How the World Really Works

2006 Fields Medalist Andrei Okounkov discussed the Law of Large Numbers and a few of its consequences. Essentially if we are looking at a random path of n steps that starts at ordered pair (0,0) and goes to (T,X), where T is some time and X is some location, then as the number of steps gets very large the probability that the path varies from a straight line is next to nothing. In other words by far the most likely thing to happen is a straight line between (0,0) and (T,X), which turns out to not be that random after all.

This result was extended to cover three dimensional walks as well. What this means, at least as far as I understand it, is that the things we see as very precise and not random at all are actually so random that they gain, as Okounkov said, an Optimality and Elegance that makes us think they are not random.

All in all I felt it was one of the most understandable of the Clay Lectures I have attended in my two years at PCMI. The speaker was funny, and to my surprise, a little bit uncomfortable at times. This seemed to be in part because English is not his native language and in part because of the challenges of addressing such a diverse crowd. He took the time to state things in a way that was very accessible, which was nice. Having the chance to interact with such individuals during mealtimes and in such intimate settings is a feature that I have only experienced at PCMI.

And so we see that our world is, in fact, very random.