Preface Why did I write this book? I am frequently struck by the fact that students with an undergraduate degree in mathematics know so little about the in- terrelations between various parts of mathematics. Even those at the start of graduate school so suffer. They may know a lot of algebra and analysis, for example, but few of them have a clue that there are bridges that connect these two subjects and that each affects the other. This book is an attempt to address this situation. Having a broad view of mathematics is an advantage whether you are a high school teacher, an industrial practitioner, a professor at a liberal arts college, or a research mathematician. Everyone, including the author, has much to learn about the interconnections between various parts of mathe- matics though those engaged in exploring the boundaries of mathematics seem to eventually discover several of these connections, at least those as- sociated with their own research. I still remember my delight, as a young assistant professor, at discovering the true test for the nature of a critical point of a function of two variables. That delight turned to a feeling that my education was at fault and then to the realization that I was one of many in the same boat going upstream in the river of mathematics. This book presents material for a senior-level course for mathematics majors, including those who intend to become school teachers. Most chap- ters explore relations between different parts of mathematics. The chapters are reasonably self-contained, but some require more sophistication than others. In fact anyone who examines this book will discover that it is far from homogeneous either in its content or in its demands on the reader in both background and effort. That’s intentional and is essentially required by the the great variation in undergraduate preparation. xi

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