Four years ago, finishing high school, I was both excited and apprehensive about attending college. I mostly worried that I had not yet learned enough to compete with my peers. There was really no way for me to have an idea of what I would be expected to know. I now find myself in almost the exact same position. This fall I will begin my graduate studies in mathematics and I am feeling both excited and nervous.
When I mentioned this feeling of anxiety to Fernando Gouvêa, he showed me the recently published book All the Mathematics You Missed [But Need to Know for Graduate School] by Thomas Garrity. Fernando promptly asked me to review it. My immediate reaction was a sort of reassurance. The simple fact that this book has been published makes me feel that there are other students out there just like me. This book has been written for us.
It is clear that Garrity's intention was not simply to reassure nervous students. The author's hope, in fact, is to provide a general outline and broad understanding of the ideas presented; "the goal of this book is to give people at least a rough idea of the many topics that beginning graduate students at the best graduate schools are assumed to know" (xiv). These topics have been condensed into 347 pages, including the index, thus making it impossible for any subject to be treated in a detailed manner. The goal of giving beginning graduate students a broad understanding of each of these topics, without outside sources, seems much too ambitious. Garrity, in fact, provides a list of sources to further understanding of the topics presented.
I have found this book to be very readable. The book includes the following chapters: Linear Algebra, Real Analysis, Differentiating Vector-Valued Functions, Point Set Topology, Classical Stokes' Theorems, Differential Forms and Stokes' Theorem, Curvature for curves and Surfaces, Geometry, Complex Analysis, Countability and the Axiom of Choice, Algebra, Lebesgue Integration, Fourier Analysis, Differential Equations, Combinatorics and Probability Theory, and finally, Algorithms. Each chapter begins by defining the basic objects, basic maps and basic goals of the subject. For example, the chapter titled Lebesgue Integration begins with a box:
Basic Object: Measure Spaces Basic Map: Integrable Functions Basic Goal: Lebesgue Dominating Convergence Theorem
The chapter begins with a brief introduction to the motivation behind the subject matter and is then divided into the following sections: Lebesgue Measure, The Cantor Set, Lebesgue Integration, Convergence Theorems, Books, and Exercises. After having taken a course on Lebesgue Measure, this chapter serves as a quick way to refresh what I have learned. Further, the Books section, a concise bibliography, is helpful since it provides a starting place for me to learn more. The definitions and theorems are very clear and easy to understand. I appreciate that Garrity's text is not intimidating.
I found this chapter to be an effective method of refreshing the math that I didn't miss as an undergraduate. While reading the chapter I began to feel comfortable with the material again. The structure of this book, outlining the main concepts and theorems, allows a fast reading and thus aids a student to focus further study on theorems that require the most attention.
What about a subject that I know much less about? During the spring semester (my last semester as an undergraduate), I took a course on groups and representations. I decided to see how helpful All the Mathematics You Missed would be for such a situation. The section on representation theory is just two pages long. In fact, all of Abstract Algebra is contained in one chapter (17 pages), while there are 10 chapters related to Analysis. As I was taking the course, this book was not especially helpful. The lack of detail didn't answer any of the questions that arose as I was learning the subject matter for the first time. For example, the section on representation theory consists of three definitions: a representation of a group, the direct sum of representations, and an irreducible representation. The one example given in this section is a representation of S3. Looking back, I can see how these definitions were chosen, however, at the time, I did not feel that the book provided an adequate explanation of the structure of representation theory.
This experience made me realize that Garrity's book is really an outline, and cannot substitute for the mathematics that I missed as an undergraduate. Instead, the book will serve as a starting point that will help focus learning. Although, this specific example is algebra, I have gotten the impression that throughout, the book touches on the history, gives a few examples and outlines the theory. Reading the chapter on geometry (I have not taken a geometry course), I have found the same type scenario to be true. Although, I really have no way to judge what should be included in such a chapter.
After reading most of the book, I still feel a little anxious about graduate school, however, I don't think that the book made me feel this way. In fact, I have found the book to be calming in a way that leads me to believe that I will be able to handle the courses that I take. I also think that the book is really interesting, which reassures my decision to go to graduate school, a decision that I have doubted at times. I think that it will be useful to have a guide outlining the theory that I have missed and I am planning on using this book to pursue certain topics. At this point, I would recommend this book to other students planning on attending graduate school.
Mariah Hamel (firstname.lastname@example.org) graduated from Colby College in May 2002 and is now a graduate student at the University of British Columbia.