A famous theorem of Artin says that every formal solution of a system of analytic equations can be
approximated by analytic solutions of that system. In the theory of Cauchy-Riemann manifolds,
which originated with Poincaré, there are surprising applications of Artin’s theorem, which imply
similar results for solutions of certain partial differential equations. The project aimed on the one
hand at making the theory of approximation accessible to researchers in Cauchy-Riemann
geometry and on the other hand, at understanding the geometric results in an algebraic context.
In the course of the project, we developed a thorough geometric understanding of the theory of the
approximation theorem; in particular, points at which the solution set of a system of analytic
equations has a nice form were characterized in a geometric manner. Some of the questions we
encountered were brought to a negative resolution: For example, we were able to show that there
are (more general) systems of equations which arise in the applications, which have the property
that even though they do not possess any analytic solutions, they permit a formal solution