Calendar

I will present some criteria (necessary or sufficient) for the action on the affine space of a group Gamma of affine transformations to be proper. This is joint work with Fanny Kassel.

The main of these criteria links properness of action to the divergence of a parameter called the Margulis invariant. This invariant measures roughly the translation part of an affine transformation, but in a way that is invariant by conjugation.

This link was already known in some special cases (and has often been exploited to construct proper actions). We tried to establish it in as general setting as possible. We proved it in particular if Gamma has some suitable Anosov property (with respect to some natural parabolic subgroup, that depends on the affine group we are working in).

I will possibly also evoke some other invariants similar to the Margulis invariant, that could lead to criteria that work in even more general settings.

The talk is concerned with the rigorous mathematical description of propagation and localisation of waves in a particular class of composite materials with random microscopic geometry, called micro-resonant (or high-contrast) random media: small inclusions of a “soft” material are randomly dispersed in a “stiff” matrix.  The highly contrasting physical properties of the two constituents,  combined with a particular scaling of the inclusions,  result in microscopic resonances, which manifest macroscopically by allowing propagation of waves in the material only within certain ranges of frequencies (band-gap spectrum).

High-contrast media with periodically distributed inclusions have been extensively studied and numerous results are available in the literature.  However,  their stochastic counterparts, which model more realistic scenarios and may exhibit localisation,  are far from being well understood from a mathematical viewpoint.  In my talk I will give an overview of existing results through the prism of stochastic homogenisation and spectral theory,  and discuss recent advances and ongoing work.

Based on joint works with M. Cherdantsev, I. Velčić, P. Bella and M. Täufer.