Felix-Klein Kolloquium des Fachbereichs

This talk explores a question reminiscent of Mark Kac’s famous problem, “Can one hear the
shape of a drum?” — namely: can one determine the shape of a fullerene, a carbon molecule
built from pentagons and hexagons, by listening to its spectrum?
We begin with the study of random eigenvalues of regular lattices, in particular the hexagonal
lattice (graphene) and its dual triangular lattice. The number of closed paths of a given length
on these lattices forms a moment sequence of certain random variables. This perspective yields
explicit formulas for the probability density and characteristic functions of their spectral distribu-
tions. These formulas are derived from a novel integral identity involving the third powers of modi-
fied Bessel functions and provide a simple stochastic method to simulate the spectra of graphene-
like structures.
In the second part, we extend this framework to nanotubes, which can be viewed as “rolled-up”
graphene sheets determined by a chiral vector (p, q). We show that the corresponding spectral
distributions converge, as the circumference p + q → ∞, to that of the infinite hexagonal lattice.
This result establishes a precise mathematical bridge between finite fullerenes, nanotubes, and
the idealized structure of graphene.
Finally, we discuss recent progress and open problems on the spectral geometry of fullerenes.
Using the concept of local weak convergence, we conjecture that large random fullerenes appro-
ximate the spectral distribution of graphene, linking finite molecular cages to infinite lattices. This
raises fundamental questions: what is the structural role of the twelve pentagons, and to what ex-
tent does the spectrum determine the shape of a fullerene?
The talk is based on joint papers with Victor Buchstaber, Simon Coste, Pavel Ievlev, Svyatoslav
Novikov, Satoshi Kuriki, and illustrates how from simple lattices one can build a rich spectral the-
ory of carbon allotropes at the crossroads of combinatorics, probability, and mathematical physics.