  We report on recent results and a new line of research at the crossroad of
two major theories in the analysis of partial differential equations. The
celebrated De Giorgi-Nash-Moser theory shows H{\"o}lder estimates and the
Harnack inequality for uniformly elliptic or parabolic equations with rough
coefficients in divergence form. The theory of hypoellipticity of H{\"o}rmander
shows, under general "bracket" conditions, the regularity of solutions to
partial differential equations combining first and second order derivative
operators when ellipticity fails in some directions. We discuss recent
extensions of the De Giorgi-Nash-Moser theory to hypoelliptic equations of
Kolmogorov (kinetic) type with rough coefficients. These equations combine a
first-order skew-symmetric operator with a second-order elliptic operator
involving derivatives in only certain variables, and with rough coefficients.
We then discuss applications to the Boltzmann and Landau equations in kinetic
theory and present a program of research with some open questions.
