
02/26 11:10am 
MILN 317 
Robert Jacobson Texas A&M University 
Characterizing pseudoconvex domains
There are dozens of ways to characterize pseudoconvex domains which are also known as domains of holomorphy. For some characterizations, there exist formally weaker statements which nonetheless imply the same conclusion. This motivates exploring what happens when the hypotheses of some of the characterizations of pseudoconvex domains are slightly loosened. We explore some results in a recent paper by Nikolov, Pflug, Thomas, and Zwonek along these lines. In particular, a domain D is pseudoconvex if and only if for any $a \in \partial D$ there is a $u_a \in \mathcal{PSH}(D)$ with $\lim_{z \to a}u_a(z) = \infty$. We discuss what happens when we replace $\lim$ with $\limsup$ 

04/09 11:10am 
MILN 317 
Joe Perez Texas A&M University, Kingsville 
The Levi Problem On Strongly Pseudoconvex GBundles
Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be the total space of a principal bundle $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold and $G$ acts by holomorphic transformations. Our ultimate goal is to determine if, in this setting, every point of the boundary is a peak point for the space of squareintegrable holomorphic functions on $M$. This question is related to the Levi problem, posed in 1911, which was to characterize domains of holomorphy in $\mathbb C^n$.
First, we established that $\square_{0,1}$ on $M$ is a $G$Fredholm operator (J Geom Analy, 19, 87106, 2009).
Now we exploit the previous result (roughly that the Laplacian is almost onto $L2(M,\Lambda^{0,1})$) by constructing large, closed, smooth invariant subspaces in $L2$ that the image of $\square$ must meet. A noncommutative generalization of the PaleyWiener theorem is used. Finally, we show that if $G$ acts by transformations satisfying a local property, then the space of squareintegrable holomorphic functions on $M$ is infinite $G$dimensional and the functions in this space have the property that they lack holomorphic extensions beyond the boundary, almost as Levi desired. See http://arxiv.org/find/all/1/all:+AND+joe+perez/0/1/0/all/0/1
