Working together
Before listing some concrete problems that could make a good bachelor's or master's thesis, I would like to comment on our mutual expectations.
A student thesis depends not only on my own areas of interest, but also on the background and interests of the student. So, in principle, we can find a common problem of interest to work on that is quite different from what is listed here.
Depending on your level, the outcome of a thesis can vary from understanding a research-level project (bachelor's thesis) to becoming an expert in a particular area of research (PhD thesis). Some theses can be finished by implementing research ideas in concrete software.
Independent of the scale and final outcome of a thesis that we may mutually agree upon, it is expected that at the end you have learnt about methods and objects that are of theoretical as well as practical importance. In general, they will relate to geometry and polyhedra.
Here are some concrete problems for a bachelor's or master's thesis.
Uniform polytopes generalize the notion of regular polygons in the plane. A way to construct a large class of them is known (the Wythoff construction). Construct small extended formulations of uniform polytopes, or at least the ones obtainable from the Wythoff construction.
The goal of the project would be to study the perfect graph theorem and try to work out an extended formulation for the convex hull of all independent sets. For general graphs, no small extension exists. Does one exist for perfect graphs?
Write a software library that can be used to efficiently do computations over polytopes by exploiting extended formulations. Ideally, your library would extend the functionality of another popular tool for polyhedral computations, such as lrs, cdd, or polymake.