Computational topology primarily focuses on the algorithmic study of the topological properties of spaces. Among these, topological invariants, i.e. properties of a space that are preserved under smooth deformation, play a primary role, and allow a characterization of topological spaces.
The lecture will gently introduce participants to topological concepts – such as topological spaces, topological equivalence, and invariants – and discuss approaches for their algorithmic realization. A strong focus is placed on applications of topological techniques in diverse application, ranging from Computer Graphics over molecular modeling to scientific data analysis. As a central tool, we will study so-called simplicial complexes, which can be viewed as a generalization of triangulations known from Computer Graphics, and discuss how topological algorithms can be efficiently realized on these. We will see that they serve as a powerful tool and data structure to model real world problems in a topological sense. Having studied these basic concepts, the lecture will highlight the role of topological techniques in data analysis of both scientific and non-scientific data. Here, the Reeb graph and other related constructs, its properties and construction, will be the final topic of the lecture.
Exercises will consist of both homework assignments that will familiarize participants with topological techniques and further illustrate the concepts introduced in the lecture. Moreover, short programming projects will provide insights into practical application of computational topology.