VIGRE2 Vertical Intergration of Research and Education Department of Mathematics, University of Utah

MATH 4950

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Spring 2007 course

Topic: Fractals
Instructor: Elena Cherkaev
Catalog Number: 4950-1
Credits: 3

Registering requires obtaining the class number from the instructor.

Course Outline:
This course is part of the REU (Research Experiences for Undergraduates) program, as an important component of the VIGRE grant funded by NSF. The subject varies from one semester to another, in which a faculty member, a graduate assistant, and up to 10 students explore a topic of significant mathematical interest. The students help to present the material or the results of their own investigations, and write a report on their findings. The format of this course is rather flexible, with the class divided into groups, each with 2-3 students and a particular topic to study over the semester.

Fractals are seen everywhere - in shapes of leaves and trees, moutain ranges and coastlines, structure of lungs and porous rocks, and many other physical and biological objects. Fractals are used to design cell phone antennas, describe forest fires, model diffusion of oils and gases, to store data and describe complex systems. The main features characterizing fractals are self-similarity, non-integer fractal dimention, and iteration. Billions of fractal shapes can be created using iterated function systems and constructing strange attractors of nonlinear dynamical systems.

Prerequisities:
We expect backgrounds beyond calculus (such as differential equations, basic complex variables and linear algebra). However, due to the variety of topics available in the class, we do not have a uniform set of requirements. Instead, each prospective student is required to visit the instructor to determine the suitability before registering for the class.

Possible topics for the projects are:
  1. Generate fractals using iterated function systems.

    Fractals are made of an initial shape, a rule, and iterations. Starting with an initial seed image, the rule is applied iteratively - again and again - to give in the limit statistically self-similar fractal shape or structure. Generate fractal curves, fractal landscapes, and many other fractal images using this technique.

  2. Fractal dimension.

    Calculate fractal dimensions of of mathematical fractals such as Serpinsky triangle gasket and Menger sponge, or of physical objects, such as composite materials with complicated microstructure.

  3. Can fractals bring an objectivity to the aesthetic evaluation of paintings in modern art?

    Can paintings be characterized by a number? Calculate and compare fractal dimentions of paintings belonging to different artists.

  4. Butterfly effect in chaotic systems.

    Fractals are closely related to iterative and dynamical systems which could exibit orderly or chaotic behavior in very close regions of parameters. Chaotic behavior stems from sensitive dependence on initial conditions which leads to gross magnification of small errors in the initial conditions. The Lyapunov exponent can be used to estimate how much small errors are magnified.

  5. Basins of attraction and Julia sets.

    The Julia set separates the points whose orbits of complex iterated maps converge to infinity from the points whose orbits converge to the origin. The boundary of the basin of attraction is fractal. Generate fractals using Julia sets. Use Newton's method and draw basins of attraction for the fixed points to generate a fractal set.