ИСТИНА

The subject of the talk lies in 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. Their connections 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. Introduction to the theory will be given including the modern applications. In particular, we will discuss the generalized Tchebyshev polynomials and their geometry, visualization of the Galois group action and its invariants, Grothendieck dessins d'enfants on reducible curves, numerous examples. This talk is based on the joint works with Natalia Amburg and George Shabat.