Voronoi’s conjecture about parallelohedra

Voronoi’s conjecture states that any parallelohedron, i.e., a polyhedron that allows for a partitioning of a space into its parallel copies, is affine equivalent to a Dirichlet-Voronoi polyhedron for some lattice.
Certain results concerning Voronoi’s conjecture will be presented in this talk. In particular, the talk will include a plan and possibly some details of the proof of Voronoi’s conjecture in a new special case where the surface of the parallelohedron remains connected when a certain class of facets of codimension 2 is removed.
The talk is based on the results obtained jointly by the speaker, A. Gavrilyuk and A. Magazinov.