In discrete geometry, linear programming bounds are important for bounding the quality of geometric configurations. These bounds have been used to show that constructions are optimal, for example the sphere packings coming from the E8 and Leech lattices. However, for many instances the linear programming bounds are not sharp and research is being done into semidefinite programming bounds. For many problems, these higher-order bounds lead to significant improvements, including new optimality proofs via sharp bounds.
The kissing number is the maximum number of non-overlapping unit spheres in n-dimensional Euclidean space that can simultaneously touch a central unit sphere. Determining the kissing number is an open problem except for dimensions 1, 2, 3, 4, 8 and 24. Bachoc and Vallentin gave improved upper bounds for the kissing number problem by computing a three-point semidefinite programming bound. We compute higher steps of the Lasserre hierarchy for this problem, which is also a semidefinite programming bound and which may be viewed as a four-point bound. To calculate this bound in practice, it is crucial to use symmetry reduction. This requires calculations related to the representation theory of the orthogonal group and invariants thereof. The resulting semidefinite program is challenging to implement and solve. Our results give improvements in upper bounds for the kissing number.

The "practical" problem we are interested in is this: Given data obtained from a physics experiment, can one decide whether they are compatible with a quantum-mechanical model subject to causality constraints? A "causality constraint" here demands that certain subsystems be statistically independent. Mathematically, this problem is related to non-commutative (NC) polynomial optimization: Given generators and relations, optimize over all states of the universal $C^*$ algebra they generate. The causal conditions give rise to polynomial constraints in the states, which until recently could not be handled in this setting. I'll report on results both on the mathematical and the physical problem.

Joint work with Laurens Ligthart and Mariami Gachechiladze. [2110.14659 (CMP), 2212.11299 (JMP)]

We provide a brief description on some (surprising) links of the Christoffel function with seemingly unrelated fields like algebraic positivity certificates, equilibrium measures of compact sets, and a generalized polynomial Pell's equation.