Multiple orthogonal polynomials generalize the notion of standard orthogonality. Originated in approximation and number theory, they find applications in several branches of analysis. However, their study requires the development of new tools and notions. In particular, it is know that their asymptotics is governed by measures on the complex plane that constitute saddle points for certain energy functionals. Furthermore, these measures can be characterized by the fact that their Cauchy transform satisfies the same algebraic equation almost everywhere. This connects the topic with another exciting topic of the geometric function theory: quadratic differentials on a Riemann surface.
We illustrate these ideas with an example of multiple orthogonal polynomials with respect to the cubic weight.This is a joint work with Guilherme L. F. Silva (KU Leuven, Belgium).