My next post is about a very interesting class:
Chaos Theory with Denny Gulick
You're probably wondering what we actually learn about in "chaos theory." The math behind chaos theory has to do mostly with iterated function systems. Meaning, if you have a function f(x) and an initial value x_0, then if you form the sequence x_1 = f(x_0), x_2 = f(x_1), x_3 = f(x_2) etc. you can look at properties of this sequence - whether it converges to one value, osciallates between two or more values, goes off to infinity, etc. Even simple functions like the logistic map (f(x) = kx(1-x) for a constant k) can produce very complicated behavior.
One interesting project we are doing is reading one of a selection of articles on applications of chaos theory, and writing a 1-2 page discussion of it. The article I chose is titled "Chaos on the Trading Floor" and is about the use of chaos theory to model the stock market. I imagine this is particularly relevant given the ongoing financial crisis.
Ironically, the use of mathematical techniques to model the stock market may have actually contributed to the financial crisis. According to Paul Krugman's book "The Return of Depression Economics and the Crisis of 2008," with the development of these techniques came the rise of hedge funds, which try to take advantage of statistical patterns in the stock market by making highly leveraged bets. This means for example that a hedge fund might start with $50 million in capital, then sell short (i.e. borrow and then sell) $950 million worth of one stock so they can buy $1 billion worth of another stock. This means that if the second stock goes up by 5%, they've doubled their money. Of course if it goes the other way around, then they lose all their money - and the people who they borrowed the original $950 million worth of stock from also lose money, because they're going to have a hard time getting their stock back. This kind of "ripple effect" was part of what made the financial crisis so severe.