Running vacuum against the $H_0$ and $\sigma_8$ tensions
  The cosmological term, $\Lambda$, was introduced $104$ years ago by Einstein
in his gravitational field equations. Whether $\Lambda$ is a rigid quantity or
a dynamical variable in cosmology has been a matter of debate for many years,
especially after the introduction of the general notion of dark energy (DE).
$\Lambda$ is associated to the vacuum energy density, $\rho_{\rm vac}$, and one
may expect that it evolves slowly with the cosmological expansion. Herein we
present a devoted study testing this possibility using the promising class of
running vacuum models (RVM's). We use a large string $SNIa+BAO+H(z)+LSS+CMB$ of
modern cosmological data, in which for the first time the CMB part involves the
full Planck 2018 likelihood for these models. We test the dependence of the
results on the threshold redshift $z_*$ at which the vacuum dynamics is
activated in the recent past and find positive signals up to $\sim4.0\sigma$
for $z_*\simeq 1$. The RVM's prove very competitive against the standard
$\Lambda$CDM model and give a handle for solving the $\sigma_8$ tension and
alleviating the $H_0$ one.
