Alexander Gaifullin (Steklov Mathematical Institute of RAS, Moscow)
  On the bellows conjecture in spaces of constant curvature.

A flexible polyhedron in an n-dimensional space of constant curvature is an (n-1)-dimensional closed polyhedral surface that can be deformed continuously so that every its face remains congruent to itself during the deformation, but the deformation is not induced by an ambient rotation of the space. Intuitively, one may think of a flexible polyhedron as of a polyhedral surface with faces made of some rigid material and with hinges at codimension 2 faces that allow dihedral angles to change continuously.

The bellows conjecture stated by Connelly in 1978 asserts that the volume of any flexible polyhedron in dimensions greater than or equal to 3 is constant during the flexion. (Originally this conjecture was stated for the three-dimensional Euclidean space.) The bellows conjecture was proved in Euclidean spaces of all dimensions (Sabitov for n=3, and the author for n≥ 4). In the talk, we shall present the author's recent results on the bellows conjecture in non-Euclidean spaces. Namely, it turns out that the bellows conjecture is false in spheres of all dimensions and is true in odd-dimensional Lobachevsky spaces.