V51-1
Spring 2000
This page discusses possible projects for the course Introduction to Mathematical Modeling. It contains some rough project ideas and references to material that might be helpful. You will find that there is no substitute to browsing in the library for information. In the spring of 2000, netsurfing will probably be less profitable, because what's on the web is likely to be less reliable and less complete. The various suggestions are numbered to help people talk about them.
The American Mathematical Monthly has lots of articles that are accessible to undergraduates (and lots that are not). A good article will have good references. If you are interested in something, track down some references. Some interesting articles I came across in the last two years are:
1. "The Velocity Dependence of Aerodynic Drag: a Primer for Mathematicians", Feb, 1999, vol. 106 #2, pp. 127-135 (for someone with an interest in engineering and airplanes)
2. "Large Torsional Oscillations of Suspension Bridges Revisited: Fixing an Old Approximation", Jan, 1999, vol 106, #1, pp. 1-18 (with a discussion of the Tacoma Narrows bridge collapse, also for an engineering oriented person)
3. "What is the Correct wwy to Seed a Knockout Tournament?", Feb. 2000, vol 107, #2, pp. 140-150. (An interesting combination of probability, such as the probability of the best team winning, and what it means for a tournament to be fair)
4. "Which Tanks Empty Faster?", Dec. 1999, vol 106, #10, pp. 943-947. (For the ingineering oriented or the merely curious: how do shape a tank so that when you pull the plug, you don't have to wait long. If you're interested in this, you will have to go a bit further than this article, but it won't take much imagination to think up related optimization problems that you can treat using similar methods.)
5. "The Forced Damped Pendulum: Chaos, Complication, and Control", Oct, 1999, vol 106, #8, pp. 741-758. (Chaos is one of the most interesting aspects of dynamical systems, and one that is the least understood. Some "cultural critics of science" (an academic field unfortunately well represented on the NYU faculty) have said that only bull headed western male linear thinkers are incapable of understanding chaos. I invite the steer headed alternatives to explain it to me. This paper might be a bit too challenging, but it has a great collection of references.)
Mathematical modeling in biology and medicine is less developed and often more accessible, particularly if you are more interesting in living things than dead things. There are several very nice textbooks or books of introductory articles. A good project could be based on a chapter of one of these.
6. Charles Peskin (NYU math profressor): Mathematics in Medicine and the Life Sciences (Peskin has a knack for finding interesting and surprising conclusions from simple models.)
7. Modeling the Dynamics of Biological Systems, E. Mosedilde and O. G. Mourtisen editors, call number QH 323.5.M628. (Most of the articles here use mathematics that is too sophisticated. However, the chapters "Bone remodeling", and "Heart Rate Variability" are interesting and not too complicated.)
8. Mathematical Biology, J. D. Muray, call number QH323.5.M88. (On the title page you find "J. D. Murray, FRS". The "FRS" stands for "Fellow of the Royal Society". Membership in this society, founded by Isaac Newton, is one of the highest honors an English scientist can achieve (knighthood or the Nobel Prize excluded). This book has well written "classroom tested" chapters on a veriety of biological phenomena.)
9. MAA Series in Mathematics, vol 15: Studies in Mathematical Biology, part I and part II: S. Al Levin, editor call number QH323.5.S78. A collection of expository articles by and for applied mathematicians. One of the articles is by NYU professor John Rinzel. The MAA is the Mathematical Association of America, the outfit that publishes the American Mathematical Monthly.)
Several people have expressed an interest in finance. I will try to come up with some articles on modeling markets. In the mean time, I can suggest two books that might have an interesting chapeter or two:
10. Futures, Options, and Derivative Securities, by Hull (There is an elegant mathematical way to decide how much you should pay for a stock option. Chapters 9 and 10 of this book explain it. If you can program the "binomial tree" method, and understand it, you can start your career on Wall Street in the 6 figures.)
11. Theory of Financial Decisions, by Jonathan Ingersoll.(There is a theory that people make decisions to "maximise expected utility". If you're interested, read the first two chapters of this book and I'll give you pointers to experiments that show that actual people do no such thing.)