Une variété compacte est appelée de Bieberbach si elle porte une métrique riemannienne plate. Les variétés de Bieberbach sont asphériques, par conséquent le supremum de leur quotient systolique, sur l'ensemble des métriques riemanniennes, est fini d'après un résultat fondamental de M. Gromov.

On étudie le quotient systolique optimal des $3$-variétés de Bieberbach compactes et orientables qui ne sont pas des tores, et on démontre qu'il n'est pas réalisé par une métrique plate. De plus, on met en évidence une métrique que l'on construit sur un type de telles variétés (C2) qui a une géométrie intéressante : elle est extrêmale dans sa classe conforme et possède de nombreuses géodésiques systoliques.

The purpose of the talk is to explain the problem of isometric embeddability à la Nash for Lorentzian manifolds. This includes:

(a) Simple ways to reduce the problem of embeddability to the (positive definite) Riemannian case, when a semi-Euclidean space of arbitrary signature is allowed as embedding space.

(b) A sharp characterization of the class of Lorentzian manifolds isometrically embeddable when the embedding space is Lorentz-Minkowski one.

(c) To sketch a proof that globally hyperbolic spacetimes (i.e., the most important class of Lorentzian manifolds in Mathematical Relativity) belong to this class. With this aim, some flavour of the Lorentzian tools and related problems of splitting of spacetimes will be given.

The talk is based in joint work with Olaf Müller (Trans. Amer. Math. Soc. 363, 2011).

We explicitly describe a structure of a regular cell complex $CWM(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space $M(L)$ is a sphere, the complex $CWM(L)$ is dual to the boundary complex of the permutahedron. The dual complex $CWM^*$ is patched of Cartesian products of permutohedra and carries a natural PL-structure. It can be explicitly realized as a polyhedron in the Euclidean space via a surgery on the permutohedron.

We consider the problem of finding a conformal metric on a compact surface with constant Gaussian curvature and a prescribed conical structure at a given number of points. The problem has a variational structure, and differently from the "regular" case, the Euler-Lagrange functional might be unbounded from below. We will look for critical points of saddle type using a combination of improved geometric inequalities and topological methods.

This is joint work with D. Bartolucci, A. Carlotto, F. De Marchis and D. Ruiz.