Liviu Nicolaescu : Counting Morse functions on S^2

Posted by Laurentiu Maxim , part of the Departmental Colloquium.

At

March 16, 2007, 3 p.m.

In

SEO 636

Abstract

Two Morse functions on a smooth manifold M are called equivalent if we can obtain one from the other by global changes of coordinates on M and R (the real numbers). V.I. Arnold has computed the number of such equivalence classes when M=S^1, and formulated a precise conjecture concerning the asymptotics of these numbers when M= S^2. I will explain how to describe the generating function of the numbers of Morse functions on S^2 in terms of certain elliptic integrals, and then how to use this to prove Arnold's conjectured asymptotics.