Talk information

Date: Sunday, March 10, 2024
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Elena Kreines (TAU)
Title: Combinatorial questions in dessin d'enfant theory

Abstract:

In the talk we consider Grothendieck dessins d’enfants, i.e., connected embedded graphs on surfaces such that after extracting the graph from the surfaces it is homeomorphic to a disjoint union of open discs. For example, any plain tree is a dessin d’enfant on a Riemann sphere. However, the only regular dessins d’enfants on the sphere are Platonic solids.

Dessins d’enfants are interesting by themselves, and also due to their natural relation to so-called Belyi pairs, i.e., non-constant meromorphic functions with at most 3 critical values defined on algebraic curves. This relation was firstly discovered by Alexander Grothendieck, see [1,3] for the details. It provides plenty of non-trivial applications in algebra, geometry and mathematical physics.

I plan to give an introduction to the theory, emphasizing various combinatorial questions. Among the other results I present the solution of the problem from [2] concerning extremal Hamiltonian covering by polygonal paths for some special class of dessins d’enfants appearing in genetics.

The talk is based on the joint results with A. Guterman, N. Jonoska, A. Maksaev, N. Ostroukhova.

[1] N. Adrianov, F. Pakovich, A. Zvonkin, Davenport, Zannier Polynomials and Weighted Trees, Mathematical Surveys and Monographs 249, American Mathematical Society, Providence (2020).
[2] J. Burns, E. Dolzhenko, N. Jonoska, T. Muche, M. Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, 161 (2013), 1378-1394.
[3] S. K. Lando, A. K. Zvonkin, Graphs on surfaces and their applications, with an appendix by D. Zagier. Encycl. of Math. Sciences 141, Springer (2004).