15:30
Given a triangulated 3-manifold, can we decide whether it is hyperbolic? In general, no efficient algorithm for answering this question is known; however, the problem becomes more manageable if we restrict our attention to specific classes of 3-manifolds. In this talk, I will discuss how to certify that a triangulated fibred 3-manifold is hyperbolic, in polynomial time in the size of the triangulation and in the Euler characteristic of the fibre. The argument relies on the theory of normal surfaces, as well as several previously known certification algorithms, of which I will give a survey. I will also mention, time permitting, a recent algorithm to decide if an element of the mapping class group of a surface is pseudo-Anosov in polynomial time, which is used in the certification procedure.