Computation as an Experimental Tool
FYS 103A (Fall 2007)
Instructor: Manil Suri
- Manil Suri, Math/Psych 419, (410) 455-2311, firstname.lastname@example.org
- Office hours: MW 1-2. Also by appointment.
- Lectures: MW 2:30-3:45 Room ENG 333 (Computer Lab)
The goal of this seminar is to make you comfortable with the idea of using mathematical computation as a tool towards answering questions and embarking on new explorations. Through this seminar, you will experience an "experimental side" to mathematics, which does not come through in regular courses. Since this is a seminar (rather than a knowledge-based course), the idea is more to gain new insights and to expand your way of looking at things. However, you will also see some interesting topics in mathematics and get an introduction to Matlab, a computer code that performs mathematical calculations. Moreover, you will gain experience in presenting mathematical/scientific topics in non-technical terms to a general audience. Finally, class discussions will enhance the learning experience. The seminar will include a field trip. (In the past, this has been to a local high school to present some of the computer ideas developed in the class.)
We will begin with some simple (or rather, simple-sounding) problems that become surprisingly difficult to solve when they grow in size. (Bring some coins to the first lecture for these experiments.) Computer simulations will lead us to the so-called Monte Carlo method, which can be used for everything from calculating areas of figures to investigating the behavior of subatomic particles. This will serve as an introduction to the Matlab code, which is very useful to know, both for future math courses and for research purposes.
Next, we will use a web-based applet http://math.bu.edu/DYSYS/applets/nonlinear-web.html to do a project on population growth and chaos. This project is quite open-ended, since it allows you to simulate various population models from a menu and determine the characteristics of each of them. This very much gets into the flavor of experimental mathematics, since the answer will not be known in advance, and will be different, depending on the models you select.
The above project will help ease you into writing your own Matlab code, which will be needed to answer some of the questions. Further projects will be on fractals as well as (if there is time) topics suggested by some of you from your fields of interest. The idea will be to deal with open-ended problems, where you will have the opportunity to explore different avenues.
Along the way, we will cover some mathematics in class - that needed to "attack" these problems - such as elementary complex analysis for understanding fractals, vectors and matrices for understanding Matlab, complexity theory for understanding the limits to computational power. We will also spend some time examining the phenomenon of scientific computation in a historical and philosophical context (scientific experiment vs paper and pencil mathematical analysis vs computer simulation). An interesting opposing viewpoint is presented in the paper by Truesdell, "The Computer: Ruin of Science and Threat to Mankind."
You will be asked to submit written reports of your projects (and possibly give oral presentations in class on them - some of these may be group projects). There will also be opportunities to work on a Powerpoint version of the project on Chaos, for presentation to a general audience (for those interested in doing so).
This is a permission required course. Please e-mail me at email@example.com to inquire about it. The course is suitable for those who are genuinely interested in (and comfortable with) mathematics. I would like to make sure that you're taking it for the right reasons. If your ALGEBRA PLACEMENT TEST SCORE said you have to take MATH 106, YOU ARE NOT ELIGIBLE FOR THIS COURSE
Computers and MATLAB
You should have a computer account (see the UMBC webpage for starting up). Matlab is available on UMBC computers, and also can be purchased in student versions if you want it for your home computer. (We will be only using fairly elementary commands.)
There will be no tests or quizzes. You grade will be based on the
following: (percentages are approximate)
Satisfactory Completion of In-class work: 50%
HW assignments and projects: 40%-50%
Other: (possibly) 10%
Although there is no textbook for this course, we will be using the following as references:
- MATLAB users' manual.
- Mathematics by Experiment: Plausible reasoning in the 21st century (J. Borwein and D. Bailey). (The first chapter, "What is Experimental Mathematics?" will be used in discussions.)
- Encounters with Chaos (Denny Gulick). The theory is too advanced for this course, but a simplified version will be used for the experimentation and expository material in fractals.
- A New Kind of Science (Stephen Wolfram). This 1200 page book has several interesting things to say about looking at scientific theory through the lens of automata. It is a rich source of problems for mathematics experiments. Selections will be included for discussion towards the end of the course.
- The Computer: Ruin of Science and Threat to Mankind (an essay from "An Idiot's Fugitive Essays on Science") by C. Truesdell. (Discussion - depending on time).