Groups of Möbius Transformations

Course Info

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(Tutorial Poster: pdf ps - Warning: these are huge files!)

Meeting Time: Tuesdays and Thursdays, 5:00 - 6:30pm, in 530 Science Center, from July 2 to August 9. (Exception: July 4 meeting postponed to July 5 at the same time.)

Indra's Pearls

Indra's Pearls is a popular book about Kleinian groups (which are groups of Möbius transformations) and will be the main reference for the course. The mathematical content of the qualitative exploration described in the book will be discussed in lecture and supplemented with other readings.

From the publisher, Cambridge University Press:

In Hindu mythology, the heaven of Indra contained a net of pearls. Each pearl was reflected in its neighbour so that the whole universe was mirrored in each pearl. This idea was rediscovered by mathematicians, first by Felix Klein, one of the great geometers of the late nineteenth century, who started with infinitely repeated reflections and was led to forms which are the chaotic images of symmetry generated by interacting spiral flows. For a century the images, painstakingly drawn by hand, barely existed outside the mathematical mind. In the 1980's, the authors embarked on the first computer exploration of Klein's vision, and in doing so, found further extraordinary images of their own. This book leads the reader on a journey from the arithmetic of complex numbers to the simple algorithms which create these delicate fractal filigrees in extraordinarily beautiful forms. It explains the pictures at a variety of levels, starting with basic algebra, continuing through do-it-yourself programs and explorations of the mathematics behind them, to the forefront of modern research.

Indra's Pearls is now in print!

More information:



Readings and homework are assigned approximately once per week. References:

Final Project Presentations

Aug 6 Aug 8
5:00 - 5:30 Guberman
Möbius Transformations in Higher Dimensions
Drawing Limit Sets of Schottky Groups
5:30 - 6:00 Lawler
Anti-Möbius Transformations
Möbius Transformations and Isometries of H3
6:00 - 6:30 Spiegelman
Elementary Kleinian Groups
Inequalities for Kleinian Groups
6:30 - 7:00 Stone
Maskit Slice Pictures

David Dumas