Universite Grenoble-Alpes

Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set admitting a countable cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an exactness result about the completion functor in the category of ultrametric locally convex vector spaces, and in particular we prove that a strict morphism between these spaces has closed image if its kernel is Fréchet.

In this presentation, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to ε. This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential uε caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possibly negative) conductivity a surrounded by a dielectric medium, with unit conductivity.

If we fix a rectangle in the affine real space and if we choose at random a real polynomial with given degree d, the probability P(d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P(d) uniformly bounded from below when d increases? I will explain a positive answer to a very close question involving real analytic functions. This is a joint work with Vincent Beffara.