Fields that require both CS and pure math I'm mainly a CS major, but I also want to learn more advanced  mathematics. I see lot of cross over between CS and applied math, same can't be said about pure math.
Definition of 'advanced': Something a pure math major learns during and after late undergraduate years.
What are the fields that combines advanced mathematical topics with computer science? 
Theoretical computer science and computational group theory are good examples.

Computational algebraic geometry is the first example that comes to mind. It's used in phylogenetics, see here (you can also look up the book Algebraic Statistics for Computational Biology). It also shows up in robotics and control theory, but I know a lot less about that.
Polytopes (see Coxeter - Regular Polytopes or Grunbaum - Convex Polytopes) also appear in a lot of applications. The only specific one that comes to mind at the moment is the problem of figuring out whether an object (like a robot) can traverse a given maze. The algorithms use the convex hull of the robot to test whether or not it could traverse the maze. I'm sure there are a lot of people more knowledgeable than me on the subject.
