Seminar in Real and Complex Geometry

Thursday, December 29, 2022, 16:15-17:45, online via zoom




Elena Kreines
(Tel Aviv University)

Embedded graphs on Riemann surfaces and beyond


Abstract
             

This talk is based on the joint works with Natalia Amburg and George Shabat. The subject of the talk lies on the intersection of algebra, algebraic geometry, and topology, and produces new interrelations between different branches of mathematics and mathematical physics. The main objects of our discussion are so-called Belyi pairs and Grothendieck dessins d'enfants. Belyi pair is a smooth connected algebraic curve together with a non-constant meromorphic function on it with no more than 3 critical values. Grothendieck dessins d'enfants are tamely embedded graphs on Riemann surfaces. The interrelations between Belyi pairs and dessins d'enfants provide the new way to visualize absolute Galois group action, new compactifications of moduli spaces of algebraic curves with marked and numbered points, new way to visualize some classical objects of string theory, mathematical physics, etc. I plan to present a brief introduction to the theory with emphasize on the geometrical aspects as well as several recent results and useful examples. Among the examples we compute the Belyi pair for the dessin provided by the natural cell decomposition of the orientation covering of the moduli space of genus zero real stable curves with 5 marked points. In particular, we prove that the corresponding Belyi function lies on the Bring curve.