A Mathematician's Survival Guide is a new addition to Stephen's Krantz "How To", series, the best known of which is How To Teach Mathematics. The present book, written in the same strong and opinionated tone, is meant to guide prospective mathematicians through the process of getting a graduate degree in mathematics, getting a job, and obtaining tenure.
The book is based on the author's extensive experience as faculty at several well-known institutions, and on his role as advisor to many graduate and undergraduate students. The first eight chapters are devoted to providing the reader with step-by-step descriptions of the various stages of the process of becoming a mathematician: how to prepare for graduate school, how to choose a graduate school, how to pass the qualifying exams, how to choose an advisor and a thesis topic, how to write your dissertation, how to get a job, and how to get tenure.
The ninth chapter has a different flavor. The author gives lists of typical subject area contents for the qualifying exams, exemplified by the current requirement at Washington University — Stephen Krantz's home institution. In addition, he writes a very brief synthesis of each subject listed, highlighting the main ideas. Even though the subject areas of the exams and their contents vary somewhat from institution to institution, I found this last chapter of the book very valuable for the prospective student who wants to know what will be expected of him. For those interested in actual examples of qualifier-type problems, the authors cites Berkeley Problems in Mathematics by Silva and De Souza, but this overview of topics definitely has a place in a graduate school self-help book.
As with any guidance book, the essential question to be answered is "Is it helpful?" To summarize it in one sentence, this book tells the reader to "work hard", "communicate with others", and "plan ahead". All of these are most definitely good advice, and I find it refreshing that Stephen Krantz is actually putting it in writing that you should work hard in order to succeed (although most people either do work hard without being told, or pay no heed to this kind of advice). But the book's main value is in the "plan ahead" part, because it gives the reader the necessary information to do so. It takes tremendous effort and organization to make it to the final step of getting a job and obtaining tenure in today's world, and being uninformed can be very costly. I would definitely recommend to every mathematics department to keep a copy of this book for their undergraduate and graduate students.
Ioana Mihaila (firstname.lastname@example.org) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is interested in mathematics competitions.