Commuting difference operators, spinors on hyperelliptic curves and
asymptotics of generalized orthogonal polynomials: Part I"

Speaker: M.Y. Mo


Abstract

We review Its-Deift-Zhou steepest descent method in a more  geometrical
rather than analytical form which uses the notion  of  admissible Boutroux
curves;  this is a hyperelliptic curve endowed  with a special harmonic
function whose domains of positivity/ negativity satisfy certain
topological requirements.

Using this notion we construct strong asymptotics for certain  polynomials
orthogonal w.r.t. a complex weight $e^{-1/\hbar V(x)} dx$  (where $V(x)$ is
a (complex) polynomial)  on certain curves in the  complex plane; the
zeroes of these polynomials become dense on Jordan  arcs in the plane
forming a forest of loop-free trivalent trees.