Felix-Klein-Kolloquium: Can one hear the shape of a Fullerence?

The 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 distributions. These formulas are derived from a novel integral identity involving the third powers of modified Bessel functions and provide a simple stochastic method to simulate the spectra of graphene-like structures.

In the second part, this framework is extended to nanotubes, which can be viewed as “rolled-up” graphene sheets determined by a chiral vector (p, q). It will be shown 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, recent progress and open problems on the spectral geometry of fullerenes are discussed. Using the concept of local weak convergence, we conjecture that large random fullerenes approximate 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 extent 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 theory of carbon allotropes at the crossroads of combinatorics, probability, and mathematical physics.

Referent: Prof. Dr. Evgeny Spodarev, Universität Ulm

Zeit: 17:15 - 18:30 Uhr

Ort: Gebäude 48, Raum 210

Die Vorträge des Felix-Klein-Kolloquiums finden jeweils um 17:15 Uhr im Raum 210 des Mathematik-Gebäudes 48 statt. Zuvor gibt es ab 16.45 Uhr die Gelegenheit, die Sprecherin oder den Sprecher beim Kolloquiumstee in Raum 580 zu treffen.