Course code:

ES5040

Level:

A - Advanced

Class size limit:

10

Meets the following requirements:

  • QR - Quantitative Reasoning
  • ES - Environmental Science

Typically offered:

Upon occasion

This course will give students a modern overview of the mathematics and statistics of fractals and scaling and their interdisciplinary applications.

We will begin with mathematical fractals and use them to define several different notions of dimension, standard ways for describing the nature of fractals’ self similarity. Students will then learn modern statistical techniques for reliably estimating fractal dimensions and power law exponents. We will also look more generally at “fat-tailed” distributions, a class of distributions of which power laws are a subset.

Next we will turn our attention to learning about some of the many processes that can generate fractals.

Finally, we will critically examine some recent applications of fractals and scaling in natural and social systems, including metabolic scaling, finance, and urban studies. These are, arguably, among the most successful and surprising areas of application of fractals and scaling; they are also areas of current scientific controversy. This course can thus serve as a case study of the promises and pitfalls of interdisciplinary mathematical research.

Students who successfully complete this course will gain: a thorough, mathematically grounded understanding of fractals and scaling; increased skills in applied mathematics; experience using modern statistical techniques (maximum-likelihood estimators and goodness-of-fit-tests for discrete and continuous data); and experience reading and critiquing current literature in applied mathematics.

Course evaluation will be based on several problem sets, participation in seminar-style class sessions, a final pedagogical presentation, and a short final report and annotated bibliography. Some computer work in R will be required, but no prior R experience is necessary.

 

Prerequisites:

Calculus II or the equivalent, and at least one of the following: Linear Algebra, Differential Equations, programming experience. A class in statistics will be helpful, but is not required. Permission of instructor.

Always visit the Registrar's Office for the official course catalog and schedules.