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I will talk about our recent joint work with Xavier Blot where we related the
intersection numbers of psi-classes on the moduli space of curves to the stationary
relative Gromov--Witten invariants of the complex projective line with an insertion of
the top Chern class of the Hodge bundle. The proof is based on the theory of DR
hierarchies, which gives a direct and explicit relation between the geometry of the
moduli space of curves and integrable systems of evolutionary PDEs. I will also try to
mention a development of this result, which involves what we called quantum
intersection numbers on the moduli space of curves.
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