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Ellenberg, Venkatesh, and Westerland have established a weak form of thefunction field analogue of the Cohen--Lenstra heuristic, on the distribution ofimaginary number fields with $\ell$-parts of their class groups isomorphic to afixed group. They first explain how this follows from an asymptotic point countfor certain Hurwitz schemes, and then establish this asymptotic by using theGrothendieck--Lefschetz trace formula to translate it into a difficulthomological stability problem in algebraic topology, which they nonethelesssolve.  These are the notes accompanying my talk at the S\'eminaire Bourbaki, whichfocus on the remarkable homological stability theorem for Hurwitz spaces.

Homology of Hurwitz spaces and the Cohen--Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)