Toric topology in old and new problems from various fields of mathematics

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.