Analysis Seminar-A geometric function theory for dimensions greater than two

Speaker:  Professor David Drasin, Purdue University

Date: Wednesday, March 23 at 2:00 PM

Where: Monroe Hall B32

Title:  A geometric function theory for dimensions greater than two

Abstract:  Classical complex analysis is a high point in pure mathematics, but at the same time has had remarkable applications in theoretical physics, engineering etc.  In many ways it is a jewel in itself, which in many respects is special for two dimensions. For example, only when n=1 or 2 can a multiplication be introduced on Euclidean n-space to create a field.

An important property of the function w=f(z) being analytic is that in general it induces a conformal map: if w_0=f'(z_0)\neq 0, then angles are preserved at z_0 and (infinitesimal) circles centered at z_0 are mapped to (infinitesimal) circles centered at w_0=f(z_0). 

J. Liouville showed that this is a very special property of two dimensions: if F is a mapping on n-space (n > 2), then F will not be conformal unless it is a M\"obius transformation (it is easy to see that M\"obius transformations exist in all real dimensions).

One of the most compelling classical achievements is Picard's theorem: if f is  analytic, nonconstant and defined on the entire complex plane, then the range of f can omit at most one point; the exponential function w=e^z shows Picard's result sharp (note that this function is also everywhere (locally) conformal but is not a M\"obius transformation).

There seems to be no obvious way to extend the rich classical theory on one complex dimension to higher (real or complex) dimensions.  The most immediate choice would be to let w=f(z) be a mapping of complex n-dimensions where f has a convergent power series.  Several complex variables has become a well-established subject, but the range of an entire function can omit an open set, and unless f is linear, it will never be conformal.

I will discuss a less obvious generalization, in which power series no longer operate, but which preserves an important feature of one complex variable: the image of a small ball centered at z_0 is sent to a nice set containing f(z_0).  If we require this image to be a ball, Liouville's theorem would show that f must be a M\"obius transformation, and thus the key insight in defining a quasiregular mapping is to allow the image to be an (infinitesimal)_ ellipsoid of uniformly bounded
eccentricity. 

We will define this class of mappings and indicates a few of their general properties and pathologies.

A deep result in the theory of quasiregular mappings, due to Seppo Rickman is that the range of a nonconstant quasiregular mapping which is defined on all of n-space can omit an at most finite set of points, and in a remarkable example of 30 years vintage he showed that the complement of the range can have cardinality as large as desired when n=3.

If time permits, I will indicate some steps which Pekka Pankka and I introduced to give a new approach to Rickman's construction which shows the sharpness of his result in all dimensions.