The Kontsevich matrix integral (aka “Matrix Airy function”), in the appropriate formal limit, generates the intersection numbers on $mathcal M_{g,n}$. In the same formal limit it is also a particular tau function of the KdV hierarchy; truncation of the times yields thus tau functions of the first Painlev’e hierarchy. This, however is a purely formal manipulation that pays no attention to issues of convergence. The talk will try to address two issues: Issue 1: how to make an analytic sense of the convergence of the Kontsevich integral to a tau function for a member of the Painlev’e I hierarchy.
Which particular solution(s) does it converge to? Where (for which range of the parameters)? There exist now generalizations due originally to Penner (and used recently by Alexandrov, Buryak, Solomon, Tessler, Pandharipande, Brezin and Hikami) that generate similar intersection numbers on the moduli space of pointed Riemann surfaces with boundary (“open” intersection numbers). Some aspects are still conjectural, due mainly to the difficult definition of the moduli space itself. The approach to the integral that we propose leads to an isomonodromic description that allows a closed—form expression (although involving several level of nested sums) of all these numbers. This is joint work with Giulio Ruzza (SISSA) and based also on prior work with Boris Dubrovin and Di Yang.