A Fixed-Point Farrago

A farrago is “an assortment or a medley” as the book quotes from The Free Dictionary, though the online Merriam-Webster defines the word to mean “a confused mixture, or hodgepodge”. Joel Shapiro’s Fixed-Point Farrago is far from a hodgepodge. It is a carefully crafted conglomeration of chapters — culled, polished and enhanced from lectures by the author and other participants during the 2012-2013 Analysis Seminar at Portland State University.

Shapiro’s book succeeds on a number of fronts. Firstly, it is difficult to fault the author’s excellent writing style and expository talents. The chapters provide models for students to appreciate the art of a well-thought out seminar talk (or series of talks) on a particular topic or theme. There are also a variety of exercises integrated into the lectures, “to encourage active participation”, which allow the text to be used for a course or an independent study. It provides access to an active learning seminar in printed format. Readers who do not have similar opportunities in the standard format (a handful of souls, blackboard and chalk, the occasional question, and the silent scratching of heads) will find Shapiro’s text to be a lively and less dusty alternative.

Fixed-point theorems and their applications are inherently attractive material for students at all levels. The book appears to be written to engage a young audience, yet there is more in it to provide pleasure to mathematicians further on in their career. In the author’s words: “The level of exposition increases slowly, requiring at first some undergraduate-level proficiency, then gradually increasing to the kind of sophistication one might expect from a graduate student. Appendices at the back of the book provide introduction to (or reminder of) some of the prerequisite material”. The author has also included well-written and informative endnotes to each chapters that point the curious reader to the original literature and excellent secondary sources.

The book has four parts, the first two of which — titled Introduction to Fixed Points and From Brouwer to Nash — are “accessible to undergraduates whose background includes the standard junior-senior-level courses in linear algebra and analysis taught at American colleges, which hopefully provides some familiarity with basic set theory and metric spaces”. The last two parts, Beyond Brouwer: Dimension \(= \infty\) and Fixed Points for Families of Maps are in the context of slightly more advanced material, and the author deftly guides the reader through some of the basics of Hilbert and Banach spaces, and of measure theory.

Highlights from Parts I & II include proofs of the two-dimensional Brouwer fixed-point theorem via Sperner’s lemma; the Knaster-Tarski fixed-point theorem (providing an elegant proof of the Schröder-Bernstein theorem); the Rogers-Milnor analytic proof of the \(n\)-dimensional Brouwer theorem that leads via Kakutani’s set-valued fixed-point theorem to Nash’s celebrated “one-page” proof of the existence of his eponymous equilibrium. It is interesting to note that Nash credits the idea of using Kakutani’s theorem in this context to David Gale.

Part III promotes Brouwer’s theorem to infinite dimensions by proving Schauder’s fixed-point theorem, that in turn is applied to obtain Peano’s existence theorem for initial-value problems as well as Victor Lomonosov’s 1973 theorem on invariant subspaces for compact linear operators on Banach spaces. The infamous invariant subspace problem, is described in some detail and continues to stand its ground as a grail-like destination for the future. Shapiro’s operator-theoretic heart glows on his sleeve throughout this part of the book more than any other. The final Part IV covers fixed-point theorems for families of affine maps due to Markov, Kakutani, and Ryll-Nardzewski. This material leads naturally to the existence of Haar measure for compact groups, as well as to the important notion of amenability due to von Neumann (and so named by M. M. Day).

With the exception of the chapter on the invariant subspace problem and Lomonosov’s theorem, there seemed to be scant discussion regarding the trajectory of current research complementing each of the areas in the book. The topics discussed in Shapiro’s text are far from exhausted, and students coming away from “Shapiro’s seminar” need to engage with other mathematicians to find out where such beautiful strands have lead.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.