We will discuss the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. We study a certain weighted least squares algorithm that uses the so-called ”standard information”, i.e., finite number of function values at some points. The main advantage of our approach is that the nodes are taken at random w.r.t. a tailored distribution and simultaneously for the whole class of functions. We show that the considered deterministic algorithm leads to near optimal results in several relevant situations. We additionally apply a recently introduced sub-sampling technique to further reduce the sampling budget.

The talk concerns effects coming from a violation of time-reversal symmetryin the vertex coupling of quantum graphs, focusing on the situation when theasymmetry is maximal at a fixed energy. It is shown that such a coupling hasa topological property: scattering at the vertex in the high-energy regimedepends substantially on the vertex parity. We explore implications of thisfact for spectral and transport properties in several classes of graphs, bothfinite and infinite periodic ones. We also identify a class of vertex couplingexhibiting a nontrivialPT-symmetry despite being self-adjoint. The resultscome from a common work with Marzieh Baradaran, Jiří Lipovský, andMiloš Tater.