All the Math You Missed (but Need to Know for Graduate School)

This is the second edition of a book intended primarily for those planning to be graduate students in mathematics. The first edition was reviewed here. Incoming graduate students would find the book most useful for identifying areas where they have a gap in their background or feel that what they remember may not be sufficient. The motivating idea is to help students prepare for their first graduate courses and for any preliminary exams.

 

“All the Math You Missed (But Need to Know…)” is something of an exaggeration. No incoming graduate student is likely to have experience with all twenty of the topics discussed here. (For example, analytic number theory and category theory are not usually encountered in undergraduate work.) Nonetheless, those planning to be graduate students are likely to appreciate the opportunity to see many of the topics the author includes. I think that most graduate schools expect at least a core background that includes analysis, algebra, and topology in preparation for first-year graduate courses. The author covers these pretty well.

 

However, the author treats no topics in any detail. He is careful to include all the basics, sometimes including theorems and occasionally proofs, with plenty of examples. He also provides a small collection of exercises and suggests textbooks for filling in or refreshing the student’s background. This new edition adds four new chapters: elementary, algebraic, and analytic number theory, as well as category theory.

 

Many students contemplating graduate school are apprehensive about their level of preparation, and this book is designed to provide some useful guidance. It has a welcoming tone and is generally low-key. The author provides enough background in each of the twenty topics to give students a good general sense of those areas of mathematics, and thus a preview of some of the subjects they might study. The writing is clear and easy to read. Along the way, he also offers some good advice for reviewing more familiar material as well as for approaching topics that students might not remember as well. Those of us who have gone to graduate school might very well have appreciated what this book has to offer when we were new and uncertain.

 

The author also suggests that the book might be useful to non-mathematicians looking to understand a bit of more serious mathematics. That may lead some potential readers astray because the treatment of most topics is likely too brief to be helpful to them.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.