Prospectus - Freshman Seminar - Fall 2002

Curtis T McMullen


The seminar aims to develop a level of mathematical literacy sufficient to undertake a qualitative description of complex phenomena. Members will also be introduced to the practice of mathematical inquiry. The formulation of suggestive questions will be as important as the search for answers.

Ideally the seminar will function as a `mathematics studio'. It will be organized around the in-depth of study of a handful of simple dynamical systems with rich internal structure. These focal examples will make contact with other areas of mathematics such as algebra, number theory, geometry and probability theory, and serve as gateways to those subjects.

The seminar is experimental, and care must be taken to maintain depth and coherence as well as breadth. There may also be an opportunity to discuss resonances between modern mathematics and contemporary art, literature and popular culture.

Ideally an advanced math student, knowledgable in Mathematica and graphics, should be available as a consultant for computer projects.


  • Cantor sets, the Koch snowflake curve, the Sierpinski gasket and variations. These simple recursive sets illustrate the notion of fractional dimension as well as the definition of `wild' geometric objects by iteration.
  • Random walks. The expected distance after n steps is sqrt(n). The graph of a random walk on the line, when rescaled, also provides a source of fractal curves. What is its dimension? How many random walks stay positive for all time? Recurrence in dimensions 1 and 2, not in 3. The arcsine law (how long does a random walk spend on the positive part of the axis?) illustrates the counter-intuitive behavior of random processes.
  • Brownian motion. Financial markets.
  • Harmonic functions. The relation of random walks to potential theory. The type problem (when is a random walk on a graph transient)?
  • Irrational rotations of the circle. These provide the simplest examples of dynamical systems with dense orbits and ergodic behavior. They can be studied by renormalization, leading to continued fractions.
  • The squaring map. Here we see lots of periodic points, a rich combinatorial structure (how to find an orbit with given rotation number?), sensitive dependence on initial conditions and invariant Cantor sets. (But how thick can they be?)
  • The quadratic maps. Coexistence of stable and unstable regimes. The cascade of period doublings, renormalization and universality.
  • Random number generators. Capitalizing on chaos by using dynamics to produce random sequences.
  • Cellular automata. The game of life. Turing machines. Evolving pictures by pixel automata.
  • The paradox of chaos. Is the universe described to infinite precision, with the distant digits being revealed by dynamics?
  • Continued fractions. Using the Gauss measure we can predict the behavior of continued fractions of random numbers. Interesting special cases include quadratic and cubic irrationals and e (the base of the natural logarithm).
  • Euclidean billiards. From irrational rotations we can proceed to billiards on a rectangle. Many unsolved problems soon arise, e.g. the existence of closed trajectories in an obtuse triangle. Note that these billiards are linear yet highly complex; their complexity comes from the corners which introduce concentrated curvature and discontinuity.
  • Billiards in triangles and polyhedra. The pedal triangle. Is billiards in a regular tetrahedron chaotic? Does every acute convex polyhedron contain a billiard loop?
  • Hyperbolic billiards. Piecing together triangles with 3, 4 or 5 at a vertex leads to Platonic solids. With 6 we obtain the Euclidean plane. With 7 or more we are lead to hyperbolic geometry and tilings, and with infinitely many we reach the ideal triangulation and SL2Z. The geodesic flow on H/SL2Z, or billiards in an ideal triangle, leads again the continued fraction algorithm.
  • Hyperbolic geometry. Building surfaces for other billiards.
  • 3-dimensional hyperbolic geometry. Knot complements. Random geodesics outside the figure eight. Knot polynomials. Mostow rigidity.


We have collected written sources to support the seminar, ranging from popular accounts to research monographs. Due to the range of sources, it may be advisable to print a booklet of selected readings for the course. Seminar participants will also be encouraged to dive into the literature themselves, and locate additional sources tailored to their own interests, aided by modern search tools.


The course will be problem-oriented. Topics will be introduced with definitions and examples, followed by questions to be addressed. Members will be invited to study the literature, conduct experiments, carry out research, and participate in discussions and problem-solving. Collaboration and initiative will be encouraged.


Computer experiments will form an important facet of the course. The simulation of dynamical systems lends them a empirical reality, complementing what can be deduced by a theoretical analysis.

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