Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over finite fields

In order to study$p$-adic étale cohomology of an open subvariety$U$of a smooth proper variety$X$over a perfect field of characteristic$p>0$, we introduce new$p$-primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of$X$depending on effective divisors$D$supported in$X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of$U$and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wildly ramified class field theory for the open subvariety$U$.

Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional

The paper presents a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert–Einstein functional on the space of “warped polyhedra” with a fixed metric on the boundary.The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.In the spherical space and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert–Einstein functional and the volume, as well as between both kinds of rigidity.We review some of the related work and discuss directions for future research.