Preface

This book is a result of the MASS course in geometry in the fall

semester of 2007. MASS core courses are traditionally labeled as

analysis, algebra, and geometry, but the understanding of each area

is broad, e.g. number theory and combinatorics are allowed as algebra

courses, topology is considered as a part of geometry, and dynamical

systems as a part of analysis. No less importantly, an interaction

of ideas and concepts from different areas of mathematics is highly

valued.

The topic came to me as very natural under these conditions. Sur-

faces are among the most common and easily visualized mathematical

objects, and their study brings into focus fundamental ideas, con-

cepts, and methods from geometry proper, topology, complex anal-

ysis, Morse theory, group theory, and suchlike. At the same time,

many of those notions appear in a technically simplified and more

graphic form than in their general “natural” settings. So, here was

an opportunity to acquaint a group of bright and motivated under-

graduates with a wealth of concepts and ideas, many of which would

be diﬃcult for them to absorb if presented in a traditional fashion.

This is the central idea of the course and the book reflects it closely.

The first, primarily expository, chapter introduces many (but not

all) principal actors, such as the round sphere, flat torus, M¨ obius strip,

Klein bottle, elliptic plane, and so on, as well as various methods of

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