Abstract:

Probability theory is a branch of pure mathematics with strong ties to many areas of application such as statistics, economics, business, engineering, biology and physics. In terms of technique, it is largely part of analysis, though with a number of dimensions that are not evident in Math 245.

In this talk, I will describe some of the high points of the subject, beginning with topics covered in our graduate course (Math 275). The key words here are random series, laws of large numbers, central limit theorems, laws of the iterated logarithm, random walks, martingales, and Brownian motion. Each of these topics can be treated at various levels of generality, but I will restrict myself in this talk to the simplest --- coin tossing random variables.

One of the major developments in recent years is the incorporation of spatial information in the models considered. As an example of this, I will talk a bit about one area of current research --- the contact process --- which is a model for the spatial spread of infection, but interestingly also arises in high energy physics.

The prerequisites for this talk are only undergraduate mathematics (not including probability) and a questioning mind.