Normal Surfaces in 3-Manifolds: Algorithms, Experiments and Questions

The theory of normal surfaces, introduced by Kneser in the 1920s and further developed by Haken in the 1960s plays a crucial role in 3-manifold topology. Normal surfaces allow topological problems to be translated into algebraic problems or linear programs, and they are the key to many important advances over the last $50$ years, including the solution of the unknot recognition problem by Haken, the 3-sphere recognition problem by Rubinstein and Thompson and the homeomorphism problem by Haken, Hemion and Matveev.

In this talk, I will summarise Haken's blueprint for algorithmic 3-manifold topology, discuss the  "difficulty" of the computational problems from a
theoretical and experimental perspective and state some open questions and challenges.