My main focus is on combinatorial optimization, including the related fields integer programming and the more general convex optimization.
All kinds of discrete mathematics appeal to me, though. I also enjoy combinatorics and abstract algebra, especially the computational types.
Programming and experimentation helps a lot while doing any kind of math. I'll list some software packages, that I enjoy using. All of them are either free software or available under some kind of academic license.
Currently, I'm working on pseudo-randum enumeration of combinatorical objects. There's already some code.
I recently finished my diploma thesis in mathematics. It's about the minimum linear arrangement problem, a classic problem in combinatorial optimization. We developed a new approach to solving the problem with a branch-and-cut algorithm using betweenness variables.
A branch-and-cut algorithm with betweenness variables for the Linear Arrangement Problem
Here are some slides for talks I've given in university seminars. Most of them are in German and cover subjects that rather fit computer science, because math talks make use of the blackboard.