xii Preface

Green’s Theorem, Dehn’s Dissection Theorem, the Cauchy Rigidity

Theorem, and the Fundamental Theorem of Algebra.

All the material in the book can be found in various textbooks,

though there probably isn’t one textbook that has it all. Whenever

possible, I will point out textbooks or other sources where you can

read more about what I am talking about. The various fields of math

surrounding the concept of a surface—geometry, topology, complex

analysis, combinatorics—are deeply intertwined and often related in

surprising ways. I hope to present this tapestry of ideas in a clear

and rigorous yet informal way.

My general view of mathematics is that most of the complicated

things we learn have their origins in very simple examples and phe-

nomena. A good way to master a body of mathematics is to first

understand all the sources that lead to it. In this book, the square

torus is one of the key simple examples. A great deal of the the-

ory of surfaces is a kind of elaboration of phenomena one encounters

when studying the square torus. In the first chapter of the book, I

will introduce the square torus and describe the various ways that

its structure can be modified and generalized. I hope that this first

chapter serves as a good guide to the rest of the book.

I aimed the class at fairly advanced undergraduates, but I tried

to cover each topic from scratch. My idea is that, with some effort,

you could learn the material for the whole course without knowing

too much advanced math. You should be perfectly well prepared for

the intended version of the class if you have had a semester each of

real analysis, abstract algebra, and complex analysis. If you have just

had the first 2 items, you should still be alright, because I embedded

a kind of mini-course on complex analysis in the middle of the book.

Following an introductory chapter, this book is divided into 6

parts. The first 5 parts have to do with different aspects of the theory

of surfaces. The 6th part is a collection of several topics, loosely

related to the rest of the book, which I included because I really like

them. Here is an outline of the book.