Analysis seminar

This learners seminar is an informal discussion of topics related to Brennan's conjecture in conformal mapping. The conjecture, dating from 1978, is one of the open problems of active interest in the field. It is tied to questions about the integral means spectra and Hausdorff dimension, and to many others as well.

In one of its formulations, the conjecture asks, given a conformal map of the unit disk, for the optimal growth of the mean inverse square of the distortion around the circles of radius r as r tends to 1.

We meet on Thursdays 11-12 in 961 Evans.

February 5
February 12
February 19
February 26
March 4
March 11
March 18
April 8
April 15
April 22
April 29
May 6
Introduction and overview
Hausdorff measures
Hausdorff dimension and box counting
Basic properties of the spectrum
Theorems of Bertilsson and Wirths
Some approaches via Loewner's method
Further examples. Beta - numbers.
Carleson - Makarov approach
Carleson - Makarov approach continued
Bertilsson's results
Schwarzian derivative. Shimorin's result
Concluding lecture