Bram Bekker

HB 04.240

Mekelweg 4

Delft, the Netherlands

I am a PhD candidate at the Discrete Mathematics and Optimization group at TU Delft in the Netherlands. I obtained a bachelor’s degree in physics and a bachelor’s and master’s degree in mathematics at Radboud University, Nijmegen.

Currently, all my projects revolve around problems in extremal geometry, i.e. problems of the form “What is the biggest set in an n-dimensional euclidean space with property X?”. I work on finding upper bounds on the sizes of these sets by exposing such a problem as an independent set problem on an infinite (i.e. topological) graph.

During my master’s I visited the George Mason Geometry Lab in Virginia, USA, for an honors program. In this project, I made computer visualizations of buildings for algebraic groups over finite and p-adic fields. We wrote a nice paper on this for the Journal of Mathematics and the Arts.

selected publications

Note
On the convergence of the k-point bound for topological packing graphs

Bram Bekker, and Fernando Mário Oliveira Filho

2023

Abs arXiv Bib

We show that the k-point bound of de Laat, Machado, Oliveira, and Vallentin, a hierarchy of upper bounds for the independence number of a topological packing graph derived from the Lasserre hierarchy, converges to the independence number.
@unpublished{bekker2023convergence, title = {On the convergence of the $k$-point bound for topological packing graphs}, year = {2023}, author = {Bekker, Bram and de Oliveira Filho, Fernando Mário}, url = {https://doi.org/10.48550/arXiv.2306.02725}, doi = {2306.02725}, }
Preprint
Optimization hierarchies for distance-avoiding sets in compact spaces

Bram Bekker, Olga Kuryatnikova, Fernando Mário Olivera Filho, and Juan C. Vera

2023

Abs arXiv Bib

Witsenhausen’s problem asks for the maximum fraction αn of the n-dimensional unit sphere that can be covered by a measurable set containing no pairs of orthogonal points. The best upper bounds for αn are given by extensions of the Lovász theta number. In this paper, optimization hierarchies based on the Lovász theta number, like the Lasserre hierarchy, are extended to Witsenhausen’s problem and similar problems. These hierarchies are shown to converge and are used to compute the best upper bounds for αn in low dimensions.
@unpublished{Bekker2023OptimizationSpaces, title = {{Optimization hierarchies for distance-avoiding sets in compact spaces}}, year = {2023}, author = {Bekker, Bram and Kuryatnikova, Olga and de Olivera Filho, Fernando Mário and Vera, Juan C.}, url = {https://doi.org/10.48550/arXiv.2304.05429}, doi = {2304.05429}, }