##### Department of Mathematics,

University of California San Diego

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### Math 296 - Graduate Student Colloquium

## Peter Ebenfelt

#### UCSD

## There is no Riemann Mapping Theorem in higher dimensions! ... Or is there?

##### Abstract:

The Riemann Mapping Theorem (RMT) is a staple in complex analysis in one variable: {\it Any simply connected domain in the plane (other than the plane itself) is biholomorphically equivalent to the unit disk.} A direct analog is not true in two dimensions and higher. As discovered by Poincar\'e, the unit ball in $C^2$ is not biholomorphic to the bidisk. The reason is that in higher dimensions the boundary of a domain inherits a non-trivial structure---a CR structure--- from the ambient complex structure. We will discuss how one can formulate a version of the RTM that holds in higher dimensions as well. After this introduction, we shall mention some current fundamental problem in this area.

Organizer: Ioan Bejenaru

### February 23, 2017

### 11:00 AM

### AP&M 6402

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