In this talk, we are going to uncover a paper written by E. Artin one hundred years ago, in 1924, entitled (in German) "Ein mechanisches System mit quasi-ergodischen Bahnen". "A Mechanical System with Quasi-Ergodic Trajectories " The connection between continued fractions and geodesics in the hyperbolic plane goes back to Humbert (1916) and Smith (1877) and it is used in an ingenious way by E. Artin in the above paper provided the first example of a dense geodesic trajectory on a special Riemann surface, the so-called "modular surface". In the first part of the talk, we will try to cover some preliminaries and generalities about discrete subgroups of the group of isometries of the hyperbolic plane and their relations to the various tessellations of the hyperbolic plane / Poincare disc. In the second part, we will try to talk about a special tessellation called Farey tessellation of the hyperbolic plane and its relation to continued fractions. One of the applications of this connection is the existence of a dense geodesic on the modular surface. If time permits, we will also try to discuss the relationship between the Diophantine approximation of an irrational number and the behavior of the corresponding geodesic on the modular surface.

Online via Zoom:
Meeting ID: 897 3971 8132
Passcode: 362880