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Topology of moduli spaces of Riemann surfaces

Periodic Reporting for period 4 - MODULISPACES (Topology of moduli spaces of Riemann surfaces)

Reporting period: 2022-07-01 to 2022-12-31

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.
What I consider to be the biggest achievements of the project are:
- With Bergström, Diaconu and Westerland we have calculated the stable homology of the braid groups with symplectic coefficients, and showed that the answer agrees exactly (in a precise sense) with the expected asymptotics of moments of families of quadratic L-functions. This is a very unexpected link between two completely different areas of mathematics. From our theorem it follows that if a version of "Borel vanishing theorem" holds for the braid groups, then this proves the function field case of the conjectural formula for asymptotics of moments. The paper appeared on the arXiv just after the end of period 4.
- With Campos, Robert-Nicoud and Wierstra, we proved the following theorem: suppose given two commutative dg algebras over a field of characteristic zero, and suppose that they are quasi-isomorphic as dg algebras. Then they must also be quasi-isomorphic as commutative dg algebras. The statement is deceptively simply, and it has been a folklore problem in rational homotopy theory since the 70's. From Koszul duality, this has the following striking consequence: two nilpotent Lie algebras are isomorphic if and only if their universal enveloping algebras are isomorphic as associative algebras. This is the first major progress on the "isomorphism problem" for Lie algebras, which also has been open since the 70's. In a followup paper we proved a closely related statement, also resolving a folklore open problem: the forgetful functor from the homotopy category of CDGA's to the homotopy category of DGA's is faithful, over a field of characteristic zero. Although seemingly outside the scope of the project, the proofs go by carefully studying the geometry of the moduli spaces of CDGA/DGA structures on a given chain complex.
- Erik Lindell (PhD student) has proved a decade-old conjecture of Djament, giving a complete calculation of the stable homology of Aut(F_n) with bivariant polynomial coefficients. The proof is beautiful and shows the strength of topological and homotopical methods.
- Josefien Kuijper (PhD student) has proved a really fundamental result in algebraic geometry, more specifically regarding cohomology of algebraic varieties. In algebraic geometry, cohomology with compact support is often defined by choosing an arbitrary compactification of your variety, but the groups are independent of this choice. When can "compact support invariants" be defined in this way in general? When does it make sense on the chain level? Informally it was understood that this is well defined if your invariant satisfies descent for the "abstract blow-up topology" on the category of varieties. Josefien turned this into a precise statement -- a very elegant equivalence of infinity-categories -- and proved it.
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