On 3-nondegenerate CR manifolds in dimension 7

I will report on joint works with B. Kruglikov on CR hypersurfaces in $\mathbb C^4$ with a degenerate Levi form. I will discuss the symmetry dimension bound 8 for all the 3-nondegenerate 7-dimensional CR real-analytic structures and present the classification of the locally homogeneous ones. The bound 8 is achieved on the homogeneous model, which is locally the only homogeneous 3-nondegenerate 7-dimensional CR manifold.