RTG WORKSHOPS & LECTURE SERIES Topics in Several Complex Variables and CR Geometry October 8-9, 2011 Room B844, East Hall University of Michigan Ann Arbor, MI The main program will begin on Saturday, October 8, 2011 in the morning at 9:30am and end on Sunday, October 9. We may have some talks on Friday afternoon for those around. Please email John Erik or Dusty if you would like to give a talk on Friday afternoon.. Speakers Peter Ebenfelt   CR submanifolds of low codimension in spheres and hyperquadrics I, II, III. The study of CR submanifolds of hypersurface type embedded transversally in spheres and hyperquadrics is a relatively old subject that has attracted considerable attention in recent years. The embeddings of a given strictly pseudoconvex CR manifold into a sphere (or those of a Levi nondegenerate one into a hyperquadric of the same signature) exhibit strong rigidity properties when the codimension of the embedding is sufficiently low compared to the CR dimension of the manifold: If the codimension is less than the CR dimension, then any two embeddings are equal modulo composition with an automorphism of the target sphere (or hyperquadric). Recently, new rigidity phenomena have been discovered for manifolds whose CR complexity is low relative to the signature (in the positive signature case), and we are starting to understand embeddings in codimensions past the rigidity regime, but still low in a suitable sense. In the first talk in this series, I will give an introduction to this subject and survey the results, starting from old and leading up to very recent results and work in progress. In the last two talks, I will attempt to describe the techniques involved and the theory behind them (Cartan-Chern-Moser theory), and explain how the proofs lead to interesting and difficult questions about sums (and differences) of squares and related questions for Hermitian and complex polynomials. There are still many open questions of this nature... Xianghong Gong   Regularity of the Frobenius-Nirenberg theorem with parameter The Newlander-Nirenberg theorem says that a formally integrable complex structure can be locally transformed into the standard complex structure in the complex Euclidean space. After presenting a proof of real Frobenius theorem by F. and R. Nevanlinna via the Picard iteration, we will show two results about Newlander-Nirenberg theorem with parameters. The proof of the first result extends Webster's proof of the Newlander-Nirenberg theorem, and it produces a sharp regularity result. The second result concerns a sharp version of Nirenberg's complex Frobenius theorem. We will demonstrate a result which shows a loss of regularity. The loss of regularity is probably due to the use of Nash-Moser methods, and such a loss of regularity has also occurred in the local CR embedding problem. Son Duong   Transversality of holomorphic mappings between CR submanifolds of complex spaces We will discuss some recent results on the CR transversality of a holomorphic mapping to a CR submanifold $N$ of a complex space when the preimage of $N$ contains another CR submanifold $M$ of the same codimension. More precisely, we will provide some conditions involving finite type property of $M$ and the generic rank of the mappings which are sufficient for the CR transversality to hold. Furthermore, we will also show that under certain conditions, finite holomorphic mappings are CR transversal. This is a joint work with Peter Ebenfelt. Jiri Lebl   Polynomials constant on a hyperplane and CR maps of spheres Joint work with Han Peters. We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D'Angelo and proved in two dimensions by D'Angelo, Kos and Riehl. The speaker together with Han Peters has proved the bound for dimensions 3 and greater. In dimensions 4 and higher we in fact have a complete classification of the sharp polynomials. As a corollary we obtain a > sharp degree bound on monomial CR maps of spheres. Funding Limited funding is available for graduate students and post-docs. Please note your request on the registration form and send an e-mail to John Erik Fornaess. > Register for the Workshop   Sponsors National Science Foundation, Department of Mathematics at the University of Michigan