Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p>1$. We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact then the number of their critical points is infinite (we use Lusternik-Schnirelmann category combined with the Hurewicz property). In the special case where the control system is subriemannian this result can be read as the corresponding version of Serre's theorem, on the existence of infinitely many geodesics between two points on a compact riemannian manifold. In the subriemannian case we show that the Hurewicz property holds for all $p\geq1$ and the horizontal-loop space with the $W^{1,2}$ topology has the homotopy type of a CW-complex (as long as the endpoint map has at least one regular value); in particular the inclusion of the horizontal-loop space in the ordinary one is a homotopy equivalence

On the multiplicity of non-iterated periodic billiard trajectories

We introduce the iteration theory for periodic billiard trajectories in a compact and convex domain of the Euclidean space, and we apply it to establish a multiplicity result for non-iterated trajectories.Comment: 21 pages, 2 figures; v3: final version, as publishe

Morse theory for the Allen-Cahn functional

In this article, we use Morse-theoretic techniques to construct connections between low energy critical submanifolds of the Allen-Cahn energy functional in the 3-sphere via the negative gradient flow.Comment: 23 pages. All comments welcom

Material Theories

The subject of this meeting was mathematical modeling of strongly interacting multi-particle systems that can be interpreted as advanced materials. The main emphasis was placed on contributions attempting to bridge the gap between discrete and continuum approaches, focusing on the multi-scale nature of physical phenomena and bringing new and nontrivial mathematics. The mathematical debates concentrated on nonlinear PDE, stochastic dynamical systems, optimal transportation, calculus of variations and large deviations theory