Cycles of curves, cover counts, and central invariants

The first topic of this dissertation is the moduli space of curves. I define half-spin relations, specialising Pandharipande-Pixton-Zvonkine’s spin relations, to reprove Buryak-Shadrin-Zvonkine’s result on the dimension of the top Chow group for the open moduli space, and give new bounds for the lower groups. I also use these relations to reduce Faber’s intersection number conjecture to a combinatorial identity.
The second is Hurwitz numbers. I prove quasi-polynomiality for orbifold simple, weakly and strictly monotone, and spin Hurwitz numbers. In the first case, this was already known, via the Johnson-Pandharipande-Tseng formula, and in the other cases this is new. I also prove that triply mixed double Hurwitz numbers satisfy piecewise polynomiality.
For monotone and for orbifold spin Hurwitz numbers, I derive a cut-and-join-equation and prove topological recursion (for r = 2 and for genus zero, general r in the latter case). The former was known by Goulden-Guay-Paquet-Novak. The latter proves Zvonkine’s r-ELSV formula for these cases.
The third is integrable hierarchies. I reprove a result by Alexandrov, showing that triple Hodge integrals on the moduli space of curves satisfy the Kadomtsev-Petviashvili hierarchy, generalising Kazarian’s single Hodge integral proof.
I give a new and purely cohomological proof of Dubrovin-Liu-Zhang’s theorem, that integrable hierarchies described by dispersive semi-simple Poisson pencils are classified by central invariants, and I streamline Carlet-Posthuma-Shadrin’s proof that this is an equivalence.