"A drunk man will find his way home, but a drunk bird may get lost forever" -Shizuo Kakutani
Hindy Drillick
hindy.drillick@columbia.edu
Course Meeting: Wednesdays from 4pm - 6pm in the Math Department, Room 417
Office Hours: Email me to make an appointment
In this seminar we will learn about random walks and Brownian Motion. In particular, we will focus on methods and techniques to answer questions such as:
Will a random walk (or Brownian motion) ever return to its starting location?
When will a random walk (or Brownian motion) first reach a certain location?
We will learn how to answer these questions for random walks on differently shaped lattices and in different dimensions. In particular we will prove Polya's famous result that a random walker in one or two dimensions will eventually return to their starting location, but a random walker in three dimensions might not.
To do this, we will explore the rich connection between random walks and the heat equation, which is a partial differential equation used to model heat flow. We will show how a random walk can be viewed as a discretized model of heat flow and how a Brownian motion can be viewed as a continuous model of heat diffusion. Inspired by this interpretation, we will then use concepts drawn from the field of partial differential equations to study random walks. This connection will give us greater insight into the nature of both random walks and heat flow.
Students should be familiar with some basic probability (e.g. the concepts of independent events, expectation and variance), linear algebra and advanced calculus. We will develop some needed probabilistic tools such as the central limit theorem along the way.
Random walks and electric networks by Doyle and Snell
Probability with Martingales by David Williams
This is a nice undergraduate introduction to measure theory and probability theory.
Random Walk: A Modern Introduction by Lawler and Limic
A more advanced book by the same author as our primary reference. (Follow the e-link to read book online.)
Grading will be based on attendance and participation. Starting from the second unexcused absence, each unexcused absence will result in the deduction of a half a letter grade (e.g. A to A-). If you are sick or have another important reason for missing class then please let me know via email so that you can be excused.
February 2: Tiber and Casey
February 9: Gavin and Eitan
February 16: Daekyun and Zara
February 23: Joseph and Daniel
March 2: Emma and Jennifer
March 9: Sterling and TBD
March: 16: spring break
March 23: TBD
March 30: TBD