## Texts

There are no required texts for this course. The following books may be helpful to supplement lectures and the book draft.

- David P. Feldman. Chaos and Fractals: An Elementary Introduction. Oxford University Press. 2012.

## Assignments

- Due Friday, September 18. Do intermediate and advanced problems from Unit 3 of the MOOC.
- Due Friday, October 2. Do intermediate and advanced problems from Unit 4 of the MOOC.

## Code

- simple regression
- bootstrap example, GPAfull.txt (Data from http://www.usna.edu/Users/math/jager/courses/sm439/

## Readings on Mandelbrot and his(?) set

- John Horgan, Who
Discovered the Mandelbrot Set?
*Scientific American*, March 13, 2009.

## Urban Scaling Papers

- Bettencourt, Luís MA, et al. Growth, innovation, scaling, and the pace of life in cities. Proceedings of the national academy of sciences 104.17 (2007): 7301-7306
- Schläpfer, Markus, et al. The scaling of human interactions with city size. Journal of The Royal Society Interface 11.98 (2014): 20130789. Also available here.
- Bettencourt, Luís
MA. The
origins of scaling in cities.
*Science*340.6139 (2013): 1438-1441. pdf here. - Gomez-Lievano, Andres, HyeJin Youn, and Luis MA Bettencourt. The statistics of urban scaling and their connection to Zipf's law. PLoS One 7.7 (2012): e40393.

*Smithsonian*article and podcast. (Victoria Jaggard, smithsonian.com, September 22, 2015.)

## Evaluation

- Several short problem sets and exercises. Some of these will be handed in; others will be presented in class. 33%
- Participation in class discussions. Students are expected to attend class and contribute to discussion. Students will also take turns leading discussions of papers and articles. 33%
- Project. Students will learn about an application or aspect of fractals or scaling and lead a class on their topic. 33%

## Goals

- Get a solid mathematical introduction to fractals, power-laws, and scaling. Understand what these terms mean and how they are applied across the sciences.
- Gain skills in reading and critiquing interdisciplinary mathematical research.
- Gain basic familiarity with the R statistics environment.
- Get an introduction to modern statistical techniques, including bootstrapping and maximum likelihood estimators.
- Gain experience learning about a topic, synthesizing material, and presenting it to peers.
- Grapple with the strengths and limitations of interdisciplinary mathematical research.

## Links

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. Level: Advanced. 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.