# Spherical quadrilaterals and non-algebraic dessins d'enfants

A spherical polygon (membrane) is a bordered surface homeomorphic to a closed disc, with $n$ distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. Spherical polygons with $n=3$ and $n=4$ are called spherical triangles and quadrilaterals, respectively. Classification of spherical quadrilaterals is a very old problem, related to the properties of solutions of the Heun equation (second order linear ordinary differential equation with four regular singular points). The corresponding problem for generic spherical triangles, related to the hypergeometric equation, was solved by Felix Klein more than 100 years ago, while non-generic cases were completely classified as late as 2011.

When all angles at the corners are integer multiples of $\pi$, classification of spherical quadrilaterals is equivalent to classification of rational functions with four real critical points, a special case of the B. and M. Shapiro conjecture, related to real Schubert Calculus and control theory. Rational functions with real critical points can be characterized by their nets, combinatorial objects similar to Grothendieck's dessins d'enfants. Similarly, spherical polygons can be characterized by their multi-colored nets, which can be understood as non-algebraic dessins d'enfants. This reduces the problem to a combinatorial problem of classification of nets.

If time permits, I'll tell about recent progress in classification of circular quadrilaterals (with the sides mapped to some circles on the sphere, not necessarily geodesic), and about connection between isomonodromic deformations of the Heun equation and solutions of the Painlevé VI equation. This connection, discovered by Richard Fuchs in 1905, allows one to study real solutions of the Painlevé VI equation by using sequences of the nets of special (with one angle equal $2\pi$) circular pentagons.