So, just out of curiosity, I would like to know, whether my impression is wrong and what current hot research-topics of practical relevance in numeric mathematics are.

The question of practical relevance of numerical mathematics has very little to do with current hot research topics. Well established topics like cubic splines, basis splines, finite element methods, spectral methods, boundary integral methods, Krylov-subspace methods, sparse matrices, nested dissection, stiff ordinary differential equations, differential-algebraic equations, domain decomposition methods, sparse grids, FFT techniques, Monte Carlo methods, stochastic differential equations, Gaussian quadrature, Chebyshev methods, ..., ..., ... all remain relevant today.

This doesn't mean that the current hot research-topics are not of practical relevance, but they are the more challenging problems. For example, in 1998 R. Hiptmair finally managed to construct working multigrid methods for Maxwell's equations, which was challenging before due to a non-trivial kernel of the corresponding operator. In the following years, many similar methods were proposed. Unrelated to this, methods to efficiently deal with the indefinite systems arising from time harmonic Maxwell or wave equations (Helmholtz equation) have occurred only recently. However, there is a difference in that Hiptmair really was able to overcome all theoretical and practical difficulties due to better theory, while the stability of fast solvers for indefinite systems (time harmonic case) remains challenging in practice. An earlier case of hot research activity following a breakthrough was caused when Sethian developed level set methods and fast marching methods in 1988. (I know that naming Hiptmair or Sethian as the only inventors of the methods they promoted is historically inaccurate, but it nicely simplify things and gives precise dates.)