  The Fermi-Hubbard model (FHM) is a cornerstone of modern condensed matter
theory. Developed for interacting electrons in solids, which typically exhibit
SU($2$) symmetry, it describes a wide range of phenomena, such as metal to
insulator transitions and magnetic order. Its generalized SU($N$)-symmetric
form, originally applied to multi-orbital materials such as transition-metal
oxides, has recently attracted much interest owing to the availability of
ultracold SU($N$)-symmetric atomic gases. Here we report on a detailed
experimental investigation of the SU($N$)-symmetric FHM using local probing of
an atomic gas of ytterbium in an optical lattice to determine the equation of
state through different interaction regimes. We prepare a low-temperature
SU($N$)-symmetric Mott insulator and characterize the Mott crossover,
representing important steps towards probing predicted novel SU($N$)-magnetic
phases.
