Algebraic geometry endows the cohomology groups of moduli spaces of curves with additional structures, such as (mixed) Hodge structures and Galois representations. Standard conjectures from arithmetic, regarding analytic continuations of L-functions attached to these Galois representations, lead to striking predictions, by Chenevier and Lannes, about which such structures can appear. I will survey recent results unconditionally confirming several of these predictions and studying patterns in the appearances of motives of low weight. The latter are governed by the operadic structures induced by tautological morphisms and the cohomology of graph complexes.
Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.
For more information see: https://personal.math.ubc.ca/~jbryan/Zoominar-UBC-ETH/