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.
|