Dynamics, Geometry and Randomness

Agenda Archive


Meeting 13: 17 Dec 2002

Reading for Everyone:
  • Adams, The Knot Book
  • Mumford et al, Indra's Pearls
  • Hayles, Complex Dynamics in Literature and Science
Projects for Everyone:
  • Why are knots knotted?
  • What surfaces can you make from triangles coming together 7 to a vertex?
  • Work on your final project, and prepare its documentation to hand in.
Individual topics and projects:
  • Final projects: Josi, Jason, Ben
New and pending questions and topics.
  • Why are knots knotted?
  • Quasifuchsian groups
  • How does it all fit together?
  • Thursday, 9 Jan, 2 pm: reunion
Meeting 12: 10 Dec 2002

Reading for Everyone:
  • Thurston, Three-Dimensional Geometry and Topology
  • Mumford et al, Indra's Pearls
  • Adams, The Knot Book
Projects for Everyone:
  • Try to visualize the 3-torus, the 3-sphere and some other 3-manifolds.
  • Why are knots knotted?
  • Work on your final project.
Individual topics and projects:
  • Dynamical systems: Brendan
  • Final projects: Brendan, Walker, Dina, Will
New and pending questions and topics.
  • What happens if you glue together a mass of triangles, 7 to a vertex?
  • The Borromean rings and the cube
  • 3-manifolds, knots and Not Knot
  • Quasifuchsian groups
  • Hayles, Complex Dynamics in Literature and Science
Schedule of final presentations:
  • 17 Dec: Josi, Jason, Ben
  • Thursday, 9 Jan: reunion
Meeting 11: 3 Dec 2002

Reading for Everyone:
  • Rademacher and Toeplitz, Pedal triangle
  • Mumford et al, Indra's Pearls
  • Thurston, Three-Dimensional Geometry and Topology
Projects for Everyone:
  • Prepare to continue discussion of RLR, Memento and Borges.
  • Work on your final project!
Individual topics and projects:
  • Dynamical systems: Brendan
  • The pedal triangle: Brendan
  • Final projects: Susannah, Andrew, Brendan
New and pending questions and topics.
  • The hexponential
  • 3-dimensional manifolds
  • Knots and Not Knot
Schedule of final presentations:
  • 10 Dec: Walker, Dina, Ben
  • 17 Dec: Josi, Jason, Will
Meeting 10: 26 Nov 2002

Reading for Everyone:
  • Borges, The garden of forking paths
  • Series, Continued fractions
  • Rademacher and Toeplitz, Pedal triangle
  • Mumford et al, Indra's Pearls
Projects for Everyone:
  • Compute some continued fractions (e.g. sqrt[n], e, pi, ...)
  • Prepare to discuss RLR, Memento and Borges.
  • Work on your final project!
Individual topics and projects:
  • 3D life and movies - Brendan
  • Dynamical systems: Brendan, Susannah and Will
  • The pedal triangle: Brendan
New and pending questions and topics.
  • Is life a garden of forking paths?
  • Points of confusion: what is the significance of the Feigenbaum constant? What does T(f(x)) = g(T(x)) have to do with chaos?
  • What does `sensitive dependence on initial conditions' mean?
  • Expansion factors for 2x mod 1, x^2-2, ax + x^2
  • Are billiards and irrational rotations chaotic?
  • In the square, not all itineraries are possible
  • The hyperbolic plane; surfaces of higher genus
  • In the curved torus, all itineraries are possible
  • Tilings, continued fractions.
  • The hexponential
Schedule of final presentations:
  • 3 Dec: Susannah, Andrew, Brendan
  • 10 Dec: Walker, Dina, Ben
  • 17 Dec: Josi, Jason, Will
Meeting 9: 19 Nov 2002

Projects for Everyone:
  • Read Tabachnikov on Billiards.
  • Study the dynamics of rotation: f(x) = x + a mod 1. Show when a is irrational, all orbits are dense.
  • Invent and investigate your own dynamical system.
  • See "Memento" and "Run Lola Run".
Individual topics and projects:
  • 3D life and movies - Brendan
  • Escher and the Droste effect - Walker
New and pending questions and topics.
  • Peano animation
  • What does `sensitive dependence on initial conditions' mean?
  • x + a mod 1: are orbits dense? Closest returns?
  • Billiards.
  • Continued fractions; find your own!
  • Is life necessarily balanced on the edge of chaos?
  • Is life a garden of forking paths, as in Run Lola Run and Borges?
  • Points of confusion: what is the significance of the Feigenbaum constant? What does T(f(x)) = g(T(x)) have to do with chaos?
Upcoming reading:
  • Borges, The garden of forking paths
  • Series, Continued fractions
  • Rademacher and Toeplitz, Pedal triangle
Organization:
  • Schedule final presentations, Dec. 3-10-17.
Meeting 8: 12 Nov 2002

Projects for Everyone:
  • Write up your midterm report.
  • Invent and investigate your own dynamical system. Questions to answer:
    • What is (or are) the attractor(s)?
    • What are the periodic points?
    • Which are attracting? Which repelling?
    • What is the doubling time?
  • Study the dynamics of rotation: f(x) = x + a mod 1.
  • How does the bifurcation picture for f(x) = ax(1-x) relate to the Mandelbrot set?
  • See "Memento" and "Run Lola Run" by 19 Nov 2002.
Individual topics and projects:
  • Complexity of English - Josi
  • 3D life and movies - Brendan
  • Escher and the Droste effect - Walker
  • What is a Julia set? - Dina
  • What is the Mandelbrot set? - Susannah
  • More on the halting problem - Ben
New and pending questions and topics.
  • Peano animation
  • What does `sensitive dependence on initial conditions' mean?
  • x + a mod 1: are orbits dense? Closest returns?
  • What is a Julia set?
  • What is the Mandelbrot set?
  • Can a computer be programmed to discover every mathematical theorem?
  • Is the world -even the solar system- "random and unpredictable"?
  • Is life necessarily balanced on the edge of chaos?
  • Billiards.
  • Continued fractions.
Upcoming reading:
  • Tabachnikov, Billiards
  • Hardy and Wright, Continued Fractions
Meeting 7: 5 Nov 2002

Projects for Everyone:
  • Invent and investigate your own dynamical system. Questions to answer:
    • What is (or are) the attractor(s)?
    • What are the periodic points?
    • Which are attracting? Which repelling?
    • What is the doubling time?
  • Study the dynamics of rotation: f(x) = x + a mod 1.
  • Read Milnor on Julia sets. Browse Marmi on chaos in the solar system.
  • How does the bifurcation picture for f(x) = ax(1-x) relate to the Mandelbrot set?
  • See "Memento" and "Run Lola Run" by 19 Nov 2002.
Individual topics and projects:
  • Complexity of English - Josi
  • 3D life and movies - Brendan
  • Escher and the Droste effect - Walker
  • Chaotic Newton's methods - Will
  • Chaos of 4x(1-x) - Jason
  • What is a Julia set? - Dina
  • What is the Mandelbrot set? - Susannah
  • More on the halting problem - Ben
New and pending questions and topics.
  • Peano animation
  • The halting problem
  • Can a computer be programmed to discover every mathematical theorem?
  • Universality
  • What is a Julia set?
  • What is the Mandelbrot set?
  • Behavior of x+a mod 1; closest returns
  • Pappus's theorem
  • Memento and Lola Rennt
  • Is the world -even the solar system- "random and unpredictable"?
  • Is life necessarily balanced on the edge of chaos?
Meeting 6: 29 Oct 2002

Readings: May; Feigenbaum; browse Marmi; read "What is Life?"
Topics/Projects for Everyone:
  • Graph the `attractor' of f(x) = ax(1-x) for a in [0,4]. (The attractor can be drawn as the set of points f^n(1/2) for 100 < n < 1000, for example.)
  • Investigate what happens in the windows of calm.
  • Investigate the rate of period doubling.
  • What happens for other functions like f(x) = a sin(x)?
Individual topics and projects:
  • Fractal ruler - Josi
  • Curves of dimension 1.9 or so - Walker, Will
  • 3D fractals and movies - Brendan
  • Life and computers - Brendan
  • Iteration of Newton's methods for x^2+1 - Will
  • Iteration of 4x(1-x) - Jason
  • What is the complexity of the English language?
  • Generation of babble.
New questions and topics.
  • Peano animation
  • Impossible questions - the halting problem
  • Chaos: 2x mod 1, Newton's method for x^2+1, 4 x (1-x), z^2
  • Attractor of a x (1-x), cascade of period doublings
  • Expansion, contraction and borderlines
  • Behavior of x+a mod 1; closest returns
  • RATS
New project for Everyone:
Invent your own dynamical system and investigate its behavior.
Your dynamical system might be an automaton, a sequence like RATS, or an iteration like f(x) = x^2+c.
Questions to answer:
  • What is (or are) the attractor(s)?
  • What are the periodic points?
  • Which are attracting? Which repelling?
  • What is the doubling time?
Meeting 5: 22 Oct 2002

Readings: May; Feigenbaum; browse Marmi.
A good book on matrices: Linear Algebra with Applications, O. Bretscher.
Topics/Projects for Everyone:
  • Graph the `attractor' of f(x) = ax(1-x) for a in [0,4]. (The attractor can be drawn as the set of points f^n(1/2) for 100 < n < 1000, for example.)
  • Invent your own dynamical system and investigate its behavior.
Individual topics and projects:
  • Fractal ruler - Josi
  • Curves of dimension 1.9 or so - Walker, Will
  • 3D fractals - Brendan
  • Dimension of Brownian graph - Andrew, Walker
  • How are random numbers generated (Knuth)? - Will
  • Cellular automata and randomness (Wolfram) - Susannah, Dina, Jason
  • How to map [0,1] onto a cube - Ben
  • What is the complexity of the English language?
  • Iteration of Newton's methods for x^2+1 - Will
  • Iteration of 4x(1-x) - Jason
Question to keep in mind on dynamics.
  • What is (or are) the attractor(s)?
  • What are the periodic points?
  • Which are attracting? Which repelling?
  • What is the doubling time?
New questions and topics.
  • Impossible questions - the halting problem
  • Invent your own random number generator.
  • Life and computers
  • Behavior of x^2 and 2x mod 1
  • Attractor of a x (1-x), cascade of period doublings
  • Universality
  • Chaos: 2x mod 1, Newton's method for x^2+1, 4 x (1-x), z^2
  • What is a Julia set?
  • What is the Mandelbrot set?
  • Behavior of x+a mod 1; closest returns
Meeting 4: 15 Oct 2002

Readings: Feller 67-97; Knuth: How to generate random numbers (1-37); What is a random sequence (142-177).
Another good probability book: Jim Pitman, Probability

Topics/Projects for Everyone:
  • What are the chances you never see a run of 5 0's in a random sequence of length N?
  • Figure out the relation between positive paths and returning paths (Feller, Prob. 7, p.96).
Other topics and projects:
  • What is the dimension of an actual computer image, or an actual coastline? - Andrew, Susannah
  • Fractal ruler - Josi
  • Curves of dimension 1.9 or so - Walker, Will
  • 3D fractals - Brendan
  • Random walks in 2 and 3 dimensions - Josi, Andrew, Dina
  • Dimension of Brownian graph - Andrew, Walker
  • Bias - Susannah
Further topics:
  • Does a random walker in the plane visit every point?
  • How complex is the Enligsh language?
  • What is a random number?
  • Dynamics of 2x mod 1
  • The map a x (1-x), e.g. a=2, 3, 4; stability of cycles
  • How does Newton's method behave for p(x) = x 2 + 1?
  • Bifurcations and x^2 + c
  • What is the Mandelbrot set? a Julia set?
Meeting 3: 8 Oct 2002

Readings: Feller 67-97; Knuth: How to generate random numbers (1-37); What is a random sequence (142-177).
What are the chances you never see a run of 5 0's in a random sequence of length N?
Invent your own random number generator.
Draw a picture of a long random walk on the line, in the plane or in space.
  • `The dilema of probability theory' - Brendan
  • Random walks - Will
  • Does a random walker always return home? - Jason
  • Is it hard to write down a random sequence? - Dina, Susannah, Walker
Questions about random walks to keep in mind:
  • Does randomness = ignorance?
  • Why is the distance from home after n steps expected to be Sqrt[n]?
  • What is the probability the walker is home at the 2nth step?
  • What is the probability the walker is home for the first time at the 2nth step?
  • Why is it common for a random walker to spend a great deal of time to one side of home?
  • What is the dimension of the graph of Brownian motion on a line?
Further topics:
  • Random walks and Pascal's triangle
  • Long one-sided excursions
  • Harmonic functions via random walks.
  • Recurrence in dimensions one, two and three
  • Positive paths and returning paths (Prob. 7, p.96)
  • Hausdorff dimension of graph of Brownian motion on a line.
  • Computer-generated random numbers (Knuth)
  • Generating random numbers via automata (Wolfram)
  • What is a random number?
  • Dimension, information, entropy and language

Other fractal projects: Compute or estimate the dimension of an actual coastline or boundary of a geographic entity. Construct a fractal ruler. Draw some 3D fractals. Draw a curve of dimension > 1.9 and a surface of dimension > 2.9. Why doesn't the Koch curve meet itself?
Meeting 2: 1 Oct 2002

Readings: pp.51-73 of the Mandelbrot selection; Littlewood, `The Dilema of Probability Theory'; Doyle and Snell, 3-7, 119-121. Browse Feller 67-97.
Invent your own fractal -- a curve, a surface or another type of object. Compute its dimension. Render it graphically.
Be ready to discuss the following topics. (Leader follows topic.)
  • Cantor function - Dumas
  • The Gosper snowball (and self-similar tilings?) - Josi
  • Windows in the Sierpinski triangle - Ben

Find an image showing the self-similar nature of an internal biological structure, such as the lung, the intestine or the capillary system.
Meeting 1: 24 Sept 2002

Read pp.25-50 of the Mandelbrot selection. This begins with `5. How long is the coast of Britain?', and continues into `6. Snowflakes and other Koch curves.'
(Readings are from coursepacks 249 and 250, available in the Science Center basement and on reserve in the Birkhoff library.)

Pick an example related to the seminar topics: a fractal, a dynamical system, a geometric object, a random process -- mathematical or from nature -- to briefly present to the class. (You can present your example in a variety of ways, e.g. on the blackboard, with a magazine clipping, with a natural object, etc.)

Pick an image from among those on the web page math.harvard.edu/~ctm/gallery that you would like to discuss -- for example, one that is familiar to you already, or one you would like to know more about. Print out your picture and bring it to the seminar.

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