Andrei Gabrielov, Purdue University – Special Colloquium

Title: Classification of Spherical and Circular Quadrilaterals

Abstract: 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 where it has conical singularities, 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. This is a very old problem, related to the properties of real solutions of the real Heun equation (a Fuchsian differential equation with four real singular points and real coefficients). 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 interior 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.

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 multi-colored nets, non-algebraic analogs of dessins d’enfants. If time permits, I’ll tell about classification of circular quadrilaterals (with the sides mapped to not necessarily geodesic circles on the sphere), and about connection between isomonodromic deformation of a Fuchsian differential equation with 5 singular points and solutions of the Painleve VI equation. This connection allows one to represent a real solution of the real Painleve VI equation by a sequence of nets of circular quadrilaterals and special (with one angle equal to $2\pi$) circular pentagons.​