  We present our investigations of SU($N$) adjoint QCD in two dimensions with
one Majorana fermion on the lattice. We determine the relevant parameter range
for the simulations with Wilson fermions and present results for Polyakov loop,
chiral condensate, and string tension. In the theory with massive fermions, all
observables we checked show qualitative agreement between numerical lattice
data and theory, while the massless limit is more subtle since chiral and
non-invertible symmetry of the continuum theory are explicitly broken by
lattice regularization. In thermal compactification, we observe $N$
perturbative vacua for the holonomy potential at high-$T$ with instanton events
connecting them, and a unique vacuum at low-$T$. At finite-$N$, this is a
cross-over and it turns to a phase transition at large-$N$ thermodynamic limit.
In circle compactification with periodic boundary conditions, we observe a
unique center-symmetric minimum at any radius. In continuum, the instantons in
the thermal case carry zero modes (for even $N$) and indeed, in the lattice
simulations, we observe that chiral condensate is dominated by instanton
centers, where zero modes are localized. We present lattice results on the
issue of confinement vs. screening in the theory and comment on the roles of
chiral symmetry and non-invertible symmetry.
