Fuchsian convex bodies: basics of Brunn--Minkowski theory

The hyperbolic space \H^d can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $\Gamma$ is a group of linear isometries of the Minkowski space such that \H^d/\Gamma is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn--Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov--Fenchel and Brunn--Minkowski inequalities. Here the inequalities are reversed

Shapes of polyhedra, mixed volumes and hyperbolic geometry

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2

A glimpse into Thurston's work

We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book "In the tradition of Thurston: Geometry and topology", edited by K. Ohshika and A. Papadopoulos (Springer, 2020)

Criticality safety control at CEA Paris-Saclay Abstract

International audienceThe CEA's organization regarding criticality-safety is the same for each of its nuclear center. First, one or more qualified criticality engineer (QCE) is located directly on nuclear installations containing fissile material. The QCE manages criticality safety in the installations. He develops criticality safety analyses for the installations and participates in the implementation of the resulting rules. Second, a center's criticality engineer (CCE) is in charge of the verification for every modification in each nuclear facilities. He is also in charge to support qualified criticality engineer, and capitalise the knowledge of all the installations. Third, a Criticality Specialist (CS) controls the good functioning of the criticality organization in the center. Furthermore, a criticality safety experts group could assist every criticality engineer of each center's organisation for the criticality calculations and analysis. This paper details the role of the Criticality Specialist, which is to define an independent opinion on nuclear facilities modifications that could affect the criticality analysis. Its role is also to be an interface between CEA and the French nuclear safety authority for criticality risks in the center. Its action is thus to ensure the quality and consistency of the files sent to the authorities, but without participating in the technical solutions choices. Then, this paper presents an example of a control carried out on one of the nuclear facilities in Paris-Saclay center. This control is about an implementation of a future operation that may present a criticality risk. The criticality specialist did several further requirements for this operation. In the end, this paper presents how this second line control has been taken into account by the nuclear installation. Finally, this paper establishes a feedback on the positioning of the criticality specialist regarding the controls performed on nuclear facilities