The research interests of the group of Aichholzer includes discrete and computational geometry, data structures and algorithms, combinatorial properties of geometric and topological graphs, and enumeration and reconfiguration algorithms.

On the algorithmic side we are especially interested in combinatorial properties of triangulations and related data structures to obtain efficient algorithms for transforming or counting/enumerating these configurations. In the area of discrete geometry we consider typical Erdős-type problems on empty and non-empty convex polygons spanned by (colored and uncolored) point sets in the plane.  And on graphs the focus lies on investigating the similarities and especially differences between geometric graphs (vertices are points in the Euclidean plane and edges are segments connecting two points) and general (topological) drawings of graphs. Here the minimum crossing number of complete graphs is one of the central open questions. Moreover, the relation of crossing minimal drawings to combinatorial structures, like k-edges and order types in the geometric case, or rotation systems in the topological setting, are of special interest.

On the teaching side our group covers algorithms, datastructurs, discrete geometry and combinatorial ascpects, but also lectures like "Algorithms and Games" or "Enumerative Combinatoric Algorithms" and "Discrete and Computatioal Geometry" are covered. This is complemented with the usual Seminar and Project courses.

And finally a more philosophical statement on how we see our scientific work (see also TU-Graz researh 2024-1): Basic research is timeless and largely free from passing fads. The knowledge gained remains valid in the long term – there is something very reassuring about this, especially in today’s fast-moving world.