The Bézier representation of homogenous polynomials has little and not the usual geometric meaning if we consider the graph of these polynomials over the sphere. However the graph can be seen as a rational surface and has an ordinary rational Bézier representation. As I will show, both Bézier representations are closely related. Further I consider rational spline constructions for spherical surfaces and other closed manifolds with a projective or hyperbolic structure.[-]

A kinetic description of a plasma in external and self-consistent fields is given by the Vlasov equation for the particle distribution functions coupled to Maxwell's equation. Numerical schemes that preserve the structure of the kinetic equations can provide new insights into the long time behavior of fusion plasmas. In this talk, I
will present a structure-preserving particle-in-cell scheme for the Vlasov-Maxwell equations based on a finite difference description of the fields. Moreover, I will discuss the parallel implementation of this method based on the AMReX framework. This is joint work with Irene Garnelo and Eric Sonnendrücker.[-]