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Advanced Mathematics: An Invitation in Preparation for Graduate School

Patrick Guidotti
De Gruyter
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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The intended readers of this book are advanced undergraduates or beginning graduate students.  The author’s goal is to present a view of the broad scope of mathematics. In content, it has some similarities to All the Math You Missed, but the latter is aimed somewhat more at preparing future graduate students for what they will encounter in graduate school and what they might be expected to know.  The current book aims more to describe the overall landscape with enough detail to give prospective students some notion of what they might be getting themselves into. The author notes in his foreword that he wanted to incorporate a balance of intuition and formalism into his treatment and to illustrate the power of mathematics by presenting the resolution of certain problems.
After a short chapter on sets and functions, the author presents background, theorems, and some proofs in nine different areas. Each area gets approximately the same level of treatment.  The topics range from the topology of metric spaces, manifolds, and differentiation to optimization, partial differential equations, and probability. (Real analysis, such as measure theory or the Lebesgue integral, is not included.  The author explains this by saying analysis is a standard first-year graduate course.)  Although vector spaces and some linear algebra are described, there is no discussion of group theory or related abstract algebra.  Each chapter has some exercises, with selected answers in an appendix.
The author succeeds at giving the reader a good notion of the content and feel of these basic areas in a fairly short book.  Will prospective students find it of value?  For some students I have known – ones who expect to see only more things like the calculus they really like – it offers a sort of warning.  For students desiring a more realistic picture of the mathematics they might see in advanced courses, this offers a solid and appealing treatment. 
Bill Satzer (, 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.