The Dark-Matter Problem of the Moduli Space of Curves and its (partial) Tropical Solution

The moduli space of curves, M_g, is an object of pivotal interest in mathematics and theoretical physics. Even though it has been studied with tremendous effort, little is known about its topology. For instance, the total dimension of the cohomology of M_g grows quickly with g, but the known constructions for cohomology classes only account for a negligibly small part of this growth. The known existence of huge amounts of cohomology and the lack of explanation for it, is dubbed the 'dark-matter problem' of M_g. In this talk, I will review the state-of-the-art and a recent breakthrough by Chan-Galatius-Payne who used a tropicalization of the moduli space of curves to construct a new source of cohomology in M_g which quickly drafts all previously known sources.  Eventually, I will present new results which show that this new source still only accounts for a small part of the total cohomology.  The proof makes use of QFT inspired path-integrals.



Michael Borinsky


ETH Zurich