This book is a continuation of the series of books by Terence Tao which derive from his mathematical blog. The preceding books were Structure and Randomness and Poincaré’s legacies, both published under the American Mathematical Society. The current review corresponds to An Epsilon of Room, volume 2. The content of this volume has been divided into two large chapters: the first one is devoted to expository articles covering a wide variety of topics, while the second one deals with more technical articles closer to the research interests of the author.
The reader will appreciate from the very first page the exquisite expository style of the author and the beauty of the proofs and ideas that are presented in the text. Though elaborated at certain steps, the technicalities in the arguments are kept to a minimum, while the important ideas are clearly highlighted, in an attempt to make them available in different contexts. The genuine insight of the author provides a perfect approach to each of the topics that are considered in the book.
The articles deal with different aspects of analysis, probability theory, mathematical physics, logic, complexity, combinatorics, number theory… The subjects covered in this volume include, for instance, Talagrand’s concentration inequality and its connection with random matrices, the Agrawal-Kayal-Saxena primality test, Grothendieck’s definition of a group, or the prime number theorem in arithmetic progressions and dueling conspiracies. This is just to mention a few of the subjects corresponding to the first part of the book. Besides, among the topics included in the second half, we find articles devoted to Szemerédi’s regularity lemma via random partitions and the correspondence principle, the Kakeya maximal function conjecture, the prevalence of determinantal point processes, and approximate bases, sunflowers, and nonstandard analysis in additive combinatorics.
Each of the topics is introduced in a very elegant way, including the appropriate motivation, but immediately, the author gets to the point of each problem, revealing where the difficulties lie. He recalls the main contributions to every issue and frequently provides examples and toy versions of the theorems that are considered. Informal overviews of some of the important arguments and/or heuristic proofs are often given in order to clarify the ideas without getting extremely technical.
All these ingredients make the book a wonderful reading to anyone with an interest in current approaches to both classical problems and emerging areas of mathematics. The author really succeeds in transmitting his passion for mathematics, and I am pretty sure that every reader will feel the necessity of broadening his/her area of interest after reading this excellent compilation of mathematical ideas.