Hurwitz numbers with completed cycles

Completed cycles are some very natural objects in the representation theory of symmetric group. Hurwitz numbers with completed cycles is a natural generalization of the usual Hurwitz numbers, where the simple critical points are replaced by more complicated singularities. These new singularities are especially important from the point of view of the Gromov-Witten theory of target curves, where they occur naturally.

I'll try to explain several possible definitions of Hurwitz numbers with completed cycles and to make a survey of known results and open questions for single and double Hurwitz numbers with completed cycles. In particular, I'll talk about
1) polynomiality properties;
2) cut-and-join operators and, as a consequence, a tropical interpretation;
3) relation to Gromov-Witten theory of target curves;
4) relation to the intersection theory of the moduli space of $r$-spin structure on curves (Zvonkine's conjecture)
5) if time permits, relation to random matrix models

The talk will follow recent joint papers with Loek Spitz and Dimitri Zvonkine.