Many natural decision questions that arise in 3-dimensional topology involve the existence of certain embedded surfaces in a 3-manifold $M$. (Is a given knot trivial? Is a given link splittable? Is a given 3-manifold Haken?) While the collection of embedded surfaces (modulo isotopy) in $M$ is an infinite set, the techniques of normal surface theory allow us to algorithmically construct a finite set of these surfaces in $M$ that generate all others in a meaningful way. Hence algorithmically answering those kinds of questions can often be reduced to checking a finite set of possibilities.

This talk will focus on the standard exposition of normal surface theory, which entails showing how most "interesting" surfaces in a 3-manifold can be isotoped into a normal form and that the set of normal surfaces corresponds to the finitely generated solution set of a certain system of integer linear equations. I will then explain the general pattern for algorithms that use normal surface theory, and, if there is time, I will give an example application of these techniques to a specific 3-manifold decision problem.

I will do my best to define any terminology not covered by the core graduate math classes. The only requirement of the audience is that they be comfortable with pictures being used as proofs.