Embedded graphs on Riemann surfaces and their applications
This talk is based on the joint works with Suren Danielian,
Alexander Guterman, and Fedor Pakovich.
Grothendieck dessins d'enfants are tamely embedded graphs on Riemann surfaces. Up to the appropriate isomorphism, they are in one-to-one correspondence with Belyi pairs, which are smooth connected algebraic curves with non-constant meromorphic functions with no more than 3 critical values. The interrelations between Belyi pairs and dessins d'enfants provide the way to visualize absolute Galois group action, compactifications of moduli spaces of algebraic curves with marked and numbered points, and many other branches of mathematics and mathematical physics.
I plan to present a brief introduction to the theory with the emphasize on the combinatorial problems and then present one new application arising from the geometry of polynomials.