This is the first of the two secondments, which are planned during the PhD project of Tobias Metzlaff and his supervisor Evelyne Hubert. The supervisor at the hosting institution was Cordian Riener. The purpose of this secondment is to train the ESR’s integrability in a new workgroup, establish scientific contacts and develop a new perspective on the so far carried out research work.
During the secondment, Tobias presented his research on group theoretic polynomial bases for global optimization to the workgroup of the department of mathematics followed by discussions with the host supervisor. A member of the workgroup and postdoc, Philippe Moustrou, suggested to use the so far obtained research results to improve a bound for the chromatic number of lattices originating from root systems, which are the infinite families An, Bn, Cn, Dn and the finite families En, Fn and Gn. The computation of a chromatic number is easily motivated by the problem of painting a map, such that neighboring countries don’t share the same color. This can be generalized from two dimensional maps to a Euclidean space of arbitrary (finite) dimension: Given a distance on this space, how many colors are needed, such that two points with distance exactly one don’t share the same color?The bound for the chromatic number can be obtained from an optimization problem on functions, that were studied by Tobias and his advisor before the secondment. The best possible bound is known, but it is not proven or refuted that this bound can always be obtained. The whole team had weekly discussions about the work on this problem with a new technique, using generalized Chebyshev polynomials. This new technique simplifies the original problem in some cases from a high degree optimization problem to a linear or quadratic one.
The first result was a new proof of the already known best possible result for the family Cn. After that, the group studied the family Dn and obtained a result close to the best possible one, when the dimension is even. Furthermore, Tobias and Cordian worked on a new formulation of the theorem of Chevalley Shephard Todd and of the theorem of Procesi Schwarz in the new setup. The work on the proof of the bound for Dn and also the proof of these theorems is in progress and will continue after the secondment.
Tobias also participated in an online conference organised by the hosting institution for the community of Norwegian mathematicians. Due to the low number of active Covid-19 cases in Tromso, it was possible to carry out the work in the offices of the UiT and so Tobias and the hosting professor were able to have discussions from face to face.
The workgroup that established during the secondment, collected the obtained results in an article and plans to publish them in the near future.