In our joint work with Shafikov (2016), we proved that the formal versus holomorphic equivalence problems in CR geometry are distinct.

This shows, in particular, that CR geometry enjoys the Stokes Phenomenon: the existence of additional holomorphic invariants supplementing a formal normal form, arising as transition functions between different geometric realizations of the formal map. Describing the Stokes phenomenon explicitly has been an open problem since then.

In our joint work with Laurent Stolovitch, we make a significant progress in solving this problem. In particular, we show that the respective (formal versus holomorphic) moduli space of analytic CR hypersurfaces in C^2 is, somewhat surprisingly, finite-dimensional (and generically empty). The construction that we use is based on the Multisummability theory in Dynamical Systems.