Boris Shapiro: On spectral polynomials of the Heun equation and their generalizations

The famous Heun equation has the form:

Q(x) y" + P(x) y' + V(x) y = 0,

where Q, P, V are polynomials of degree 3, 2 and 1 respectively. Fixing Q and P one can show that for a given positive integer N there exist N + 1 linear polynomials V such that the latter equation has a polynomial solution of degree N. One can plot their N + 1 roots in the complex plane and ask what happens with this (N + 1)-tuple of points when N → ∞. The answer is very intriguing and will be explained in the lecture. (This recent result is due to Kouichi Takemura, Milos Tater and myself.) An analogous problem for operators of higher order is widely open.