Title: Algebraic aspects of the Penrose tiling

Abstract: Penrose tiling is one of the most famous examples of aperiodic order. Unlike for periodic tilings, there is no obvious way to associate a group with it, and partial symmetries, defined by inverse semigroups, are more natural. I will show that a natural semigroup associated with Penrose tiling, as defined by J. Kellendonk, is generated by a finite automaton and is an example of a self-similar contracting inverse semigroup. The semigroup acts on a Cantor set, and one can consider its topological full group, which is, in this case, a finitely generated virtually simple group. I will give other interpretations of this group, and discuss some open problems related to it.