We discuss the recent Hodge--GUE correspondence conjecture on an explicit
relationship between special cubic Hodge integrals over the moduli space of stable
algebraic curves and enumeration of ribbon graphs with even valencies. We sketch
a proof of this conjecture based on the Virasoro constraints. We also discuss the
conjectural relationship between the cubic Hodge integrals satisfying the local
Calabi--Yau condition and the Bogoyavlensky--Toda hierarchy (aka fractional
KdV). The talk is based on a series of joint works with B. Dubrovin, S.-Q. Liu
and Y. Zhang.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/YRSM2017