Enumeration of planar maps: from the quadratic method to quantum gravity
Among the many contributions of William Tutte is the enumerative theory
of planar maps, which he laid down in his series of "Census" papers in
1962-1963. Unexpectedly, his results had an echo in theoretical physics
in the eighties, where the enumeration of maps was first connected with
random matrix theory, and then applied successfully to the study of
two-dimensional quantum gravity (maps being viewed as discrete random
surfaces). Since the turn of this century, the subject further developed
into the bijective theory of planar maps: in addition to giving
elementary proofs of Tutte's results, this allowed to gain insight into
the geometry of random planar maps, culminating in the definition of the
Brownian map and related deep concepts in probability theory. I will
attempt to present a (partial and personal) overview of this fascinating
story.