Taras Panov (Moscow State University)
A family of closed manifolds is cohomologically rigid if the manifolds in the family are distinguished up to homeomorphism by their cohomology rings. Generally being a rare phenomenon, cohomological rigidity may be established for some families of manifolds arising in toric topology (Bott towers, toric and quasitoric manifolds, and moment-angle manifolds). There is also a related combinatorial concept of cohomological rigidity for simple polytopes: a polytope P is cohomologically rigid if its combinatorial type is determined by the integral cohomology ring of any (quasi)toric manifold over P.
We shall discuss several results on cohomological rigidity of toric families and certain polytopes, and suggest some open problems.