Past

The main goal of this talk is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. First, I will discuss a relation between results on cubature formulas and on discrepancy. In particular, I'll show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. Second, I'll present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Third, I'll give some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this talk is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative.

The Riemann zeta function involves, for Re s>1, the summation of the inverse s-th powers of the integers. A class of zeta-like functions is obtained if the s-th powers of integers which contain specified digits are omitted from the summation. The numerical summation of such series, their convergence properties and analytic continuation are considered in this lecture.

Parallel sparse direct solvers are an interesting alternative to iterative methods for some classes of large sparse systems of linear equations. In the context of a parallel sparse multifrontal solver (MUMPS), we describe a new dynamic scheduling strategy aiming at balancing both the workload and the memory usage. More precisely, this hybrid approach balances the workload under memory constraints. We show that the peak of memory can be significantly reduced, while we have also improved the performance of the solver.

Then, we present preliminary work concerning a parallel out-of-core extension of the solver MUMPS, enabling to solve increasingly large simulation problems.

This is joint work with P.Amestoy, A.Guermouche, S.Pralet and E.Agullo.

The concept of the slow invariant manifold is the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the constructive methods of invariant manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space. The equation of motion for immersed manifolds is obtained.

Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.

A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.

The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for nudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; model reduction in chemical kinetics.

Using the dyadic parametrization of curves, and elementary theorems and

probability theory, examples are constructed of domains having bad properties on

boundary sets of large Hausdorff dimension (joint work with F.D. Lesley).

Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (1999). These processes describe the evolution of particles that

undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Lambda has the Beta$(2-\alpha,\alpha)$ they are also known to describe the genealogies of large populations where a single individual can produce a large number of offsprings. Here we use a recent result of Birkner et al. (2005) to prove that Beta-coalescents can be embedded in continuous stable random trees, for which much is known due to recent progress of Duquesne and Le Gall. This produces a number of results concerning the small-time behaviour of Beta-coalescents. Most notably, we recover an almost sure limit theorem for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the infinite site frequency spectrum associated with mutations in the context of population genetics.

It is suggested that topological membranes play a fundamental role

in the recently proposed topological M-theory. We formulate a topological theory

of membranes wrapping associative three-cycles in a seven-dimensional target

space with G_2 holonomy. The topological BRST rules and BRST invariant action

are constructed via the Mathai-Quillen formalism. We construct a set of local

and non-local observables for the topological membrane theory. As the BRST

cohomology of local operators turns out to be isomorphic to the de Rham

cohomology of the G_2 manifold, our observables agree with the spectrum of

d=4, N=1 G_2 compactifications of M-theory.