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Topology of moduli spaces of Riemann surfaces

Periodic Reporting for period 2 - MODULISPACES (Topology of moduli spaces of Riemann surfaces)

Reporting period: 2019-07-01 to 2020-12-31

"The goal of this project is to understand the geometry and topology of the space of Riemann surfaces. This space, a so-called ""moduli space"", has been studied for more than 150 years and is still largely mysterious. The specific questions outlined in the proposal are all about relating the moduli space to other long-standing mathematical problems: the irrationality of zeta values, the arithmetic theory of modular forms, and the mathematics of string theory."
"Johan Alm (postdoc) has given a geometric reinterpretation of what is known as Drinfeld's conjecture. This result significantly elucidates the (in part still conjectural) relation between the Grothendieck-Teichmueller Lie algebra and (motivic) multiple zeta values.

Erik Lindell (PhD student) has made progress on understanding the homology of the Torelli group. This group and its (co)homology has been studied for a very long time from many perspectives and progress has been notoriously scarce; many basic questions are widely open. Lindell has obtained a ""lower bound"" for how large the Torelli group can be, by using so-called ""abelian cycles"" to construct lots and lots of nonvanishing homology classes. Lindell's construction is remarkably simple and flexible, and vastly improves upon existing constructions of stable homology classes.

With Adrian Diaconu (Minnesota), I have realized a surprising connection between the cohomology of certain moduli spaces with coefficients of multiple Dirichlet series. Moduli spaces are in general closely intertwined with automorphic objects but the link between the two that we have found here seems to be of an entirely different nature. If it all works out, theorems in topology will provide precise asymptotics of these Dirichlet series and error terms."
"The biggest goal from now to the end of the project is that together with Adrian Diaconu, Craig Westerland and Jonas Bergström we will hopefully derive asymptotics for coefficients of multiple Dirichlet series from the stable topology of the moduli of hyperelliptic curves. This would be a great result if it all works out. I also plan to write down my own results about the stable cohomology of the moduli space of curves with twisted coefficients (which is long overdue). Lindell is at the moment trying to generalize his results about the Torelli group of mapping class groups to other ""Johnson homomorphisms""."