Seminar given to the IWR section at University of Heidelberg.

Fullerenes are carbon molecules that form polyhedral cages with surfaces similar to graphene sheets, and whose bond structures are exactly those planar cubic graphs that have only pentagon and hexagon faces. These simple rules induce a mathematical description of fullerenes that is both simple and extensive, and which ties together combinatorics, graph theory, and discrete di↵erential geometry. Fullerenes are also potentially very interesting from a practical stand point: Their geometrical properties make some fullerenes incredibly strong, while the 12-pentagon rule allows opening the hollow shell by changing a single pentagon to a hexagon. This makes them interesting as delivery systems, reflected by their occurrence in nature in other forms in addition to carbon structures. For example, a number of virus capsids, which protect and deliver genetic material, are fullerene analogs. Many chemical and physical properties of fullerenes are uniquely determined by their bond graphs, and can be approximated very rapidly directly from the graph. This is important when we wish to identify from millions of fullerene isomers a select few that possess desired properties, such as stability, shapes, malleability, electrical properties, etc. In this talk, I will describe a method that derives one such property, the three-dimensional shape of a fullerene, from its graph by way of its induced Riemannian surface metric, at speeds that allows automatic study of millions of fullerene isomers. The method is in an experimental stage, but has shown promising preliminary results.