Pepijn researched topics related to Persistent Homology, a central tool in Topological Data Analysis (TDA). His thesis consists of three papers touching on three different aspects of TDA: interpretability, stability and time-varying data.
The first paper addresses the question of interpreting the output of persistent homology in terms of the original data. Interpreting what persistent homology outputs is difficult, particularly in higher dimensions when human intuition falters. The ideas in this project were inspired by classical ideas such as the Thurston norm in geometric topology and can be seen as "persistent" extensions. This project is a joint effort with VU mathematician Magnus Botnan, who was Pepijn's daily supervisor.
The second paper addresses that persistent homology is notoriously sensitive to outliers. Together with Lucas Slot, Pepijn used the theory of Christoffel-Darboux kernels to devise a modification of persistent homology, which is provably robust to outliers and efficiently computable for low-dimensional data. This paper is also the first to bridge these two seemingly unrelated fields of mathematics.
The final project Pepijn has been working on deals with developing persistent homology for time-series data. Building on previous work by one of his coauthors, the goal of this project has been to develop a persistence module for time-series data with better representation-theoretical properties than previous approaches.
During his PhD, Pepijn has had the opportunity to go on research visits to Munich and Oslo and attend various workshops and conferences worldwide, including Kyoto, Lausanne, Vienna, Oxford, and Dallas.
In February, Pepijn will begin his postdoctoral work at KTH in Stockholm.
Pepijn's thesis is available from this link: https://research.vu.nl/en/publications/topologically-optimal-bounding-chains-for-persistent-homology-and