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Motives, mapping class groups, and monodromy

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arxiv 2409.02234 v1 pith:QNMGA7VQ submitted 2024-09-03 math.AG math.GTmath.NT

Daniel Litt This is my paper · ORCID
classification math.AGmath.GTmath.NT
keywords algebraicclassconjecturesdevelopmentsdifferentialequationsgeometrymapping
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We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic varieties, we study mapping class group actions on character varieties and their algebro-geometric avatar -- isomonodromy differential equations -- from the point of view of both complex and arithmetic geometry. We then collect some open questions and conjectures on these topics. These notes are an extended version of my talk at the April 2024 Current Developments in Mathematics conference at Harvard.

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  1. Density of integral points in the Betti moduli of quasi-projective varieties

    math.AG 2025-06 conditional novelty 7.0 of 10

    Potential density of integral points is established for relative SL2 and PGL2 character varieties of all smooth quasi-projective complex varieties with snc compactification.

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