On the cohomology of almost complex manifolds and spaces of harmonic forms

Let (M, J) be a 2n-dimensional almost complex manifold. We will define the
Bott-Chern cohomology of (M, J). When J is integrable, we will recover the
usual definition of Bott-Chern cohomology.
Furthermore, once fixed a J-Hermitian metric g on (M, J), we will study the
kernel of self-adjoint elliptic operators naturally defined on (M, J, g), focusing
on the 4-dimensional case.
The results discussed have been obtained in some papers joint with R. Piovani,
L. Sillari, N. Tardini, X. Wang.