Pseudo-Anosov constructions and Penner's conjecture

In this first talk, we give an introduction to Penner’s construction of pseudo-Anosov mapping classes. Penner conjectured that all pseudo-Anosov maps arise from this construction up to finite power. We give an elementary proof (joint with Hyunshik Shin) that this conjecture is false. The main idea is to consider the Galois conjugates of pseudo-Anosov stretch factors.