The tube over the future light cone as a model for everywhere Levi-degenerate hypersurfaces in complex space

In 1907, H.Poincare raised the problem to classify (locally) real hypersurfaces in complex space, up to local biholomorphisms. In the case when the Levi form is everywhere nondegenerate (i.e. the distribution of complex tangents is everywhere contact), the problem was solved in the work of E. Cartan, N. Tanaka, S.-S. Chern and J. Moser. In the Levi degenerate case, the problem appears to be much more difficult. Of particular interest here is the category of everywhere Levi-degenerate hypersurfaces, which though can't be reduced to hypersurfaces of smaller dimension. Such hypersurfaces appear for the first time in the space ${\mathbb C}^3$, and the basic example is the tubular manifold over the future light cone in the space of imaginary parts. In our joint work with Martin Kolář, we classify hypersurfaces under discussion in ${\mathbb C}^3$, by using the tube over the future light cone as a model and developing then the Poincare-Moser homological approach. The classification is done by presenting a complete convergent normal form for such hypersurfaces.