Groups of Möbius Transformations

Course Info

Signing up: The tutorial is now full. For information about your enrollment status, contact Cliff Taubes (chtaubes@math.harvard.edu).

Tutorial Description: pdf ps html
(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:

• CUP Official Page
• Online Availability

Links

• My gallery of limit sets, or of slices
• Curt McMullen's Gallery (check out "Kleinian Groups")
• Dave Wright's Web Page - Co-author of Indra's Pearls; has a gallery of limit set pictures and downloadable fortran programs.

Assignments

• Lecture 1: 7/2/2002
  Reading: Indra Ch 1-3 (most of this will be review)
  Optional:
  • Complex Analysis Review: Ahlfors Ch 1-3, Bak and Newman Ch 1-3.
  • Groups and group actions: Beardon Ch 1-2
• Lecture 2: 7/5/2002
  Homework 1: ps pdf
• Lecture 3: 7/9/2002
  Reading: Indra Ch 4 pp 96-111, Ch 5 pp 121-141
  Optional:
  • Review relevant group theory: Gallian, esp. ch 26, 28, and 3; Jacobson, 0 - 1.12
  • Cayley graphs, surface topology: Stillwell, Ch 0 - 2
• Lecture 4: 7/11/2002
  Homework 2: ps pdf
• Lecture 5: 7/16/2002
  Reading: Indra Ch 6, Beardon Ch 5,6.
  Explore: Web sites related to Schottky groups
  
  • What happens when the Schottky circles meet (requires Java; applet by Pat Hooper)
  • Cantor sets and area (Java)
  • More to come.
• Lecture 6: 7/16/2002
  Homework 3: ps pdf
  Apollonian Gasket:
  • Schottky circles and circle packing - gif ps
  • Just the circle packing (limit set) - gif ps
  • Twisting animation: with Schottky circles - mpeg gif
  • Twisting animation: just the limit set - mpeg gif
  • Many more pictures and animations
• Final Project: Select a topic!
  Suggestions - ps pdf
• Lecture 7: 7/23/2002
  Reading: Indra Ch 7, and look at the section of Ch 8 devoted to the Maskit slice (pp 259 - 267).
• Lecture 8: 7/25/2002
  Homework 4: ps pdf
• Lecture 9: 7/30/2002
  Reading: Indra Ch 8 - 9

• Complex Analysis, L. Ahlfors, McGraw-Hill, 1979.
• Complex Analysis, J. Bak and D. J. Newman, Springer, 1996.
• The Geometry of Discrete Groups, A. F. Beardon, Springer, 1983.
• Contemporary Abstract Algebra, J. Gallian, D.C. Heath and Company, 1994.
• Basic Algebra I, N. Jacobson, W. H. Freeman and Company, 1996.
• Classical Toplogy and Combinatorial Group Theory, J. Stillwell, Springer, 1993.