Departmental Colloquium
John D'Angelo
UIUC
CR Mappings between spheres in different dimensions
Abstract: CR Geometry concerns CR manifolds and CR mappings between them.
CR stands for both Cauchy-Riemann and Complex-Real.
In this talk the source and target CR manifolds will be the unit spheres
in (generally different dimensional) complex Euclidean spaces. We start
by contrasting an elementary result on the unit disk with a result of
Herb Alexander (UIC, 1974) concerning proper self-mappings between balls. This
discussion leads us to analyzing the relationship between the target dimension
and how complicated the mapping can be. By making a simplifying
assumption, we obtain a beautiful combinatorial problem asking for bounds
on the degree of a real polynomial (with nonnegative coefficients, and
constant on a hyperplane) in terms of the number of its distinct
monomials. In two dimensions Lens spaces play a key role. The talk
culminates with two recent results (obtained with Jiri Lebl and Han
Peters) obtaining various bounds on the degrees of these mappings.
Friday October 27, 2006 at 3:00 PM in SEO 636