New facets: constructions of polytopes, combinatorics of fullerenes, geometry of Lobachevsky space

The article of V. M. Buchstaber and N. Ray, “An invitation to toric topology: vertex four of a remarkable tetrahedron” (Toric topology, Contemp. Math., 460, AMS, Providence, RI, 2008, 1–27) presented toric topology, as part of equivariant algebraic topology, via its relationship with combinatorics, algebraic geometry and symplectic geometry. Over the past years, these ties have been deepened, long-standing problems have been solved, and new directions of the research have been opened. Developing the idea of a toric tetrahedron, we present a polyhedron containing new vertices: discrete geometry, hyperbolic geometry, combinatorial group theory, mathematical theory of fullerenes. We will demonstrate new facets of interaction between various areas of mathematics and toric topology.