Fullerenes, geometric combinatorics and hyperbolic geometry

At present, the fullerene classification problem is well-known and is important due to the applications in chemistry, physics, biology and nanotechnology. A mathematical fullerene is a three-dimensional convex simple polytope with all 2-faces being pentagons and hexagons. The talk is devoted to fundamental connections between the mathematical theory of fullerenes and geometric combinatorics, graph theory, the four-color problem, Coxeter groups, Pogorelov polyhedra, and Lobachevsky geometry. We present applications to the fullerene classification problem and to the classical problem of three-dimensional hyperbolic manifolds.