Computation as an Experimental Tool

FYS 103A (Fall 2008)

Instructor: Manil Suri

http://www.math.umbc.edu/~suri/fys.html

Basic Information

Learning Goals

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.

Description

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 second 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 assignments and 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).

Permission Required

This is a permission required course. Please e-mail me at suri@math.umbc.edu 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.)

Grading

There will be one mid-term test on Oct 20. You grade will be based on the following: (percentages are approximate)
Satisfactory Completion of In-class Work: 45%
HW assignments and projects: 40%
Mid-term 15%
Letter grades will follow the formula A>90%, B>80%, C>65%, D>50%

Absence from Class

Since it is very difficult to make up the computer instruction, it is essential you attend all classes. You will be considered absent if you miss more than 10 minutes of class. You are allowed a maximum of two absences from class over the semester. Additional absences, without valid reason like a doctor's note, will lead to a deduction of 3% each from the "Satisfactory Completion of In-class Work" portion of your grade.

References

Although there is no textbook for this course, we will be using the following as references:

1. MATLAB users' manual.

2. 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.)

3. 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.

4. 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, time permitting.

5. 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).

Academic Conduct

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. To read the full Student Academic Conduct Policy, consult the UMBC Student Handbook, the Faculty Handbook, or the UMBC Policies section of the UMBC Directory.