The solvability of differential equations

It was a great surprise when Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain. Hörmander proved in 1960 that almost all linear partial differential equations are not locally solvable. This also has connections with the spectral instability of non-selfadjoint semiclassical operators. Nirenberg and Treves formulated their well-known conjecture in 1970: that condition (Ψ) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition (Ψ) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part. The Nirenberg-Treves conjecture was finally proved in 2006. We shall present the background, the main ideas of the proof and some open problems