I'll introduce the connection between monodromy, in the usual topological/geometric sense, and Galois theory, especially how it appears in work related to the inverse Galois problem.

 

We'll calculate a few Galois groups over C(t) using these ideas, then see how some arithmetic input allows one to carry the calculation down to Q(t) by calculating a particular outer action, and from there specialize to Q and number fields.

 

There will be pictures!