|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...