# The Cauchy-Riemann equations in several variables

June 9 to June 13, 2014

at the

American Institute of Mathematics, Palo Alto, California

organized by

John P. D'Angelo, Bernhard Lamel, and Dror Varolin

This workshop, sponsored by AIM and the NSF, will focus on the many interesting questions that remain about the interaction between estimates for solutions of the Cauchy-Riemann equations and the behavior of the Bergman kernel associated to the given norm. Does knowledge of the Bergman kernel provide Hörmander type estimates for the solution of ${\overline \partial}$ under weaker pseudoconvexity assumptions? Another question concerns the link between the off-diagonal decay of the Bergman kernel and the ability to solve ${\overline \partial}$ with better hypotheses than those given by Hörmander's theorem. The workshop will consider many test scenarios in which we can formulate and explore conjectural answers.

Most results about ${\overline \partial}$ require stringent curvature or potential-theoretic conditions. It is of great geometrical interest to find solutions for sections of bundles appearing naturally in the CR setting, such as the infinitesimal CR automorphisms. The workshop will also discuss how results about ${\overline \partial}$ can be used in Hermitian analogues of Hilbert's $17$-th problem and related ideas.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.