Colloquium Peter Smillie: Counting fullerenes with modular forms — an application of number theory to carbon chemistry

By Peter Smillie, Mathematical Institute, Heidelberg University

Fullerenes are polyhedral molecules made of carbon atoms, first discovered in 1985. A few isomers are naturally occurring and many more have since been synthesised, with numerous applications across materials science, biology, and medicine. One source of theoretical interest in fullerenes is that there are a lot of them; it is well known that if n is large, you can synthesise about n^9 (n to the 9th power) distinct isomers with exactly n carbon atoms.

As I will explain, the set of all isomers has a beautiful mathematical structure, corresponding to certain integral points in 18 dimensional space. A fundamental problem is to predict the chemical properties of an isomer from this mathematical description. This is the goal of a joint project with Ganna Gryn’ova at HITS.

I will present a surprising result, joint with Philip Engel, that there exists an exact formula for the number of distinct isomers with n atoms using modular forms, which are certain sequences with deep connections to the prime numbers. We are still working out the details, but this should reproduce tables that mathematical chemists have computed up to n= 300 or so, and let us calculate essentially instantaneously the number of isomers for any n.

Short CV:

Peter Smillie is a “Geometry Plus” Junior Professor at the University of Heidelberg. His primary research is in differential geometry, especially the study of minimal surfaces in symmetric spaces and moduli spaces of geometric structures. It was his work on the moduli space of polyhedra that led to his discovery with P. Engel in 2017 of the connection between polyhedra and modular forms. In ongoing joint work with F.

Bonsante and A. Seppi, he is developing new tools to study the asymptotic behavior of special surfaces in spacetime. In 2022, together with N. Sagman, he found a counterexample to the well-known Labourie conjecture in higher Teichmüller theory. He received his Ph.D. in mathematics from Harvard in 2018, and held post-doctoral positions at the IHES in Paris and at Caltech, before starting at Heidelberg in 2022.

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