Recent progress in Mathematics and Statistics: Alan Stapledon -- Local Motivic Monodromy

This is a combinatorics talk hiding in geometric clothing.  The clothing consists of the
local motivic monodromy conjecture, an analogue of the Weil conjectures.  Given a
polynomial f with integer coefficients, it predicts a remarkable relationship between
arithmetic properties (number of solutions to f = 0 modulo an integer) and topological
properties (eigenvalues of monodromy of the Milnor fibre of the complex hypersurface
defined by f).  The conjecture remains wide open in general.  Surprisingly, it remains
open for "generic" choices of f, even though there are well known and relatively simple
combinatorial formulas for all the relevant quantities.  This brings us to the heart of
the talk: for "generic" f, we relate the local motivic monodromy conjecture to a
long-standing open question of Stanley on triangulations of simplices.  Progress towards
the latter question then helps resolve the local motivic monodromy conjecture.  This is
joint work with Matt Larson and Sam Payne.