Mathematics ++: Selected Topics Beyond the Basic Courses

This book has its origins, we are told, in the authors’ experiences teaching graduate students in computer science, who needed background in certain mathematical topics. Since these topics were not covered in the basic courses that these students had taken, the authors undertook to introduce them in courses spanning several semesters, the lecture notes of which, suitably expanded, became this text.

There are six chapters, all of them independent of each other, each one surveying a fairly sophisticated graduate-level topic in modern mathematics. The chapters are expository and emphasize ideas rather than gory details (though there are proofs when appropriate), and lots of attention is paid to examples. It was actually something of a surprise to me to see that some of the topics discussed here would even be relevant to computer science students.

Of the six chapters, my favorite was the last (and, at about one hundred pages in length, by far the longest), which covers topology. It starts with the definition of a topological space and proceeds quickly and efficiently through the rudiments of general topology, homotopy, simplicial homology and manifolds. This is an excellent overview of the subject. Some results (for example, the Brouwer fixed point theorem) are proved, but a number of results are just stated; given the expository nature of the chapter, this seems an entirely reasonable choice.

The first five chapters cover, in order, measure theory, high dimensional geometry, Fourier analysis, group representation theory, and polynomials. The shortest of these chapters (group representation theory) is about 25 pages long and the longest (polynomials) is about 60. Naturally, in chapters of these lengths, it is impossible to give a full account of the subject; nevertheless, even with these size constraints, a lot of interesting mathematics is presented, and even professionals might find some things here that they did not previously know.

For example, the two chapters on measure theory and Fourier analysis discuss, between them, many of the topics covered in a graduate course in real analysis, as well as some that may not get covered: Lebesgue measure and also abstract measure spaces, the Lebesgue integral and convergence theorems, topological groups and the Haar measure, the measure-theoretic foundations of probability, \(L^p\) spaces, convolutions, Fourier series, Fourier transforms, the character group of a group, and other topics.

The two most algebraic chapters in the book are 4 and 5, which cover, respectively, group representation theory and polynomials. Chapter 4 gives a quick introduction to the basics, including a proof of Maschke’s theorem. There is a section on the representations of the symmetric group, with another section applying these ideas to communication complexity. However, I did not see a single character table in this chapter, an omission that I thought was unfortunate. Another account of this material, about the same number of pages in length and having substantial topic overlap with this chapter, can be found in chapter 10 of the second edition of Artin’s Algebra, though Artin does omit the material on the symmetric group and its applications.

Chapter 5 mostly concerns algebraic geometry. The discussion is elementary (no sheaves or schemes, just affine and projective varieties) but covers some interesting material: Hilbert’s basis theorem and the Nullstellensatz, for example, are not only quoted but proved, as is a simplified version of Bézout’s theorem (it is stated here as an inequality rather than an equality, thus avoiding the need to worry about multiplicity and points at infinity; these issues are mentioned, however, which struck me as a nice touch). Considering the computer-science origins of the book, it is not surprising that algorithmic aspects of algebraic geometry are considered; there is, for example, some discussion of Gröbner bases. If I have any complaint with this chapter, it is that it does not contain as many pictures as I would have liked; nevertheless, as a quick and accessible introduction to a subject that many students find quite hard, this is a valuable resource.

Chapter 2, on high-dimensional geometry, struck me as the most difficult of the chapters in the book, perhaps because the material covered in it, unlike the subjects of the other chapters, is not routinely covered in graduate school courses and seems somewhat more specialized. But there are interesting things to be found here, even for a nonspecialist. Fun fact: if two unit vectors are selected at random in \(\mathbb{R}^n\), where \(n\) is a large positive integer, then the angle between them will, with high probability, be close to 90 degrees. Probability, in fact, turns out to be an essential tool for the subject of this chapter, and it, and measure theory, are used throughout.

One inevitable question that a reviewer must ask, and attempt to answer, is: who is the intended audience for the book? In this case, the assumed prerequisites for understanding the text include reasonable familiarity with most of the standard topics of an undergraduate mathematics curriculum. It is assumed, for example, that a reader will already know the basics of analysis, abstract algebra, and linear algebra. Thus, it seems unlikely that most undergraduate students would get a lot out of this text, though a well-prepared, highly motivated undergraduate nearing graduation might enjoy looking at this text to see what comes next.

As for graduate students, the intended audience here, as previously noted, is graduate students in theoretical computer science who need these mathematical topics. In addition, this book may appeal to graduate students in mathematics who want (say, for example, for qualifier exam preparation) a quick overview of an area of mathematics without getting bogged down in technical details. In this regard, I was reminded of the MAA Guide series, the entries of which seem to be addressed to a similar group; for example, Folland’s short book A Guide to Advanced Real Analysis has significant overlap with chapters 1 and 3 of this text, and Kendig’s A Guide to Plane Algebraic Curves also provides a nice exposition of basic algebraic geometry; it has non-empty overlap with, but is about three times as long as, chapter 5 of this book.

Consistent with its intended role as a textbook, each chapter contains a reasonable supply of exercises (no solutions provided in the text), and each chapter ends with a bibliography.

Conclusion: I like expository books, because I think, particularly in these days of increasing specialization, that they serve a valuable purpose, not only for students but also professionals who want to see what’s going on in other areas, or who need some background in one area for research in another. This book is a fine example of that genre.

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Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.