DC 8 – Exploring the synergy between optimisation hierarchies and approximation kernels Project Title: Exploring the synergy between optimisation hierarchies and approximation kernelsDoctoral Candidate: N.N.Host Institution: Tilburg UniversitySupervisors: Etienne de Klerk, Frank Vallentin Objectives: Approximation kernels are ubiquitous in approximation theory, famous examples being the Dirichlet (reproducing) kernel, and the Jackson kernels. In optimisation, these kernels also play an important role in hierarchies for polynomial optimisation problems, including MINO problems involving polynomials. In this project we would like to bridge the divide between the two communities, by applying insights from approximation theory to optimisation and vice versa. Thus, we aim to construct new multivariate approximation kernels using optimisation techniques, and conversely, use kernels from the approximation theory literature to analyse related optimisation hierarchies. Finally, we intend to investigate applications to clustering in data science. Indeed, the application of kernels in this area is an important emerging research topic. Expected Results: Theoretical results and software on approximation kernels and their use in data science and optimisation. Planned secondment: 6 months at UoC (F. Vallentin) to learn about Hilbert-Schmidt kernels and their role in optimisation during the 2nd year; 1 month at Mosek (E. Andersen) to gain experience with optimisation software development, in the summer of the 2nd year. Degree awarding institution: Tilburg University Note: Due to the 4-year PhD programme at Tilburg University, the contract of DC 8 will be extended to 4 years.