L3

The B-D equations describe a mean field approximation for a many body system in relaxation to equilibrium. The two B-D equations determine the time evolution of the density c(L,t) of particles with mass L, L=1,2,... One of the equations is a discretized linear diffusion equation for c(L,t), and the other is a non-local constraint equivalent to mass conservation. Existence and uniqueness for the B-D system was established in the 1980's by Ball, Carr and Penrose. Research in the past decade has concentrated on understanding the large time behavior of solutions to the B-D system. This behavior is characterized by the phenomenon of "coarsening", whereby excess density is concentrated in large particles with mass increasing at a definite rate. An important conjecture in the field is that the coarsening rate can be obtained from a particular self- similar solution of the simpler LSW system. In this talk we shall discuss the B-D and LSW equations, and some recent progress by the speaker and others towards the resolution of this conjecture.

Multi-well relaxation problem emerges e.g. in characterising effective properties of composites and in phase transformations. This is a nonlinear problem and one approach uses its reformulation in Fourier space, known in the theory of composites as Hashin-Shtrikman approach, adapted to nonlinear composites by Talbot and Willis. Characterisation of admissible mixtures, subjected to appropriate differential constraints, leads to a quasiconvexification problem. The latter is equivalently reformulated in the Fourier space as minimisation with respect to (extremal points of) H-measures of characteristic functions (Kohn), which in a sense separates the microgeometry of mixing from the differential constraints. For three-phase mixtures in 3D we obtain a full characterisation of certain extremal H-measures. This employs Muller's Haar wavelet expansion estimates in terms of Riesz transform to establish via the tools of harmonic analysis weak lower semicontinuity of certain functionals with rank-2 convex integrands. As a by-product, this allows to fully solve the problem of characterisation of quasiconvex hulls for three arbitrary divergence-free wells. We discuss the applicability of the results to problems with other kinematic constraints, and other generalisations. Joint work with Mariapia Palombaro, Leipzig.

For a positively graded algebra A we construct a functor from the derived

category of graded A-modules to the derived category of graded modules over

the quadratic dual A^! of A. This functor is an equivalence of certain

bounded subcategories if and only if the algebra A is Koszul. In the latter

case the functor gives the classical Koszul duality. The approach I will

talk about uses the category of linear complexes of projective A-modules.

Its advantage is that the Koszul duality functor is given in a nice and

explicit way for computational applications. The applications I am going to

discuss are Koszul dualities between certain functors on the regular block

of the category O, which lead to connections between different

categorifications of certain knot invariants. (Joint work with S.Ovsienko

and C.Stroppel.)

A tree-rooted map is a planar map together with a

distinguished spanning tree. In the sixties, Mullin proved that the

number of tree-rooted maps with $n$ edges is the product $C_n C_{n+1}$

of two consecutive Catalan numbers. We will present a bijection

between tree-rooted maps (of size $n$) and pairs made of two trees (of

size $n$ and $n+1$ respectively) explaining this result.

Then, we will show that our bijection generalizes a correspondence by

Schaeffer between quandrangulations and so-called \emph{well labelled

trees}. We will also explain how this bijection can be used in order

to count bijectively several classes of planar maps

In the case of Einstein's equations coupled to a non-linear scalar field with a suitable exponential potential, there are solutions for which the expansion is accelerated and of power law type. In the talk I will discuss the future global non-linear stability of such models. The results generalize those of Mark Heinzle and Alan Rendall obtained using different methods.