  We introduce an effective method to solve the $\bar\partial$-harmonic forms
on the Kodaira-Thurston manifold endowed with an almost complex structure and
an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic
PDE system to countably many linear ODE systems. By solving a fundamental
problem on linear ODE systems, the problem of finding $\bar\partial$-harmonic
forms is equivalent to a generalised Gauss circle problem.
  We demonstrate two remarkable applications. First, the dimension of the
almost complex $\bar\partial$-Hodge numbers on the Kodaira-Thurston manifold
could be arbitrarily large. Second, Hodge numbers vary with different choices
of Hermitian metrics. This answers a question of Kodaira and Spencer in
Hirzebruch's 1954 problem list.
