Mathematical Bridges

This is a problem book, with very interesting and challenging problems. The unifying idea is that the problems are all susceptible to a “bridge” attack, that is, rather than work on a problem in the branch of mathematics where it originated, we attempt to carry it over a bridge to a different part of mathematics and work on it with the techniques there. A couple of good examples are given in Chapter 12 on the Extreme Value Theorem: the fundamental theorem of algebra and a form of the spectral theorem for matrices. Both problems are stated in algebra and appear to belong there, but by changing our viewpoint to one of analysis we can produce simple proofs. Another good example is Chapter 3 on determinants. When studying the determinant of a sum of matrices, \(\det(A+B)\), we can introduce a parameter and instead look at \(\det(A + xB)\). The determinant suddenly becomes a polynomial, and we know a lot about polynomials and can apply this knowledge to the determinants.

The bridge idea has been applied on a large scale in mathematics many times. Several hundred years ago Descartes had the idea of stating geometrical problems in algebraic terms and using algebra to solve them; we now call this analytic geometry. Generating functions are a technique that is used primarily for combinatorial problems, but solves them by using algebraic techniques to manipulate the functions, or (even more impressively) using analytic properties of the functions. Number theory has been especially lucky in the use of this idea. The problems are stated about integers, but can often be solved by considering them in the context of abstract algebra (field extensions) or complex analysis; there are whole sub-divisions of number theory for these techniques (algebraic number theory and analytic number theory respectively).

The problems considered in this book are more modest, and the authors do not build any grand theories, but the examples and exercises are still interesting. The subject matter is primarily linear algebra and real analysis, with a few other things mixed in. Each chapter is structured with a few worked examples and general theorems, followed by the exercises. None of the exercises is drill and most require a good bit of ingenuity. The authors are probably best known for their association with the Math Olympiads, but most of the problems are at a higher level; this is not a practice book for the Olympiads. The problems are a mixture of things from the problems sections of the American Mathematical Monthly and the Mathematics Magazine, along with some Olympiad and Putnam Competion problems, and a lot of math folklore results. A few well-known theorems are mixed in, such as Tauber’s theorem on power series (p. 161).

Very Bad Feature: no index.

The level of scholarship is very good, and I only spotted one error: Tannery’s limit theorem for series (p. 158) is misattributed to Lebesgue.

Bottom line: a unique problem book, full of interesting problems and new viewpoints.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.