Frank Vallentin has been awarded an NWO VIDI grant at the end of 2009 for his research project 'Semidefinite programming and harmonic analysis: Foundations and applications'. Since April 2009, Vallentin is assistant professor in the Optimization and Systems Theory research group of the Delft University of Technology. He remained part-time affiliated to the group Algorithms, Combinatorics and Optimization of the Centrum Wiskunde & Informatica (CWI) in Amsterdam.
The main objective of Vallentin's research project is the combination of semidefinite programming and harmonic analysis. Over the last two decades semidefinite programming became one of the strongest general purpose tools for the design and analysis of efficient algorithms in optimization. Over the last two centuries harmonic analysis, Fourier analysis, became the strongest general purpose tool to exploit qualitative and quantitative structure of mathematical objects, like functions and operators. Now Fourier analysis is omnipresent in our modern technological life.
The theoretical foundations for the project are still widely open. Indeed, while the current theory of semidefinite programming is mostly finite-dimensional, harmonic analysis predominantely deals with transcendental objects, i.e. objects in infinite-dimensional spaces. A main challenge will be to develop the corresponding theory of infinite-dimensional semidefinite programming.
The goal is to use this combination to solve computational difficult problems in mathematics and mathematical engineering which cannot be attacked by current techniques. These problems come from different areas: continuous combinatorial optimization, energy minimization in geometry and mathematical physics, statistics, and engineering. Intensive use of computer mathematics will be made to solve these problems, applying both symbolic algorithms from pure mathematics and numerical algorithms from applied mathematics.
First successes have already been booked by Vallentin and his colleagues. In particular, they were able to compute the best known bounds for the famous kissing number problem (see the CWI press release or the article published on the website of de Volkskrant). Vallentin co-supervised the PhD thesis of Fernando de Oliveira Filho, defended in December 2009 at the university of Amsterdam, where the classical geometric problem of colouring the Euclidean space is attacked with this type of approach - see the news item on the dissertation of Fernando de Oliveira Filho.
These results were achieved while Vallentin was fully affiliated to CWI, as part of the VIDI research project on Semidefinite programming and combinatorial optimization of Monique Laurent and of the Spinoza project of Lex Schrijver.
Illustration: Frank Vallentin, CWI