Abstracts

Han Peters (U of Wisconsin), Iterated monodromy groups and time averages

I will discuss mondromy groups in more detail. In particular I will outline how I hope to prove that a polynomial has a time average near a point z if and only z lies in a Siegel disc.

Erlend Wold (U of Oslo), Carlemann approximation on totally real manifolds

We will discuss work in progress concerning necessary and sufficient conditions for Carlemann approximation (better and better approximation as you approach infinity) by entire functions on totally real polynomially convex manifolds.

Jean Ruppenthal (U of Bonn), L^p-obstructions for the ∂-equation on a cone with isolated singularity

Let Y be a homogeneous pure dimensional variety in ℂⁿ with an isolated singularity at the origin, and D ⋐Y strongly pseudoconvex, 0∈D, D^*=D∖{0}. Moreover, let X be the projective variety defined by Y in ℂℙ^n-1. We will compute the obstructions to solving the ∂-equation in the L^p-sense on D^* for 1≤ p ≤ ∞ in terms of some cohomology classes on X, which are well-known for example if X is an elliptic curve by the Theorem of Riemann-Roch.

Anna Siano (U of M), Peak functions

Let D be a smoothly bounded pseudoconvex domain in ℂⁿ of finite type at some point p ∊ ∂D. And let A^α be the algebra of holomorphic functions on D that are C^α up to the boundary. We say that f ∊A^α is a peak function at p if f(p)=0 and | f |<1 on D∖{p}. An open question is whether there exists a function f ∊A ^α that peaks at p for some α>0. We survey some known results.

Anne-Katrin Herbig (U of Wien), Global regularity of the Bergman projection and the Diederich-Fornæss exponent

Suppose D is a smoothly bounded, pseudoconvex domain for which the Diederich-Fornæss exponent can be chosen arbitrarily close to 1. J.J. Kohn showed that this implies that the Bergman projection is globally regular. I will demonstrate a different proof of this fact which goes back to unpublished work of J.D. McNeal.

Rodrigo Parra (U of M), Volume Estimates Under Iteration of Meromorphic Mappings

We will approach the problem of estimate the volumes of (small) open sets Ω under iterations of a meromorphic self-map f of the projective complex space (or more generally a Kahler manifold). When our mapping f is dominant (which is the interesting case), we have that f is generically locally open and one-to-one, so Vol( f ^j (Ω)) has approximately the same size of Vol(Ω). This is not longer true when one of the iterates of Ω intersects the critical set of f and we are interested in to understand this behavior. The study of volumes estimates has a long history in Complex Dynamics and it has been used to construct and characterize invariant currents. This talk will be based on a paper of Vincent Guedj.

Klas Diederich (U of Wuppertal), Segre varieties and the geometric reflection principle in several complex variables II

We will continue to explain more details from the machinery of the Segre varieties as the main tool of the geometric reflection principle in several variables.

Chris Hammond (U of M), Analytic functionals and the Fantappié Transformation

We will follow Chapter 3 of the monograph "Analytic Functionals and Complex Convexity" by Andersson, et al., which among other things, describes the relationship between the dual space of O(kE) and O(kE^*), for k ≤ -1, where E is a ℂ-convex set and E^*is its dual complement.

Liz Vivas (U of M), Fatou-Bieberbach Domains with C^∞-boundary

We will read a paper by Stensønes in which she constructs Fatou-Bieberbach Domains with smooth boundary. The construction is substantially different from the usual basin of attraction approach. We will focus on the main ideas rather than in the calculations in the proof.

Lina Lee (U of M), Ohsawa-Takegoshi extension theorem

I will try to give a sketch of an alternate proof of the extension theorem by Diederich-Herbort using negligible weights. We will also discuss the application to Bergman metric estimation.