Abstracts

Han Peters (U of Wisconsin), Time averages of polynomials

I will discuss how the notion of time averages arises from the study of polynomial automorphisms, and prove some results that I did not prove in December.

Erlend Wold (U of Oslo), A counterexample to uniform approximation on totally real manifolds in ℂ ³

We will construct a closed submanifold M of ℂ³ such that M is diffeomorphic to ℝ² and has the following properties: 1) M is totally real, 2) M is polynomially convex, but 3) Continuous functions on M cannot be approximated uniformly by entire functions. Note that by 1) and 2) uniform approximation is possible on compact piecesof M.

Jean Ruppenthal (U of Bonn), A ∂-theoretical Proof of Hartogs' Extension Theorem on Stein spaces with isolated singularities

Let X be a connected normal Stein space of pure dimension d ≥2 with isolated singularities. By solving a weighted ∂-equation with compact support on a desingularization of X, we derive Hartogs' Extension Theorem on X by the ∂-idea of Ehrenpreis.

Anne-Katrin Herbig (U of Wien), Plurisubharmonic defining functions

Let D be a bounded domain with smooth boundary. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. This implies that the so-called Diederich-Fornæss exponent can be chosen arbitrarily close to 1 and that the closure of D admits a Stein neighborhood basis. I shall try to explain (some parts of) the proof of this. This is joint work with J.E. Fornæss.

Andreea Nicoara (U of Harvard), Equivalence of types

In 1979 Joseph J. Kohn showed that the subellipticity of the ∂-Neumann problem on a pseudoconvex domain with real-analytic boundary is equivalent to the termination of a certain algorithm he devised that constructed subelliptic multipliers and to the condition that all holomorphic varieties have finite order of contact with the boundary of the domain. This three-way equivalence for the more general case of a smoothly-bounded pseudoconvex domain is called the Kohn Conjecture. I will describe my solution to the Kohn Conjecture involving algebraic geometry over the non-Noetherian ring of smooth functions as well as Catlin's notions of boundary systems and multitypes.

Mattias Jonsson (U of M), Dynamical compactifications of ℂ ²

To study the dynamics of a polynomial mapping on ℂ², it is useful to have a compactification X of ℂ² that is well adapted to the dynamics. In the case of polynomial automorphisms, this was essentially achieved by Friedland and Milnor, using Jung's classification theorem. The compactification obtained is (up to a change of variables) the projective plane ℙ², and serves as a first, but crucial, step in constructing interesting invariant measures, currents etc. For general (noninvertible) maps, no classification theorem is available. I will report on joint work with Charles Favre where we use valuative techniques to construct good compactifications (in a sense to made precise in the talk). As a consequence, we obtain that the degree growth of any polynomial mapping satisfies a finite linear integral recursion formula.

Laura DeMarco (UIC), Polynomials, trees and valuations

The moduli space of polynomials can be compactified by a space of trees. These trees play the role (in the setting of dynamics of rational maps) of the R-trees which compactify spaces of hyperbolic structures. If there's time, I will describe some new work relating the compactification to a space of valuations. This is a joint project with Curt McMullen.

Jae-Seong Cho (Purdue U), Sharp Subelliptic Estimates for Special Domains

The largest possible value of the subellptic gains on (0,1)-forms for general peudoconvex domains of finite type are not well known except for some simple cases. It was known, due to David Catlin, that its upper bound is less than or equal to the reciprocal of the maximum order of contact of holomophic curves to the boundary. I will show that for various types of Special domains this value is the largest one by constructing a family of plurisubharmonic functions near the boundary with large Hessian.

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

We will consider how the so-called Segre varieties become the tools for showing that any proper holomorphic map from a bounded domain in ℂⁿ with smooth real-analytic boundary onto a domain of the same kind can be holomorphically extended to an open neighborhood of the closure of the first domain. The talk will give a survey of known result and focus on methods of proof without going too much into technical details.

Feng Rong (Syracuse U), Singularities of holomorphic maps of ℂⁿ tangent to the identity

Abate proved that every holomorphic map of ℂ² tangent to the identity at an isolated fixed point has parabolic curves. One of the main ingredients of his proof is a reduction theorem on the "singularities" of such maps. We are going to talk about similar reduction theorems in higher dimensions and related results. We will mainly focus on the basic ideas and definitions rather than the details of any proof.

David Barrett (U of M), Real hypersurfaces in two-dimensional projective space

This talk will examine geometric aspects of real hypersurfaces in two-dimensional projective space, the dual complement construction and a related characterization of the Leray integral transform.