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.