The Hermitian 1-matrix model has been known for 30 years to satisfy the
KP hierarchy. It is an infinite set of partial differential equations,
here with respect to the coupling constants of the model. At least three
proofs of this statement are known. Other systems in two-dimensional
enumerative geometry such as, famously, intersection numbers of the
moduli space of Riemann surfaces, satisfy the KP hierarchy. In this
talk, I will introduce the KP hierarchy from the point of view of the
Sato Grassmannian as an orbit of an action of GL(\infty). This will then
give us the opportunity to present the proof of Kazarian and Zograf that
the generating function of bipartite maps satisfies the KP hierarchy. I
will then focus on constellations, which are generalizations of
bipartite maps, and are also the dual to the jackets of the bipartite
edge-colored graphs of unitary-invariant tensor models. Constellations
are known to also satisfy the KP hierarchy but only a proof via
algebraic combinatorics is known. I will report some progress towards
understanding their integrable properties using
Tutte/Schwinger-Dyson--like equations. Only this last part is an
original contribution.
Joseph Ben Geloun
Fabien Vignes-Tourneret
