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 xiii

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.