractions, and obtain counter-examples to various conjectures in the literature. We also give a proof of the Lagrangian fibration conjecture (due to

Hassett-Tschinkel/Huybrechts/Sawon) via wall-crossing.

We review potential theoretic aspects of degenerate parabolic PDEs of p-Laplacian type.

Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat

equation do in the classical theory. In the parabolic potential theory, the so-called superparabolic

functions are essential. For the ordinary heat equation we have supercaloric functions. They are defined

as lower semicontinuous functions obeying the comparison principle. The superparabolic

functions are of actual interest also because they are viscosity supersolutions of the equation. We discuss

their existence, structural, convergence and Sobolev space properties. We also consider the

definition and properties of the nonlinear parabolic capacity and show that the infinity set of a superparabolic

function is of zero capacity.

I will talk about the fixed-point properties of matrix groups acting CAT(0) paces, spheres and acyclic manifolds. The matrix groups include general linear groups, sympletic groups, orthogonal groups and classical unitary groups over general rings. We will show that for lower dimensional CAT(0) spaces, the group action of a matrix group always has a global fixed point and that for lower dimensional spheres and acyclic manifolds, a group action by homeomorphisms is always trivial. These results give generalizations of results of Farb concerning Chevalley groups over commutative rings and those of Bridson-Vogtmann, Parwani and Zimmermann concerning the special linear groups SL_{n}(Z) and symplectic groups Sp_{2n}(Z).

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'.