Many American universities have their graduate students go through “qualifying examinations” before they go on to the thesis portion of their PhDs. Most of these exams can now be found online, so one can easily collect a good selection of interesting mathematics problems. Supply solutions, and you have this book. It’s a good idea, and the authors/editors have executed it well.
The first edition of this book appeared over a decade ago; for this second edition, about one hundred problems from the last decade of quals have been added. Overall there are more than 500 problems.
The organization of the problems is somewhat idiosyncratic: analysis is featured in three different “parts”, on real analysis, complex analysis, and PDEs, while algebra gets just one part, topology (both point-set and algebraic) gets another, and geometry is represented only by its differential incarnation. I’m not sure whether this means that few qualifying examinations include questions in algebraic geometry, ODEs, and combinatorics; perhaps it’s just a feature of the selection.
Each problem is presented with an indication of the department that originally posed it. Then a solution is presented. The solutions were created by a team of Chinese mathematicians; they are generally well done, though the English is occasionally a little bit exotic, as in “a set of infinite points” for “an infinite set of points” (page 165, solution to problem 2221).
At times the solutions are a little too fancy. The very first problem in the book, for example, asks for a proof that any endomorphism of a three-dimensional real vector space V has an invariant subspace. The proof uses the R[x]-module structure of V, but I suspect all that the problem-poser wanted was the observation that the characteristic polynomial, being of degree three, will necessarily have a root.
As one would expect, the solutions are terse and to the point. This is not the place to learn algebra or PDEs, but it is a great source of problems for instructors in addition to its obvious value for graduate students preparing for their qualifying exams.
Fernando Q. Gouvêa took his qualifying exams, with fear and trembling, in the fall of 1983. He is now Carter Professor of Mathematics at Colby College in Waterville, ME.