viii Preface

particular, solutions of the heat equation can be found using diago-

nalization of symmetric matrices.

The next chapter goes to continuous time and continuous space.

We start with the Brownian motion which is the limit of random walk.

This is a fascinating subject in itself and it takes a little work to show

that it exists. We have separated the treatment into Sections 2.1

and 2.6. The idea is that the latter section does not need to be read

in order to appreciate the rest of the chapter. The traditional heat

equation and Laplace equation are found by considering the Brownian

particles. Along the way, it is shown that the matrix diagonalization

of the previous chapter turns into a discussion of Fourier series.

The third chapter introduces a fundamental idea in probability,

martingales, that is closely related to harmonic functions. The view-

point here is probabilistic. The final chapter is an introduction to

fractal dimension. The goal, which is a bit ambitious, is to determine

the fractal dimension of the random Cantor set arising in Chapter 3.

This book is derived from lectures given in the Research Ex-

periences for Undergraduates (REU) program at the University of

Chicago. The REU is a summer program taken in part or in full

by about eighty mathematics majors at the university. The students

take a number of mini-courses and do a research paper under the

supervision of graduate students. Many of the undergraduates also

serve as teaching assistants for one of two other summer programs,

one for bright junior high and high school students and another de-

signed for elementary and high school teachers. The first two chapters

in this book come from mini-courses in 2007 and 2008, and the last

two chapters from a 2009 course.

The intended audience for these lectures was advanced undergrad-

uate mathematics majors who may be considering graduate work in

mathematics or a related area. The idea was to present probability

and analysis in a more advanced way than found in undergraduate

courses. I assume the students have had the equivalent of an advanced

calculus (rigorous one variable calculus) course and some exposure to

linear algebra. I do not assume that the students have had a course in

probability, but I present the basics quickly. I do not assume measure

theory, but I introduce many of the important ideas along the way,