MacTutor

1.1. From the Publisher.

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving mathematical problems includes numerous exercises and model solutions throughout. Assuming only basic high-school mathematics, the text is ideal for general readers and students of 14 years and above with an interest in pure mathematics.

1.2. From the Preface written in 1990.

Proclus, an ancient Greek philosopher, said:

This therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which is ours by birth.

But I just like mathematics because it is fun.

Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories, and anecdotes are important to the young in understanding real life. Mathematical problems are 'sanitised' mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a 'hide-and-seek' course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it.

In this book I shall solve selected problems from various levels and branches of mathematics. Starred problems (*) indicate an additional level of difficulty, either because some higher mathematics or some clever thinking are required; double-starred questions (**) are similar, but to a greater degree. Some problems have additional exercises at the end that can be solved in a similar manner or involve a similar piece of mathematics. While solving these problems, I will try to demonstrate some tricks of the trade when problem-solving. Two of the main weapons - experience and knowledge - are not easy to put into a book: they have to be acquired over time. But there are many simpler tricks that take less time to learn. There are ways of looking at a problem that make it easier to find a feasible attack plan. There are systematic ways of reducing a problem into successively simpler sub-problems. But, on the other hand, solving the problem is not everything. To return to the gold nugget analogy, strip-mining the neighbourhood with bulldozers is clumsier than doing a careful survey, a bit of geology, and a small amount of digging. A solution should be relatively short, understandable, and hopefully have a touch of elegance. It should also be fun to discover. Transforming a nice, short little geometry question into a ravening monster of an equation by textbook coordinate geometry does not have the same taste of victory as a two-line vector solution.

1.3. From the Preface written in 2005.

This book was written 15 years ago; literally half a lifetime ago, for me, In the intervening years, I have left home, moved to a different country, gone to graduate school, taught classes, written research papers, advised graduate students. married my wife, and had a son. Clearly, my perspective on life and on mathematics is different now than it was when I was 15. I have not been involved in problem-solving competitions for a very long time now, and if I were to write a book now on the subject it would be very different from the one you are reading here.

Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience. As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non-trivial answers, As a high-school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems (such as those in this book) and searching for the right 'trick' that would unlock each one. As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today. As a graduate student, I learnt the pride of having one's own research project. and the unique satisfaction that comes from creating an original argument that resolved a previously open question. Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realising how even very complex and deep results are often at heart be guided by very simple, even common-sensical, principles. The 'Aha!' experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one. And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.

As I wrote this book before my professional mathematics career, many of these insights and experiences were not available to me, and so in many places the exposition has a certain innocence, or even naivety. I have been reluctant to tamper too much with this, as my younger self was almost certainly more attuned to the world of the high-school problem solver than I am now. However, I have made a number of organisational changes: formatting the text in LaTex, arranging the material into what I believe is a more logical order, and editing those parts of the text which were inaccurate, badly worded, confusing, or unfocused. I have also added some more exercises. In some places, the text is a bit dated (Fermat's Last theorem, for instance, has now been proved rigorously), and I now realise that several of the problems here could be handled more quickly and clearly by more 'high-tech' mathematical tools; but the point of this text is not to present the slickest solution to a problem or to provide the most up-to-date survey of results, but rather to show how one approaches a mathematical problem for the first time, and how the painstaking, systematic experience of trying some ideas, eliminating others, and steadily manipulating the problem can lead, ultimately, to a satisfying solution.

I am greatly indebted to Tony Gardiner for encouraging and supporting the reprinting of this book, and to my parents for all their support over the years. I am also touched by all the friends and acquaintances I have met over the years who had read the first edition of the book. Last, but not least, l owe a special debt to my parents and the Flinders Medical Centre computer support unit for retrieving a 15-year old electronic copy of this book from our venerable Macintosh Plus computer!

Terence Tao
Department of Mathematics,
University of California. Los Angeles
December 2005

1.4. Review by: John Baylis.
The Mathematical Gazette 93 (526) (2009), 189-190.

We learn from the preface that the author works at the University of California as a research pure mathematician, that he is now in his early thirties and that this book was first written when he was a fifteen-year old. The preface also comments that the great changes in experiences of mathematics and of life in general in the intervening years would have produced a very different book, in particular greater use of 'high tech' mathematical techniques in tackling some of the problems, had he been writing it now. I am glad that the temptation to do such a rewrite was resisted, and am confident that many Gazette readers will agree with me. It is of course not possible to detect how much editing and 'improving' has been done, but the result is a delight. The guiding light of the book is simply the joy of mathematics, and the vehicle for expressing it is a collection of problems at the level of mathematical Olympiads and some slightly less tricky warm-up problems. The personal perspective mentioned in the title is that of a successful Olympian with a continuing involvement in mathematical competitions.
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Many mathematicians have reported the efficacy of leaving an intractable problem, doing something else or going to sleep, and letting the subconscious take over. This seemed to work for me in several cases, not as dramatically as a complete solution popping up, but certainly as a useful new idea emerging.

Of the problems I have worked on but not solved I only once yielded to the temptation of reading the solution, and I recommend this resistance to readers. Reading the solution after solving a problem is much more rewarding, and you may even get the boost of 'yes, but my solution is better' !
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This book is a welcome fresh addition to the now extensive problem literature, and my evidence for this assertion is the time taken to write this review - I was too busy enjoying the problems!

1.5. Review by: Mihaela Poplicher.
Mathematical Association of America (21 October 2006).
https://maa.org/press/maa-reviews/solving-mathematical-problems-a-personal-perspective

It is hard to say anything about this book without saying first some things about the author ... so I will start there. Terence Tao is only thirty-one years old, but has done more work than many mathematicians three times his age. He was born in Australia in 1975, has competed in the International Mathematical Olympiad in 1987, 1988 and 1989 and won a bronze, silver, and gold medal respectively. He was the younger competitor ever to win a gold medal at this very tough competition. In 1996, at the age of 21, Terence Tao got his PhD in mathematics from Princeton University. Four years later, in 2000, he was the youngest full professor in the Mathematics Department at the University of California, Los Angeles. He has published over a hundred of mathematics papers already.

The year 2006 has been especially good for Terence Tao: he was awarded a Fields Medal at the International Congress of Mathematicians, as well as a "genius grant" of $500,000 from the MacArthur Foundation. When awarded the Fields Medal, Tao was called "a supreme problem solver whose spectacular work has had an impact across several mathematical areas." The MacArthur Foundation described Tao as "a mathematician bringing technical brilliance and profound insight to a host of seemingly intractable problems in such areas as partial differential equations, harmonic analysis, combinatorics, and number theory."

And now, finally, about the book! This is a second edition, the first one having been published fifteen years ago, when the author was half his present age (i.e. fifteen)! As the author mentions in the preface to the second edition, "in many places the exposition has a certain innocence, or even naivety. I have been reluctant to temper too much with this, as my younger self was almost certainly more attuned to the world of the high-school problem solver than I am now." This is probably very good: it gives the book a very special charm.

The book is a wonderful read for anybody interested in challenging mathematics problems at the high-school level, but is invaluable for the high-school students interested in participating in the mathematical Olympiads and these students' coaches.
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There are a handful of really wonderful books that can introduce a young high-school student to the beauty of mathematics. This is definitely one of them. Besides, this book is probably going to be known as the first book written by one of the best mathematicians of the twenty-first century.

2.1. From the Publisher.

Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrödinger equation, the nonlinear wave equation, the Korteweg-de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations.

Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.

As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.

2.2. From the Terence Tao blog.

These lecture notes try (perhaps ambitiously) to introduce the reader to techniques in analysing solutions to nonlinear wave, Schrödinger, and KdV equations, in as self-contained a manner as possible. It is a six-chapter book; the first three chapters and an appendix can be found here. It is based on these lectures.

2.3. The beginning of the Preface.

Politics is for the present, but an equation is something for eternity.
(Albert Einstein)

This monograph is based on (and greatly expanded from) a lecture series given at the NSF-CBMS regional conference on nonlinear and dispersive wave equations at New Mexico State University, held in June 2005. Its objective is to present some aspects of the global existence theory (and in particular, the regularity and scattering theory) for various nonlinear dispersive and wave equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, nonlinear wave, and wave maps equations. The theory here is rich and vast and we cannot hope to present a comprehensive survey of the field here; our aim is instead to present a sample of results, and to give some idea of the motivation and general philosophy underlying the problems and results in the field, rather than to focus on the technical details. We intend this monograph to be an introduction to the field rather than an advanced text; while we do include some very recent results, and we imbue some more classical results with a modern perspective, our main concern will be to develop the fundamental tools, concepts, and intuitions in as simple and as self-contained a matter as possible. This is also a pedagogical text rather than a reference; many details of arguments are left to exercises or to citations, or are sketched informally. Thus this text should be viewed as being complementary to the research literature on these topics, rather than being a substitute for them.

The analysis of PDE is a beautiful subject, combining the rigour and technique of modern analysis and geometry with the very concrete real-world intuition of physics and other sciences. Unfortunately, in some presentations of the subject (at least in pure mathematics), the former can obscure the latter, giving the impression of a fearsomely technical and difficult field to work in. To try to combat this, this book is devoted in equal parts to rigour and to intuition; the usual formalism of definitions, propositions, theorems, and proofs appear here, but will be interspersed and complemented with many informal discussions of the same material, centring around vague "Principles" and figures, appeal to physical intuition and examples, back-of-the-envelope computations, and even some whimsical quotations. Indeed, the exposition and exercises here reflect my personal philosophy that to truly under- stand a mathematical result one must view it from as many perspectives as possible (including both rigorous arguments and informal heuristics), and must also be able to translate easily from one perspective to another. The reader should thus be aware of which statements in the text are rigorous, and which ones are heuristic, but this should be clear from context in most cases.

2.4. Review by: Sebastian Herr.
Mathematical Reviews MR2233925 (2008i:35211).

This monograph is a remarkable introduction to nonlinear dispersive evolution equations, in particular to their local and global well-posedness and scattering theory.
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Terence Tao certainly succeeds in writing a vivid and instructional text on nonlinear dispersive partial differential equations. It touches on topics of recent research interest and is a valuable source both for the beginning graduate student and, to some extent, for the advanced researcher. The numerous exercises invite the reader to actively follow the exposition. Tao greatly achieves his objectives stated in the preface by writing a pedagogical text which complements and illustrates the relevant research literature. For this purpose, the book keeps a reasonable balance between informal discussions on the one hand, and rigorous arguments on the other hand, with a consistent focus on the core of the matter.

3.1. From the Publisher.

Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years, thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate level textbook will allow students and researchers easy entry into this fascinating field. Here, for the first time, the authors bring together, in a self-contained and systematic manner, the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as the Green-Tao theorem on arithmetic progressions and Erdös distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new material.

3.2. From the Prologue.

This book arose out of lecture notes developed by us while teaching courses on additive combinatorics at the University of California, Los Angeles and the University of California, San Diego. Additive combinatorics is currently a highly active area of research for several reasons, for example its many applications to additive number theory. One remarkable feature of the field is the use of tools from many diverse fields of mathematics, including elementary combinatorics, harmonic analysis, convex geometry, incidence geometry, graph theory, probability, algebraic geometry, and ergodic theory; this wealth of perspectives makes additive combinatorics a rich, fascinating, and multi-faceted subject. There are still many major problems left in the field, and it seems likely that many of these will require a combination of tools from several of the areas mentioned above in order to solve them.

The main purpose of this book is to gather all these diverse tools in one location, present them in a self-contained and introductory manner, and illustrate their application to problems in additive combinatorics. Many aspects of this material have already been covered in other papers and texts (and in particular several earlier books have focused on some of the aspects of additive combinatorics), but this book attempts to present as many perspectives and techniques as possible in a unified setting.

3.3. Review by: Leon Vaserstein.
SIAM Review 52 (1) (2010), 215-216.

The authors of this book are famous for their research in different areas of mathematics (including applied mathematics). They won many awards and prizes, e.g., a Fields Medal for Tao, a George Polya Award for Vu, and the Information Theory Society paper award for Tao for his contribution to compressed sensing, which has already had a broad impact on a diverse set of fields, including signal processing, information theory, function approximation, MRI, radar design, and sigma-delta conversion.

The book provides a detailed exposition of important old and recent results in combinatorial number theory, including recent outstanding achievements by Bourgain, Gowers, and Katz. Probably the most celebrated result is the theorem of Green Tao asserting that the set of primes contains arbitrary long arithmetic progressions. This is a more difficult result than that of Szemeredi on progressions in subsets of integers of positive density because the primes have zero density. Several proofs of Szemeredi's result are also discussed in the book.

This book covers, in a self-contained and systematic manner, the basic tools in additive combinatorics such as sum set estimates, inverse theorems, graph theory techniques, crossing numbers, and algebraic methods. A large number of exercises is provided.

Experts in different areas (probability, ergodic theory, algebraic geometry, number theory, algebra, combinatorics, harmonic analysis, etc.) will find familiar ideas and methods as well as opportunities to learn new ones. Graduate students and applied mathematicians can learn a lot of useful mathematics.

3.4. Review by: Ben Green.
Bulletin of the American Mathematical Society 46 (3) (2009), 489-497.

The term additive combinatorics was coined a few years ago by Terry Tao to describe a rapidly developing and rather exciting area of mathematics. My personal experience is that rather few people have heard the term, though they are often familiar with some of the landmark results. When asked to define the area, I often experience a little difficulty, and in this respect I perhaps have a little in common with Dr M Kirschner, head of the Harvard School of Systems Biology, who said, "Systems biology is like the old definition of pornography: I don't know what it is, but I know it when I see it." He went on to say that "it's a marriage of the natural science [sic] and computer science with biology, to try and understand complex systems." Well one might say that additive combinatorics is a marriage of number theory, harmonic analysis, combinatorics, and ideas from ergodic theory, which aims to understand very simple systems: the operations of addition and multiplication and how they interact. Even that definition is something of an oversimplification, as a glance at the choice of topics in the book under review shows.
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Some of the most classical results in additive number theory, such as Lagrange's theorem that every integer is the sum of 4 squares or Vinogradov's result that every large odd number is the sum of three primes, can be regarded as additive-combinatorial questions about multiplicatively defined sets. However those sets, referred to in the books of Nathanson as the classical bases, are very particular. Though I am shying away from a definition of additive combinatorics, the subject often concerns more general situations.
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This book is more suitable as a reference text than for a course, though it would certainly be a useful accompaniment to a graduate course designed around a carefully selected subset of the material such as that given by Gowers in Cambridge in 1999 (which is where I learnt much of this material). I plan to use parts of it myself when lecturing a similar such course in 2009. There are a few parts of the book that are rather heavy going, although this often reflects the difficulty of the underlying material.

In summary, the book under review is a vital contribution to the literature, and it has already become required reading for a new generation of students as well as for experts in adjacent areas looking to learn about additive combinatorics (Chapter 4, for example, might be found very interesting to some theoretical computer scientists). This was very much a book that needed to be written at the time it was, and the authors are to be highly commended for having done so in such an effective way. I have three copies myself: one at home, one in the office, and a spare in case either of those should become damaged.

3.5. Review by: Sergeĭ V Konyagin and Ilya D Shkredov.
Mathematical Reviews MR2289012 (2008a:11002).

The subject of the book under review is additive combinatorics - a young and extensively developing area in mathematics with many applications, especially to number theory. Roughly speaking, one can define this area as combinatorics related to an additive group structure. Modern additive combinatorics studies various groups, from the classical group of integers to abstract groups of arbitrary nature.

It is difficult to determine a starting point for additive combinatorics. Among the origins of the theory one should mention the Cauchy theorem on set addition on the group of residues modulo a prime ... I Schur's theorem on monochromatic solutions to the equation $x + y = z$ and, certainly, the famous van der Waerden theorem on monochromatic arithmetic progressions. Probably the first serious application of combinatorial methods to classical number theory was made by Shnirelʹman ...

Van der Waerden's theorem had a great influence on the development of additive combinatorics. In this connection, it is worthy of mention that the most spectacular results of additive combinatorics, namely, Szemerédi's theorem on arithmetic progressions in subsets of the set of integers of positive density, Gowers' estimates for the density of sets without arithmetic progressions, and, of course, the theorem of Green and Tao on the existence of arbitrarily long progressions in the set of primes, are directly related to van der Waerden's theorem. The last two results - and also such outstanding achievements as the theorem of Bourgain, Katz, and Tao on sums and products of sets in finite fields and Ruzsa-Chang's refinement of Freiman's theorem - have led to the extremely active development of additive combinatorics in the last decade. During this period it has become a very rich and fruitful theory that is interacting and interlacing different areas of mathematics, such as harmonic analysis, graph theory, probability theory, ergodic theory, geometry of numbers, and algebraic geometry. This theory is beautiful and contains a lot of challenging problems. It is not a surprise that it has combined the efforts of many leading mathematicians, including the authors of the book under review. However, there has been an absence of systematic exposition of contemporary additive combinatorics (earlier results are presented in the monograph by M B Nathanson. The purpose of the book under review is to fill this gap.

The monograph is designed for a wide mathematical audience and does not require any specific background from a reader. However, everybody who intends to read this book should be ready to study tools and ideas from different areas of mathematics, which are concentrated in the book and presented in an accessible, coherent, and intuitively clear manner and provided with immediate applications to problems in additive combinatorics.
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3.6. Review by: Donald L Vestal.
Mathematical Association of America (6 June 2007).
https://maa.org/press/maa-reviews/additive-combinatorics-0

This book concerns the problem of finding the sum set or difference set of a given set of integers $A$ (or, more generally, some subset of an additive group). Starting with additive sets $A$ and $B$, the sum set $A + B$ is the set of all elements of the form $a + b$, where $a$ comes from $A$ and $b$ comes from $B$. Similarly, the difference set $A - B$ is the set of all elements of the form $a - b$, where $a$ comes from $A$ and $b$ comes from $B$. The authors study questions such as

For a given additive set $A$, what can we say about the size of sum and difference sets $A + A$ and $A - A$?

When are they "large" or "small?"

Conversely, if these sum or difference sets are large or small, what does that tell us about the original set $A$?

In general, the sum set $A + B$ will have some structure; in particular, for a given set $A$, the sum sets$A + A, 3A = A + A + A, 4A, …, kA$, will have some structure, especially as $k$ increases.

This is an incredibly dense book. Although the topic being covered may seem small enough, the authors provide an amazingly rich summary of the study of these problems. (Like many advanced mathematics texts, this book came about from lecture notes.) They include 388 references, 637 exercises, and they make use of a wide array of mathematical tools: probability, geometry, Fourier analysis, graph theory, ergodic theory, abstract algebra, even a little topology.

Coming in at just around 500 pages, one might think that the authors are verbose; quite the opposite: the writing style is terse. The proofs do not give every detail, so the reader does have to pay attention ... and will need to have some expertise in the subject. This means that the audience for this book is rather limited. But if you're interested in sum and difference sets, this is a great reference to have.

4.1. From the Publisher.

This two-volume introduction to real analysis is intended for honours undergraduates, who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning - the construction of the number systems and set theory, and then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several-variable calculus and Fourier analysis, and finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. ...

The course material is deeply intertwined with the exercises, as it is intended for the student to actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

4.2. From the Preface.

This text originated from the lecture notes I gave teaching the honours undergraduate-level real analysis sequence at the University of California, Los Angeles, in 2003. Among the undergraduates here, real analysis was viewed as being one of the most difficult course to learn, not only because of the abstract concepts being introduced for the first time (e.g., topology, limits, measurability, etc .), but also because of the level of rigour and proof demanded of the course. Because of this perception of difficulty, one was often faced with the difficult choice of either reducing the level of rigour in the course in order to make it easier, or to maintain strict standards and face the prospect of many undergraduates, even many of the bright and enthusiastic ones, struggling with the course material.

Faced with this dilemma, I tried a somewhat unusual approach to the subject. Typically, an introductory sequence in real analysis assumes that the students are already familiar with the real numbers, with mathematical induction, with elementary calculus, and with the basics of set theory, and then quickly launches into the heart of the subject, for instance the concept of a limit. Normally, students entering this sequence do indeed have a fair bit of exposure to these prerequisite topics, though in most cases the material is not covered in a thorough manner. For instance, very few students were able to actually define a real number, or even an integer, properly, even though they could visualise these numbers intuitively and manipulate them algebraically, This seemed to me to be a missed opportunity. Real analysis is one of the first subjects (together with linear algebra and abstract algebra) that a student encounters, in which one truly has to grapple with the subtleties of a truly rigorous mathematical proof. As such, the course offered an excellent chance to go back to the foundations of mathematics, and in particular the opportunity to do a proper and thorough construction of the real numbers.

Thus the course was structured as follows. In the first week, I described some well-known "paradoxes" in analysis, in which standard laws of the subject (e.g., interchange of limits and sums, or sums and integrals) were applied in a non-rigorous way to give nonsensical results such as 0 = 1. This motivated the need to go back to the very beginning of the subject, even to the very definition of the natural numbers, and check all the foundations from scratch. For instance, one of the first homework assignments was to check (using only the Peano axioms) that addition was associative for natural numbers (i.e., that $(a + b) + c = a + (b + c)$ for all natural numbers $a, b, c$). Thus even in the first week, the students had to write rigorous proofs using mathematical induction. After we had derived all the basic properties of the natural numbers, we then moved on to the integers (initially defined as formal differences of natural numbers); once the students had verified all the basic properties of the integers, we moved on to the rationals (initially defined as formal quotients of integers); and then from there we moved on (via formal limits of Cauchy sequences) to the reals. Around the same time, we covered the basics or set theory, for instance demonstrating the uncountability or the reals. Only then (after about ten lectures) did we begin what one normally considers the heart of undergraduate real analysis - limits, continuity, differentiability, and so forth.

The response to this format was quite interesting. In the first few weeks, the students found the material very easy on a conceptual level, as we were dealing only with the basic properties of the standard number systems. But on an intellectual level it was very challenging, as one was analysing these number systems from a foundational viewpoint, in order to rigorously derive the more advanced facts about these number systems from the more primitive ones. One student told me how difficult it was to explain to his friends in the non-honours real analysis sequence (a) why he was still learning how to show why all rational numbers are either positive, negative, or zero, while the non-honours sequence was already distinguishing absolutely convergent and conditionally convergent series, and (b) why, despite this, he thought his homework was significantly harder than that of his friends. Another student commented to me, quite wryly, that while she could obviously see why one could always divide a natural number $n$ into a positive integer $q$ to give a quotient $a$ and a remainder $r$ less than $n$, she still had, to her frustration, much difficulty in writing down a proof of this fact. (I told her that later in the course she would have to prove statements for which it would not be as obvious to see that the statements were true; she did not, seem to be particularly consoled by this.) Nevertheless, these students greatly enjoyed the homework, as when they did persevere and obtain a rigorous proof of an intuitive fact, it solidified the link in their minds between the abstract manipulations of formal mathematics and their informal intuition of mathematics (and of the real world), often in a very satisfying way. By the time they were assigned the task of giving the infamous "epsilon and delta" proofs in real analysis, they had already had so much experience with formalising intuition, and in discerning the subtleties of mathematical logic (such as the distinction between the "for all" quantifier and the "there exists" quantifier), that the transition to these proofs was fairly smooth, and we were able to cover material both thoroughly and rapidly. By the tenth week, we had caught up with the non-honours class, and the students were verifying the change of variables formula for Riemann-Stieltjes integrals, and showing that piecewise continuous functions were Riemann integrable. By the conclusion of the sequence in the twentieth week, we had covered (both in lecture and in homework) the convergence theory of Taylor and Fourier series, the inverse and implicit function theorem for continuously differentiable functions of several variables, and established the dominated convergence theorem for the Lebesgue integral.

5.1. Note.

The Publisher's information and the Preface are as Analysis I.

6.1. From the Publisher.

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007, Terry Tao began a mathematical blog, as an outgrowth of his own website at UCLA. This book is based on a selection of articles from the first year of that blog. These articles discuss a wide range of mathematics and its applications, ranging from expository articles on quantum mechanics, Einstein's equation , or compressed sensing, to open problems in analysis, combinatorics, geometry, number theory, and algebra, to lecture series on random matrices, Fourier analysis, or the dichotomy between structure and randomness that is present in many subfields of mathematics, to more philosophical discussions on such topics as the interplay between finitary and infinitary in analysis. Some selected commentary from readers of the blog has also been included at the end of each article. While the articles vary widely in subject matter and level, they should be broadly accessible to readers with a general graduate mathematics background; the focus in many articles is on the "big picture" and on informal discussion, with technical details largely being left to the referenced literature.

6.2. From the Preface.

Almost nine years ago, in 1999, I began a "What's new?" page on my UCLA home page in order to keep track of various new additions to that page (e.g. papers, slides, lecture notes, expository "short stories", etc.). At first, these additions were simply listed without any commentary, but after a while I realised that this page was a good place to put a brief description and commentary on each of the mathematical articles that I was uploading to the page. (In short, I had begun blogging on my research, though I did not know this term at the time.)

Every now and then, I received an email from someone who had just read the most recent entry on my "What's new?" page and wanted to make some mathematical or bibliographic comment; this type of valuable feedback was one of the main reasons why I kept maintaining the page. But I did not think to try to encourage more of this feedback until late in 2006, when I posed a question on my "What's new?" page and got a complete solution to that problem within a matter of days. It was then that I began thinking about modernising my web page to a blog format (which a few other mathematicians had already begun doing). On 22 February 2007, I started a blog with the unimaginative name of
"What's new" ...

It soon became clear that the potential of this blog went beyond my original aim of merely continuing to announce my own papers and research. For instance, by far the most widely read and commented article in my blog in the first month was a non-technical article, "Quantum Mechanics and Tomb Raider", which had absolutely nothing to do with my own mathematical work. Encouraged by this, I began to experiment with other types of mathematical content on the blog; discussions of my favourite open problems, informal discussions of mathematical phenomena, principles, or tricks, guest posts by some of my colleagues, and presentations of various lectures and talks, both by myself and by others; and various bits and pieces of advice on pursuing a mathematical career and on mathematical writing. This year, I also have begun placing lecture notes for my graduate classes on my blog.

After a year of mathematical blogging, I can say that the experience has been positive, both for the readers of the blog and for myself. Firstly, the very act of writing a blog article helped me organise and clarify my thoughts on a given mathematical topic, to practice my mathematical writing and exposition skills, and also to inspect the references and other details more carefully. From insightful comments by experts in other fields of mathematics, I have learned of unexpected connections between different fields; in one or two cases, these even led to new research projects and collaborations. From the feedback from readers I obtained a substantial amount of free proofreading, while also discovering what parts of my exposition were unclear or otherwise poorly worded, helping me improve my writing style in the future. It is a truism that one of the best ways to learn a subject is to teach it; and it seems that blogging about a subject comes in as a close second.

In the last year (2007) alone, at least a dozen new active blogs in research mathematics have sprung up. I believe this is an exciting new development in mathematical exposition; research blogs seem to fill an important niche that neither traditional print media (textbooks, research articles, surveys, etc.) nor informal communications (lectures, seminars, conversations at a blackboard, etc.) adequately cover at present. Indeed, the research blog medium is in some ways the "best of both worlds"; informal, dynamic, and interactive, as with talks and lectures, but also coming with a permanent record, a well defined author, and links to further references, as with the print media. ...

6.3. Review by: Cosma Shalizi.
American Scientist 97 (2) (2009), 160-162.

Tao is an almost ridiculously distinguished young mathematician, perhaps best known for his work in combinatorics and number theory, especially the theory of arithmetic progressions of prime numbers. In early 2007, he turned the "what's new" section of his home page into a blog, and his new book, Structure and Randomness, collects some of the writings that first appeared there: expository notes on mathematical results that are or ought to be well known, sketches of unusual proofs for classical theorems, the texts of three invited lectures, a selection of discussions of open problems, and a few curiosities, including a famous - or infamous - attempt to explain quantum mechanics in terms of the video game Tomb Raider. What should we make of this?

The first thing to say is that Tao is a mathematician writing for other mathematicians. The knowledge of modern mathematics needed to follow every thing in this book, or on his blog, is very broad. The implied reader of the expository notes is familiar with abstract algebra, algebraic geometry, functional analysis, graph theory, harmonic analysis, Lie algebras, mathematical logic, measure theory, number theory, partial differential equations, real analysis and representation theory, among other topics; other fields (most notably ergodic theory) appear as background to the lectures and as open problems. Readers needn't have very deep knowledge of any of these subjects, and no one chapter uses them all, but Tao is certainly not writing for neophytes. (Online, he usually links terms to their Wikipedia definitions, but that doesn't work in a book, of course.) Given that background, however, Tao does a fine job of providing new insights into old ideas, building intuition about why results come out the way they do, exploring why certain problems are at once interesting and hard, and explaining tricks.
...
This brings us, finally, to the fact that this book began as a series of blog posts. The essential fact about blogs is that they are extremely cheap to produce, costing only Internet access and the writer's time. This means that there is no minimum size for publications, and little or no role for the filters (peer review, editors) used in scholarly and commercial publishing to keep resources from being wasted on complete rubbish. Conceivably, Tao could have persuaded a maths journal to publish a pedagogical note on the amplification trick, but maybe not, even with his considerable authority within the discipline. With a blog, valuable material that fails to make it past the filters of traditional media can nonetheless find an outlet. Moreover, that material finds a public outlet, one that makes the traditional "invisible colleges" of scientific disciplines visible to the wider world, including those who would like to join them and contribute to the disciplines. Here Tao's blog has been exemplary, and the endnotes to these chapters record many improvements that are credited to online commenters and interactions.

Of course, filters exist for a reason, and blogs that lack them may, as a con sequence, disseminate plenty of value less material. In a well-functioning intellectual ecology, we would figure out a way to combine the advantages of the traditional filtering mechanisms with the advantages of nearly free online dis semination. Having the imprimatur of publication follow for blogs that have proved themselves might be a reason able way forward.

But such suppositions are almost al ways overtaken by events, the future being stranger than anyone really expects. What's important is that Tao has a book, and a blog, that mathematicians will definitely want to read, either on their screens or on dead trees, and it will be of interest to mathematically sophisticated readers coming from physics, statistics, economics, computer science and doubtless other disciplines. In Structure and Randomness we have a fascinating glimpse into the mind of one the best mathematicians working today.

6.4. Review by: William T Gowers.
Mathematical Reviews MR2459552 (2010h:00002).

Traditionally, there have been two kinds of book about research-level mathematics. One kind is the textbook or monograph, which chooses a certain well-defined area of mathematics and presents some of the main definitions and results in that area, giving detailed proofs and usually trying to do so in as efficient and coherent a way as possible. The other kind is the popular mathematics book, which attempts to convey to the mathematical layperson some of the excitement of mathematics without going into any details that would risk putting off the reader.

There are problems with both kinds of book. Generally, one must work extremely hard to read books of the first kind. Indeed, many mathematicians become adept at not reading them in the way that their writers hope, and instead scouring the pages for what the main ideas are, or what the result is that they actually need. At the other end of the spectrum, reading a popular mathematics book can be easy and enjoyable, but also frustrating for the experienced mathematician who will often long for more detail.

Textbooks and popular science are still the two obvious niches for mathematics in the book market, but the advent of the Internet has brought about a sudden change in the possibilities for mathematical exposition, because now anybody can put anything they like on the Web. As a result, there has been a rapid rise in a form of mathematical exposition that is too technical for the layperson, but much easier to read and enjoy for mathematicians than a textbook. A medium that is particularly well suited to this is the blog, and the undisputed king of all mathematics blogs, with thousands of regular readers, is that of Terence Tao.

Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others. Some of these are areas to which he has made fundamental contributions. Others are areas that he appears to understand at the deep intuitive level of an expert despite officially not working in those areas. How he does all this, as well as writing papers and books at a prodigious rate, is a complete mystery. It has been said that Hilbert was the last person to know all of mathematics, but it is not easy to find gaps in Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later.

Now, in an interesting experiment, several of Tao's blog posts have been tidied up (partly in response to comments from others on the posts) and published as books. So far there are three books, but there is no reason to suppose that the sequence will stop there. It is natural to ask what you get from the books that you do not get from reading the blog itself. One answer is that reading a book is still, in many ways, easier than reading from a computer screen: you can flick through pages, you can quickly see what the book contains, you can carry the book around, and so on. And while some of the material in these books is in the form of short, self-contained articles that could be read quite easily online (if you were ready to dig around amongst the very large number of posts that are there now on the blog), there are also some long sequences of articles on various topics that are more like traditional textbooks, though with more chat and with some of the gorier details left out. If you happen to want to learn about some of the areas that Tao covers in depth, then reading what Tao has to say about them is much easier than diving straight into a textbook (if you actually manage to find one that covers similar ground), and gives you a wonderful overview. Reading these extended discussions in book form will, for many people at least, be easier than reading them on the blog.

6.5. Review by: Fernando Q Gouvêa.
Mathematical Association of America (30 July 2009).
https://maa.org/press/maa-reviews/structure-and-randomness-pages-from-year-one-of-a-mathematical-blog

Blogging has created a whole new way of cultivating one's obsessions. There are many mathematical bloggers, from enthusiastic high school students to professional journalists. Active researchers have also gotten into the game, producing blogs that focus on mathematical ideas and intuitions, open problems, and recent work. These allow public access to conversations that used to be limited to a small circle; phrases such as "what's really going on" and "here's the essential idea," long banished from journals, have found a home online. Terry Tao's mathematical research blog, entitled What's New, is one of the best of these.

Much of what appears in the blogosphere is ephemeral, of course, and this is true of mathematical research blogs as well. But there is quite a bit that is worth preserving, citing, and re-reading; for those purposes, books are a better vehicle than the web. And there are old fogeys like me, for whom reading a book is far more comfortable than reading online. Hence Structure and Randomness, a collection of "pages from year one of a mathematical blog." There are more volumes to come.

Tao has selected 32 of the 93 articles he posted during 2007 and polished them for publication. The articles are grouped into three sections, corresponding to different kinds of writing about mathematical research:

(i) Informal expository writing about mathematical ideas, focusing especially (but not exclusively) on topics and techniques close to Tao's research interests.

(ii) Write-ups of lectures.

(iii) Discussions of open problems, focusing on potential strategies and the difficulties they must overcome.

According to the preface, a possible fourth category, namely discussion of recent work by others, has been left out.

I was particularly impressed by the first group of articles, which range from playful to deeply serious but are always clear and full of good ideas. There is a fascinating discussion of "Soft analysis, hard analysis, and the finite convergence principle" that would be accessible to (good) undergraduate math majors. It wrings real insight out of a distinction that often just generates heated disagreement. The chapter on non-standard analysis is also very good, providing an incredibly lucid explanation of what is going on.
...
In one article, Tao gives a fairly elementary proof of the Hilbert Nullstellensatz. "I was a little unsatisfied," he says, "with the proofs I was able to locate - they were fairly abstract and used a certain amount of algebraic machinery, which I was terribly rusty on - so, as an exercise, I tried to find a more computational proof that avoided as much abstract machinery as possible." Another article is a technical discussion of how to use "arbitrage" to strengthen inequalities.

My least favourite of the expository articles in this section is the one that seems to have started it all, an article illustrating the weirdness of the quantum world via the computer game Tomb Raider. Maybe that just reconfirms my old fogey status.

The articles in the second group, while still interesting, represent a more standard way of writing about mathematics: lecture notes. Tao has preserved some of the informality of his lectures, so the notes are pleasant to read and informative. They deal mostly with Tao's own work, so there is (necessarily) some repetition. The dominant theme of these lectures, the tension between structure and randomness and techniques for taking advantage of each, provides the title for the whole book.

The third section contains discussions of unsolved problems, focusing, in most cases, on what makes these problems hard. Tao's approach is to explain the problem, consider possible strategies for solution, and then to explain why no one has (so far) been able to get those strategies to work. As he puts it, this is the kind of conversation students have with their advisors and experts have among themselves. They should both spur new work on the problems and provide helpful guidance. (It is commonly said that in mathematics one cannot publish articles like "How I Tried and Failed to Prove the Poincaré Conjecture." Tao shows us how it can be done.)

7.1. From the Publisher.

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007, Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to non-technical puzzles and expository articles. The articles from the first year of that blog have already been published by the American Mathematical Society. The posts from 2008 are being published in two volumes. This book is Part I of the second-year posts, focusing on ergodic theory, combinatorics, and number theory. Chapter 2 consists of lecture notes from Tao's course on topological dynamics and ergodic theory. By means of various correspondence principles, recurrence theorems about dynamical systems are used to prove some deep theorems in combinatorics and other areas of mathematics. The lectures are as self-contained as possible, focusing more on the 'big picture' than on technical details. In addition to these lectures, a variety of other topics are discussed, ranging from recent developments in additive prime number theory to expository articles on individual mathematical topics such as the law of large numbers and the Lucas-Lehmer test for Mersenne primes. Some selected comments and feedback from blog readers have also been incorporated into the articles. The book is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.

7.2. From the Preface.

In February of 2007, I converted my "What's new" web page of research updates into a blog. This blog has since grown and evolved to cover a wide variety of mathematical topics, ranging from my own research updates, to lectures and guest posts by other mathematicians, to open problems, to class lecture notes, to expository articles at both basic and advanced levels.

With the encouragement of my blog readers, and also of the American Mathematical Society, I published many of the mathematical articles from the first year (2007) of the blog, which will henceforth be referred to as Structure and Randomness throughout this book. This gave me the opportunity to improve and update these articles to a publishable (and citeable) standard, and also to record some of the substantive feedback I had received on these articles from the readers of the blog. Given the success of the blog experiment so far, I am now doing the same for the second year (2008) of articles from the blog. This year, the amount of material is large enough that the blog will be published in two volumes.

As with Structure and Randomness, each part begins with a collection of expository articles, ranging in level from completely elementary logic puzzles to remarks on recent research, which are only loosely related to each other and to the rest of the book. However, in contrast to the previous book, the bulk of these volumes is dominated by the lecture notes for two graduate courses I gave during the year. The two courses stemmed from two very different but fundamental contributions to mathematics by Henri Poincaré, which explains the title of the book.

This is the first of the two volumes, and it focuses on ergodic theory, combinatorics, and number theory.

7.3. Review by: William T Gowers.
Mathematical Reviews MR2459552 (2010h:00002).

In the second book, there is a section of about 150 pages on ergodic theory, which is derived from a set of lecture notes for a course that Tao gave, and there is a chapter of about 35 pages on additive prime number theory, based on an invited lecture and an invited lecture series.
...
As for the shorter articles, these make excellent mathematical "bedtime reading". Although they are quite diverse, there are also some common themes. For example, a central interest of Tao's is the relationship between hard analysis and soft analysis. ... Most of them are connected in one way or another with Tao's research interests, so if you have research interests that overlap with those of Tao, then it is a great pleasure to read through this random (but not completely random) sample of what is going on in his mind. Given how broad Tao's research interests are, that applies to just about everyone.

8.1. From the Publisher.

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007, Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to non-technical puzzles and expository articles. The articles from the first year of that blog have already been published by the AMS. The posts from 2008 are being published in two volumes. This book is Part II of the second-year posts, focusing on geometry, topology, and partial differential equations. The major part of the book consists of lecture notes from Tao's course on the Poincare conjecture and its recent spectacular solution by Perelman. The course incorporates a review of many of the basic concepts and results needed from Riemannian geometry and, to a lesser extent, from parabolic PDE. The aim is to cover in detail the high-level features of the argument, along with selected specific components of that argument, while sketching the remaining elements, with ample references to more complete treatments. The lectures are as self-contained as possible, focusing more on the 'big picture' than on technical details. In addition to these lectures, a variety of other topics are discussed, including expository articles on topics such as gauge theory, the Kakeya needle problem, and the Black-Scholes equation. Some selected comments and feedback from blog readers have also been incorporated into the articles. The book is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.

8.2. From the Preface.

This is the second of the two volumes, and it focuses on geometry, topology, and partial differential equations. In particular, Chapter 2 contains the lecture notes for my course on the famous Poincaré conjecture that every simply connected compact three-dimensional manifold is homeomorphic to a sphere, and its recent spectacular solution by Perelman. This conjecture is purely topological in nature, and yet Perelman's proof uses remarkably little topology, instead working almost entirely in the realm of Riemannian geometry and partial differential equations, and specifically in a detailed analysis of solutions to Ricci flows on three-dimensional manifolds, and the singularities formed by these flows. As such, the course will incorporate, along the way, a review of many of the basic concepts and results from Riemannian geometry (and to a lesser extent, from parabolic PDE), while being focused primarily on the single objective of proving the Poincaré conjecture. Due to the complexity and technical intricacy of the argument, we will not be providing a fully complete proof of this conjecture here but we will be able to cover the high-level features of the argument, as well as many of the specific components of that argument, in full detail, and the remaining components are sketched and motivated, with references to more complete arguments given. In principle, the course material is sufficiently self-contained that prior exposure to Riemannian geometry, PDE, or topology at the graduate level is not strictly necessary, but in practice, one would probably need some comfort with at least one of these three areas in order to not be totally overwhelmed by the material. (I ran this course as a topics course; in particular, I did not assign homework.)

8.3. Review by: William T Gowers.
Mathematical Reviews MR2459552 (2010h:00002).

Most of the third book is devoted to Perelman's proof of the Poincaré conjecture, which Tao covered in a graduate course. I remember when I first heard that Tao had written a detailed exposition of this proof: all I could do was laugh. He might say that the tools that Perelman used, from PDEs and Riemannian geometry, were tools that he (Tao) knew a great deal about, but this is nevertheless a breath-taking achievement.

9. Terence Tao, Analysis. I (Second edition) (Hindustan Book Agency, New Delhi, 2009).

Since the publication of the first edition, many students and lecturers have communicated a number of minor typos and other corrections to me. There was also some demand for a hardcover edition of the texts. Because of this, the publishers and I have decided to incorporate the corrections and issue a hardcover second edition of the textbooks. The layout, page numbering, and indexing of the texts have also been changed; in particular the two volumes are now numbered and indexed separately. However, the chapter and exercise numbering, as well as the mathematical content, remains the same as the first edition, and so the two editions can be used more or less interchangeably for homework and study purposes.

10.1. Note.

The Publisher's information and the Preface are as Analysis I.

11.1. From the Publisher.

In 2007 Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to nontechnical puzzles and expository articles. The first two years of the blog have already been published by the American Mathematical Society. The posts from the third year are being published in two volumes. The present volume consists of a second course in real analysis, together with related material from the blog. The real analysis course assumes some familiarity with general measure theory, as well as fundamental notions from undergraduate analysis. The text then covers more advanced topics in measure theory, notably the Lebesgue-Radon-Nikodym theorem and the Riesz representation theorem, topics in functional analysis, such as Hilbert spaces and Banach spaces, and the study of spaces of distributions and key function spaces, including Lebesgue's $L^{p}$ spaces and Sobolev spaces. There is also a discussion of the general theory of the Fourier transform. The second part of the book addresses a number of auxiliary topics, such as Zorn's lemma, the Carathéodory extension theorem, and the Banach-Tarski paradox. Tao also discusses the epsilon regularisation argument - a fundamental trick from soft analysis, from which the book gets its title. Taken together, the book presents more than enough material for a second graduate course in real analysis. The second volume consists of technical and expository articles on a variety of topics and can be read independently.

11.2. From the Preface.

The current text contains many (though not all) of the posts for the third year (2009) of the blog, focusing primarily on those posts of a mathematical nature which were not contributed primarily by other authors, and which are not published elsewhere. It has been split into two volumes.

The current volume consists of lecture notes from my graduate real analysis courses that I taught at UCLA, together with some related material. These notes cover the second part of the graduate real analysis sequence here, and therefore assume some familiarity with general measure theory (in particular, the construction of Lebesgue measure and the Lebesgue integral), as well as undergraduate real analysis (e.g., various notions of limits and convergence). The notes then cover more advanced topics in measure theory (notably, the Lebesgue-Radon-Nikodym and Riesz representation theorems) as well as a number of topics in functional analysis, such as the theory of Hilbert and Banach spaces, and the study of key function spaces such as the Lebesgue and Sobolev spaces, or spaces of distributions. The general theory of the Fourier transform is also discussed. In addition, a number of auxiliary (but optional) topics, such as Zorn's lemma, are discussed

11.3. Review by: Oscar Blasco.
Mathematical Reviews MR2760403 (2012b:42002).

The volume under review contains the lecture notes from the graduate real analysis course taught by the author at UCLA, together with some material related to it. The contents of the book, which is expected to have a continuation under the name of "Volume II", as mentioned in the introduction by the author, is done using the posts from the third year of the blog started by Terence Tao under the title "What's new", from which several mathematical articles were published in two previous volumes ...
...
It is divided into two main chapters. In the first chapter the author revises several topics in real analysis. Each section contains the required definitions, theorems and proofs and it is full of examples and exercises to help the reader get a quick (but quite strong in the case that all the exercises are solved) knowledge of most of the basic subjects in real analysis, being a beautiful compound of measure theory, functional analysis and harmonic analysis.
...
The book also contains a second chapter including some related and more specific articles completing the previous sections. Most of these articles contain new ideas and points of view of the author on some classical notions and results. I list them all: An alternate approach to the Carathéodory extension theorem; Amenability, the ping-pong lemma and the Banach-Tarski paradox; The Stone and Loomis-Sikorski representation theorems; Well-ordered sets, ordinals, and Zorn's lemma; Compactification and metrisation; Hardy's uncertainty principle; Create an epsilon of room (which gives the title of the book and where Tao discusses a trick using an epsilon regularisation argument and uses it for some applications); Amenability.

11.4. Review by: Allen Stenger.
Mathematical Association of America (4 March 2011).
https://maa.org/press/maa-reviews/an-epsilon-of-room-i-real-analysis-pages-from-year-three-of-a-mathematical-blog

As real analysis books go, this one is very abstract. Much of it deals with topics that would usually be considered functional analysis, such as the Hahn-Banach Theorem and the Open Mapping Theorem, along with quite a lot on dual spaces. The most concrete topic in it is the Fourier transform, and even this is done on locally-compact abelian groups rather than on the real line.

The book is intended as a second graduate course in analysis, after a course on measure and integration. It originated as a blog that posts lecture notes for a course 245C in real analysis at UCLA that uses Folland's Real Analysis: Modern Techniques and Their Applications as a secondary text. The blog entries have been cleaned up somewhat here and the result does indeed read like a textbook, not like lecture notes or a blog. ...

Overall I was disappointed in this book. It is a competent treatment and does include a number of interesting things, but it is also ordinary. It doesn't have the kind of brilliancies and insights that you would expect from a mathematician of this author's calibre. The first half of the book is a fairly conventional course in functional analysis. Things improve somewhat in the next quarter of the book, which turns these very general theorems to more particular structures, in particular a long chapter on Fourier transforms and a shorter one on distributions. The last quarter of the book is a potpourri of sidelines from the main text and you may or may not like it based on your interests; it has quite a lot on the Axiom of Choice, and several short articles on specialised topics.

12.1. From the Publisher.

In 2007 Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to nontechnical puzzles and expository articles. This second volume contains a broad selection of mathematical expositions and self-contained technical notes in many areas of mathematics, such as logic, mathematical physics, combinatorics, number theory, statistics, theoretical computer science, and group theory.

12.2. From the Preface.

The current volume consists of sundry articles on a variety of mathematical topics, which I have divided (somewhat arbitrarily) into expository articles (Chapter 1) which are introductory articles on topics of relatively broad interest, and more technical articles (Chapter 2) which are narrower in scope, and often related to one of my current research interests. These can be read in any order, although they often reference each other, as well as articles from previous volumes in this series.

12.3. Review by: D R Heath-Brown.
Mathematical Reviews MR2780010 (2012c:00001).

This book is an improved and updated version of part of the author's blog.

The year 2007 has appeared as a single book while 2008 was dealt with in two volumes. The posts from 2009 have again been split into two volumes, the first of which consists primarily of lecture notes on real analysis.

This, the second volume for 2009, is divided into "expository articles" and "technical articles", though in reality the latter are largely expository as well. The style of the exposition is relatively informal, concentrating on the key ideas and leaving many of the details to the reader. This will make the volume particularly useful for anyone wanting to read some elegant mathematics in an unfamiliar area. However, the serious student will have quite a bit of homework to do.

The blog covers a remarkable spread of topics. There are articles on sailing faster The current volume consists of sundry articles on a variety of mathematical topics, which I have divided (somewhat arbitrarily) into expository articles (Chapter 1) which are introductory articles on topics of relatively broad interest, and more technical articles (Chapter 2) which are narrower in scope, and often related to one of my current research interests. These can be read in any order, although they often reference each other, as well as articles from previous volumes in this series. than the wind, on Benford's Law, on Bose-Einstein condensates, and on algebraic topology, for example. However, there is perhaps a concentration around themes in the spectrum running from combinatorics to analytic number theory.

Overall, this is a fascinating book in which to dabble, with much elegance, and much that will inspire the reader. It is recommended to all readers from graduate students up.

13.1. Note.

The fact that a paperback edition was published shows the popularity of the book.

14.1. From the Publisher.

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Radamacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis.

There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasised. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text.

As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

14.2. From the Preface.

In the fall of 2010, I taught an introductory one-quarter course on graduate real analysis, focusing in particular on the basics of measure and integration theory, both in Euclidean spaces and in abstract measure spaces. This text is based on my lecture notes of that course, which are also available online on my blog, together with some supplementary material, such as a section on problem solving strategies in real analysis which evolved from discussions with my students.

This text is intended to form a prequel to my graduate text (An epsilon of room, Vol. I ), which is an introduction to the analysis of Hilbert and Banach spaces (such as $L^{p}$ and Sobolev spaces), point-set topology, and related topics such as Fourier analysis and the theory of distributions; together, they serve as a text for a complete first-year graduate course in real analysis.

The approach to measure theory here is inspired by the text [E Stein and R Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces], which was used as a secondary text in my course. In particular, the first half of the course is devoted almost exclusively to measure theory on Euclidean spaces $\mathbb{R}^{d}$ (starting with the more elementary Jordan-Riemann-Darboux theory, and only then moving on to the more sophisticated Lebesgue theory), deferring the abstract aspects of measure theory to the second half of the course. I found that this approach strengthened the student's intuition in the early stages of the course, and helped provide motivation for more abstract constructions, such as Carathéodory's general construction of a measure from an outer measure.

Most of the material here is self-contained, assuming only an undergraduate knowledge in real analysis (and in particular, on the Heine-Borel theorem, which we will use as the foundation for our construction of Lebesgue measure); a secondary real analysis text can be used in conjunction with this one, but it is not strictly necessary. A small number of exercises however will require some knowledge of point-set topology or of set-theoretic concepts such as cardinals and ordinals.

A large number of exercises are interspersed throughout the text, and it is intended that the reader perform a significant fraction of these exercises while going through the text. Indeed, many of the key results and examples in the subject will in fact be presented through the exercises. In my own course, I used the exercises as the basis for the examination questions, and signalled this well in advance, to encourage the students to attempt as many of the exercises as they could as preparation for the exams.

14.3. Review by: Takis Konstantopoulos.
The American Mathematical Monthly 120 (8) (2013), 762-768.

First, let me explain how I came across it. A few months ago, I visited a big book store at Harvard Square, and, as usual, headed straight for the maths section. I had heard about Tao's recent book, which grew out of his blog, where Tao offers lively instruction to students worldwide. Initially, it looked like yet another book on the topic. Strangely, however, I started reading and was quickly absorbed by its style. I observed that it starts humbly, almost trivially one might say, by talking about Jordan content and the Riemann integral and ends up with proving Rademacher's theorem on the almost everywhere differentiability of Lipschitz functions on $\mathbb{R}^{d}$, a theorem which many know of, but few have seen its proof. And it looked like the path from Jordan to Rademacher was fully justified in 200 pages. So I bought a copy of it and went through it in detail, something that was not hard to do because of its vivid style, reminiscent of a mathematical conversation (not surprising, since it is the out come of a (good) blog). In the end, I concluded that this is a book which I can (and will) use for an advanced undergraduate course, because it talks to the reader, it avoids the abstract Carathéodory measurability, it makes a smooth transition from Jordan-Riemann-Darboux concepts to Lebesgue measure, it leaves room for the reader to think, and also it goes along way!

14.4. Review by: Mahendra G Nadkarni.
Mathematical Reviews MR2827917 (2012h:28003).

The book under review is a rather complete first course on measure and integration, giving details, discussions, and often visual descriptions of interesting ideas and examples of the theory. The reviewer also finds novelty in the presentation of the deeper aspects of the topic such as differentiation theorems, which are developed via density arguments and estimates on the Hardy-Littlewood maximal function.
...
The entire book is not just an introduction to measure theory as the title says but a lively dialogue on mathematics with a focus on measure theory. Chapter 2 is a part of this dialogue with students and readers, giving strategies for problem solving, proofs of the Rademacher differentiation theorem, and five Kolmogorov consistency theorems on measures in infinite product spaces.

14.5. Review by: Mihaela Poplicher.
Mathematical Association of America (14 April 2012).
https://maa.org/press/maa-reviews/an-introduction-to-measure-theory

This text is based on the lecture notes for a one-quarter graduate course in real analysis. The course and the book focus on the basics of measure and integration theory, both in Euclidean spaces and in abstract measure spaces. The author mentions that this text is intended as a prequel to his 2010 book An Epsilon of Room I, which is an introduction to the analysis of Hilbert and Banach spaces. These two books can serve as material for a complete graduate course in real analysis.

The author used as an inspiration the book by E Stein and R Shakarchi Real Analysis. Measure Theory, Integration and Hilbert Spaces. In particular, the first half of the text (Chapter 1) is about measure theory in Euclidean spaces $\mathbb{R}^{d}$, and the abstract aspects of measure theory are deferred to the second half (Chapter 2). This is because this approach strengthens the students' intuition in the first part of the course, while providing motivation for more abstract facts, such as Carathéodory's general construction of a measure from an outer measure.

Most of the material of the text is self contained and addressed to the students with only an undergraduate knowledge of real analysis. Some exercises require also some knowledge of point-set topology or set theory.

There are many exercises; in fact, many of the results and examples are presented through exercises. The intention of the author is that the reader perform a significant portion of these exercises while going through the book; the students taking a course based on this book and working through the exercises will have the added benefit of being well-prepared for the examinations.

The first half of the book (Chapter 1) includes all the basics of Measure Theory and makes-up the material for a full course on the subject. The second half (Chapter 2) contains Related Articles (problem solving strategies, probability spaces, the Rademacher differentiation theorem, infinite product spaces and the Kolmogorov extension theorem) - all of which could be considered as "optional material".

The best part is that the author has a wonderful blog containing plenty of material, as well as a special part of the blog for the course he taught on Measure Theory.

For instructors, mathematicians and especially graduate students, this text, as well as the blog and the other books which appeared or are in preparation by Tao constitute a treasure trove of material by one of the best mathematicians of our time. It is a pleasure and a special gift to have these resources.

15.1. From the Publisher.

The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

15.2. From the Preface.

In the winter of 2010, I taught a topics graduate course on random matrix theory, the lecture notes of which then formed the basis for this text. This course was inspired by recent developments in the subject, particularly with regard to the rigorous demonstration of universal laws for eigenvalue spacing distributions of Wigner matrices. This course does not directly discuss these laws, but instead focuses on more foundational topics in random matrix theory upon which the most recent work has been based. For instance, the first part of the course is devoted to basic probabilistic tools such as concentration of measure and the central limit theorem, which are then used to establish basic results in random matrix theory, such as the Wigner semicircle law on the bulk distribution of eigenvalues of a Wigner random matrix, or the circular law on the distribution of eigenvalues of an iid matrix. Other fundamental methods, such as free probability, the theory of determinantal processes, and the method of resolvents, are also covered in the course.

This text begins in Chapter 1 with a review of the aspects of probability theory and linear algebra needed for the topics of discussion, but assumes some existing familiarity with both topics, as well as a first-year graduate-level understanding of measure theory. ...

The core of the book is Chapter 2. While the focus of this chapter is ostensibly on random matrices, the first two sections of this chapter focus more on random scalar variables, in particular, discussing extensively the concentration of measure phenomenon and the central limit theorem in this setting. These facts will be used repeatedly when we then turn our attention to random matrices, and also many of the proof techniques used in the scalar setting (such as the moment method) can be adapted to the matrix context. Several of the key results in this chapter are developed through the exercises, and the book is designed for a student who is willing to work through these exercises as an integral part of understanding the topics covered here. The material in Chapter 3 is related to the main topics of this text, but is optional reading.

This text is not intended as a comprehensive introduction to random matrix theory, which is by now a vast subject. For instance, only a small amount of attention is given to the important topic of invariant matrix ensembles, and we do not discuss connections between random matrix theory and number theory, or to physics.

15.3. Review by: Steven Joel Miller.
Mathematical Reviews MR2906465 (2012k:60023).

The last few years have been a time of exciting progress in random matrix theory, as many longstanding conjectures have finally fallen. The present book appears at an opportune moment to introduce and excite students about the field. As there are already numerous good survey articles and technical books about the many aspects of the field, from classical random matrix theory to connections with number theory, the author here chooses to restrict the focus to a small but important part, the spectral distribution of random Wigner and iid matrix ensembles. This textbook is meant to introduce graduate students to the field and give them both a thorough grounding on the prerequisite material as well as a tour through the field. The book is well written, with extensive references and well-chosen problems. It can serve as both an introduction to the subject and a guide to the existing literature. As the author and his colleagues are responsible for many of the recent advances, he knows from direct experience which perspectives are fruitful, and is able to highlight these appropriately.
...
This is a well-written book, providing a very accessible introduction for those with a solid background in probability and analysis. It is a terrific introduction to the subject, as the author constantly emphasises where the various techniques can and cannot be used, and why. The author has succeeded in providing a good tour through an important part of random matrix theory, and readers will be well-prepared to continue further after reading this book.

15.4. Review by: Benjamin Schlein.
Jahresbericht der Deutschen Mathematiker-Vereinigung 115 (2013), 57-59.

The book under consideration is a basic introduction to the field of random matrix theory, which has been very active in the last years. It aims at graduate students interested in this subject or willing to start to work in this direction.

The general goal of random matrix theory is the understanding of the statistical properties of the eigenvalues and of the eigenvectors of $N \times N$ matrices, whose entries are random variables with a given probability law. Typically, one is interested in the behaviour of the spectrum in the limit of large $N$.

Random matrices have first been introduced in the fifties by Wigner, to describe the excitation spectrum of heavy nuclei. Wigner's basic idea was the following. When dealing with very complex systems, it is impossible to write down the precise Hamilton operator. Instead, it makes sense to assume the matrix entries of the Hamiltonian to be random variables, and to establish properties of the eigenvalues which hold for typical realisations of the disorder (the eigenvalues of the Hamiltonian are the energy level of the system and determine the excitation spectrum observed in experiments). It turns out that Wigner's idea was very successful and, to this day, random matrices are widely used in nuclear physics to predict the spectrum of heavy nuclei, at least as a first approximation. Since their introduction, random matrices have been linked to several other branches of mathematics and physics. The spectrum of a large class of disordered and chaotic systems shares many similarities with the one of simple ensembles of random matrices. The success of Wigner's intuition and their ubiquitous appearance are signs for one of the most remarkable properties of random matrices; universality. In vague terms, universality refers to the fact that local spectral properties of systems with randomness depend on the underlying symmetry but are largely independent of further details, like the precise distribution of the randomness.
...
The book "Topics in Random Matrix Theory" by Terence Tao is based on a graduate course that the author gave at UCLA in 2010. It contains three chapters. The first chapter contains a general introduction to probability theory and additional preparatory material related to eigenvalues of sums of Hermitian matrices. Chapter two starts with a review of the phenomenon of concentration of measure, which turns out to be an important tool for the analysis of random matrices. It continues with a discussion of basic topics in random matrix theory, including the operator norm of random matrices, the semicircle law of Wigner matrices, the circular law for matrices without symmetry and an introduction to free probability. Chapter three, on the other hand, is dedicated to a selection of related subjects; Dyson Brownian motion, the Golden-Thompson inequality and the derivation of Wigner-Dyson's sine-kernel for the (bulk) correlations of GUE (and of the Airy kernel for the edge correlations).

16.1. From the Publisher.

Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in ground-breaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemerédi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems.

This book, which is the first monograph in this area, aims to cover all of these topics in a unified manner, as well as to survey some of the most recent developments, such as the application of the theory to count linear patterns in primes. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one's knowledge.

16.2. From the Preface.

Traditionally, Fourier analysis has been focused on the analysis of functions in terms of linear phase functions such as the sequence $n \mapsto e(\alpha n) := e^{2\pi i\alpha n}$. In recent years, though, applications have arisen - particularly in connection with problems involving linear patterns such as arithmetic progressions - in which it has been necessary to go beyond the linear phases, replacing them to higher order functions such as quadratic phases $n \mapsto e(\alpha n^{2})$. This has given rise to the subject of quadratic Fourier analysis and, more generally, to higher order Fourier analysis.

The classical results of Weyl on the equidistribution of polynomials (and their generalisations to other orbits on homogeneous spaces) can be interpreted through this perspective as foundational results in this subject. However, the modern theory of higher order Fourier analysis is very recent indeed (and still incomplete to some extent), beginning with the breakthrough work of Gowers and also heavily influenced by parallel work in ergodic theory, in particular, the seminal work of Host and Kra. This area was also quickly seen to have much in common with areas of theoretical computer science related to polynomiality testing, and in joint work with Ben Green and Tamar Ziegler, applications of this theory were given to asymptotics for various linear patterns in the prime numbers.

There are already several surveys or texts in the literature that seek to cover some aspects of these developments. In this text (based on a topics graduate course I taught in the spring of 2010), I attempt to give a broad tour of this nascent field. This text is not intended to directly substitute for the core papers on the subject (many of which are quite technical and lengthy), but focuses instead on basic foundational and preparatory material, and on the simplest illustrative examples of key results, and should thus hopefully serve as a companion to the existing literature on the subject.

16.3. Review by: David Conlon.
Mathematical Reviews MR2931680.

Recent progress in additive combinatorics has shown the need for higher-order analogues of Fourier analysis.
...
The book is split into two parts. The first part is the core of the book and discusses the origins and applications of higher-order Fourier analysis, building towards a discussion of the author's work, with Green and Ziegler, on asymptotics for linear equations in the primes. Along the way, he discusses a number of topics of interest, including the classical theory of equidistribution (from multiple perspectives), Roth's theorem on three-term arithmetic progressions in dense subsets of the integers (again from multiple perspectives), and the inverse theorems for the Gowers uniformity norms. These inverse theorems play a key role both in proving Szemerédi's theorem and in deriving the correct asymptotics for linear equations in the primes, and their study has been one of the key factors behind the development of a higher-order Fourier analysis.

The second part of the book consists of edited versions of a number of related posts taken from Tao's blog. There is a lengthy discussion of ultralimit analysis (which was used in the work of Green, Tao and Ziegler on inverse theorems for the uniformity norms) and its applications to quantitative algebraic geometry, in particular a quantitative version of a theorem of Gromov on groups of polynomial growth. The author also discusses higher-order analogues of Hilbert spaces, where the usual binary inner product is replaced by a $2^{d}$-ary inner product between $2^{d}$ functions, and the classical uncertainty principle.

16.4. Review by: Tom Sanders.
Jahresbericht der Deutschen Mathematiker-Vereinigung 115 (2014), 207-209.

The book is based on a topics graduate course taught by the author in 2010 and is not intended as a substitute for the (rather formidable) core papers on the subject. Instead it focuses on basic foundational material and illustrative examples.
...
This area is a technically very demanding area and even understanding the statements of many of the results requires an investment. This book provides a number of very helpful insights from one of the architects of the theory and, as such, is a valuable resource.

17.1. From the Publisher.

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such "folklore mathematics". But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog.

The articles, essays, and notes in this book are derived from the author's mathematical blog in 2010. It contains a broad selection of mathematical expositions, commentary, and self-contained technical notes in many areas of mathematics, such as logic, group theory, analysis, and partial differential equations. The topics range from the foundations of mathematics to discussions of recent mathematical breakthroughs.

17.2. From the Preface.

In February of 2007, I converted my "What's new" web page of research updates into a blog. This blog has since grown and evolved to cover a wide variety of mathematical topics, ranging from my own research updates, to lectures and guest posts by other mathematicians, to open problems, to class lecture notes, to expository articles at both basic and advanced levels. In 2010, I also started writing shorter mathematical articles on my Google Buzz feed.

This book collects some selected articles from both my blog and my Buzz feed from 2010, continuing a series of previous books based on the blog.

The articles here are only loosely connected to each other, although many of them share common themes (such as the titular use of compactness and contradiction to connect finitary and infinitary mathematics to each other). I have grouped them loosely by the general area of mathematics they pertain to, although the dividing lines between these areas is somewhat blurry, and some articles arguably span more than one category. Each chapter is roughly organised in increasing order of length and complexity (in particular, the first half of each chapter is mostly devoted to the shorter articles from my Buzz feed, with the second half comprising the longer articles from my blog).

17.3. Review by: Grigore Ciurea.
Mathematical Reviews MR3026767.

Tao's book provides several articles, essays, and notes derived from the author's mathematical blog in 2010. In the preface Tao writes: "This blog has since grown and evolved to cover a wide variety of mathematical topics, ranging from my own research updates, to lectures and guest posts by other mathematicians, to open problems, to class lecture notes, to expository articles at both basic and advanced levels."

The book covers a lot of wonderful material, from the logic and foundations of mathematics to discussions of recent mathematical breakthroughs, and can be used in many ways. How can it cover so much? Generally, the author avoids technical issues, ignoring details, and refers to the literature when expedient.

Two hundred and fifty pages are divided into six chapters; each chapter ends with a summary or a commentary concerning the topic which has been described.
...
The book contains historical remarks, related discussion, and connections to other areas of science, especially physics. Some of these items are informal and many are interesting. At times the prose interferes with the mathematics. Of course, the book cannot be used as a text, because of the need for expedient references, but it offers a wonderful perspective on various parts of the mathematical landscape.

18.1. From the Preface.

The third edition contains a number of corrections that were reported for the second edition, together with a few new exercises, but is otherwise essentially the same text.

19.1. From the Preface.

The third edition contains a number of corrections that were reported for the second edition, together with a few new exercises, but is otherwise essentially the same text.

20.1. From the Publisher.

Winner of the 2015 Prose Award for Best Mathematics Book!

In the fifth of his famous list of 23 problems, Hilbert asked if every topological group which was locally Euclidean was in fact a Lie group. Through the work of Gleason, Montgomery-Zippin, Yamabe, and others, this question was solved affirmatively; more generally, a satisfactory description of the (mesoscopic) structure of locally compact groups was established. Subsequently, this structure theory was used to prove Gromov's theorem on groups of polynomial growth, and more recently in the work of Hrushovski, Breuillard, Green, and the author on the structure of approximate groups.

In this graduate text, all of this material is presented in a unified manner, starting with the analytic structural theory of real Lie groups and Lie algebras (emphasising the role of one-parameter groups and the Baker-Campbell-Hausdorff formula), then presenting a proof of the Gleason-Yamabe structure theorem for locally compact groups (emphasising the role of Gleason metrics), from which the solution to Hilbert's fifth problem follows as a corollary. After reviewing some model-theoretic preliminaries (most notably the theory of ultraproducts), the combinatorial applications of the Gleason-Yamabe theorem to approximate groups and groups of polynomial growth are then given. A large number of relevant exercises and other supplementary material are also provided.

20.2. From the Preface.

Hilbert's fifth problem, from his famous list of twenty-three problems in mathematics from 1900, asks for a topological description of Lie groups, without any direct reference to smooth structure. As with many of Hilbert's problems, this question can be formalised in a number of ways, but one commonly accepted formulation asks whether any locally Euclidean topological group is necessarily a Lie group. This question was answered affirmatively by Montgomery-Zippen and Gleason. As a by-product of the machinery developed to solve this problem, the structure of locally compact groups was greatly clarified, leading in particular to the very useful Gleason-Yamabe theorem describing such groups. This theorem (and related results) have since had a number of applications, most strikingly in Gromov's celebrated theorem on groups of polynomial growth, and in the classification of finite approximate groups. These results in turn have applications to the geometry of manifolds, and on related topics in geometric group theory.

In the fall of 2011, I taught a graduate topics course covering these topics, developing the machinery needed to solve Hilbert's fifth problem, and then using it to classify approximate groups and then finally to develop applications such as Gromov's theorem. Along the way, one needs to develop a number of standard mathematical tools, such as the Baker-Campbell-Hausdorff formula relating the group law of a Lie group to the associated Lie algebra, the Peter-Weyl theorem concerning the representation-theoretic structure of a compact group, or the basic facts about ultrafilters and ultraproducts that underlie nonstandard analysis.

This text is based on the lecture notes from that course, as well as from some additional posts on my blog on further topics related to Hilbert's fifth problem. The first chapter of this text can thus serve as the basis for a one-quarter or one-semester advanced graduate course, depending on how much of the optional material one wishes to cover. The material here assumes familiarity with basic graduate real analysis (such as measure theory and point set topology), and including topics such as the Riesz representation theorem, the Arzelà-Ascoli theorem, Tychonoff's theorem, and Urysohn's lemma. A basic understanding of linear algebra (including, for instance, the spectral theorem for unitary matrices) is also assumed.

20.3. Review by: Ben Joseph Green.
Mathematical Reviews MR3237440.

This book consists of two parts, the first much longer than the second. The first part, addressed in the bulk of this review, is a discussion of the structure theory of locally compact groups and related topics connected with "Hilbert's fifth problem". The second part is a miscellaneous selection of topics, in which respect it is similar to several previous texts based on the author's weblog. The reader interested in group theory and/or additive combinatorics will find much of interest in this second part, but so will others. Whilst some of the material (the Jordan-Schur theorem, Ado's theorem, the Peter-Weyl theorem) is quite standard, other parts are far less so.
...
In summary, this book is a very useful and well-written contribution to the literature, greatly clarifying the whole area. It provides a fine introduction for anyone wishing to understand recent advances in additive combinatorics, written by the main player in the field, but it will also be of great help to those wishing to learn about the theory of locally compact groups for its own sake.

20.4. Review by: Isaac Goldbring.
Department of Mathematics, University of California, Irvine.
https://www.math.uci.edu/~isaac/Taobookreview-final.pdf

The book under review (based primarily on the author's lecture notes and blog posts) centres around three major theorems: the positive resolution of Hilbert's fifth problem, due to the combined efforts of Gleason, Montgomery, and Zippin; the structure theorem for finite approximate groups, in its full generality due to Breuillard, Green, and Tao, building upon a major breakthrough by Hrushovski; and Gromov's theorem for polynomial growth. Each of these theorems are widely considered to be jewels in their respective areas of mathematics: the theory of locally compact groups, additive combinatorics, and geometric group theory respectively. The book proves these theorems in full details, draws analogies and connections between them, and, most importantly to the logic audience, explains the use of ultraproduct/nonstandard techniques in deriving these results.

The cornerstone around which the rest of the book revolves is the positive solution to Hilbert's fifth problem. Recall that a Lie group is a smooth manifold G endowed with the structure of a group for which the group operations are smooth functions. (In other words, a Lie group is a group object in the category of smooth manifolds.) In particular, a Lie group is a locally Euclidean topological group, that is, a topological space equipped with a group structure for which the group operations are continuous and which further possesses an open neighbourhood of the identity homeomorphic to some open subset of some (finite-dimensional) Euclidean space. Perhaps the most common interpretation of Hilbert's fifth problem asks whether or not this a priori weaker structure, namely that of being a locally Euclidean topological group, already implies the stronger structure of being a Lie group, that is, whether or not the locally Euclidean group can be equipped with the structure of a smooth manifold (compatible with the given topology on the group) for which the group operations are now upgraded from being mere continuous functions to being smooth functions.

21.1. From the Publisher.

Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemerédi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.

21.2. From the Preface.

Expander graphs are a remarkable type of graph (or more precisely, a family of graphs) on finite sets of vertices that manage to simultaneously be both sparse (low-degree) and "highly connected" at the same time. They enjoy very strong mixing properties: if one starts at a fixed vertex of an (two-sided) expander graph and randomly traverses its edges, then the distribution of one's location will converge exponentially fast to the uniform distribution. For this and many other reasons, expander graphs are useful in a wide variety of areas of both pure and applied mathematics.

There are now many ways to construct expander graphs, but one of the earliest constructions was based on the Cayley graphs of a finite group (or of a finitely generated group acting on a finite set). The expansion property for such graphs turns out to be related to a rich variety of topics in group the- ory and representation theory, including Kazhdan's property (T), Gowers' notion of a quasirandom group, the sum-product phenomenon in arithmetic combinatorics, and the Larsen-Pink classification of finite subgroups of a linear group. Expansion properties of Cayley graphs have also been applied in analytic number theory through what is now known as the affine sieve of Bourgain, Gamburd, and Sarnak, which can count almost prime points in thin groups.

This text is based on the lecture notes from a graduate course on these topics I gave at UCLA in the winter of 2012, as well as from some additional posts on my blog on further related topics. The first chapter of this text can thus serve as the basis for a one-quarter or one-semester advanced graduate course, depending on how much of the optional material one wishes to cover. While the material here is largely self- contained, some basic graduate real analysis (in particular, measure theory, Hilbert space theory, and the theory of $L^{p}$ norms), graph theory, and linear algebra (e.g. the spectral theorem for unitary matrices) will be assumed. Some prior familiarity with the classical Lie groups (particularly the special linear group $SL_{n}$ and the unitary group $U_{n}$) and representation theory will be helpful but not absolutely necessary.

21.3. Review by: Harald A Helfgott.
Mathematical Reviews MR3309986.

This is a book based on the author's blog. The first half consists of lecture notes for a course taught by the author a few years ago on expansion in Cayley graphs. The second half is much more loosely organised; it is essentially a collection of blog posts having a more or less close link to the subject of the book.

The reviewer has just taught a course using this book as one of his main sources. The first thing to say is that it is undoubtedly a very useful resource. However, whoever uses the book for teaching will be well advised to collect other sources, and to take some time to reflect while reading each chapter, pondering whether matters can be taught in a shorter or cleaner way.
...
Reviewer remarks: Unfortunately, an excellent blog does not necessarily translate into a uniformly first-rate book, especially when the path from the former to the latter has clearly been very short. A blog has an element of journalism to it, and the standards of journalism and permanent literature are not the same. It is useful to have the present text in a permanent format, but one must ask oneself whether the hard cover is worth the loss (of hyperlinks, say) incurred in the change in format, particularly when little else is added in the process.

At the same time, this is a very valuable resource, though, as we have seen, there are some particulars in which it could be improved. We can hope that both the book and this review will prove useful to whoever writes a definitive work on the subject.

21.4. Review by: Alexander Lubotzky.
Bulletin of the American Mathematical Society 56 (2) (2019), 361-366.

The beautiful book of Terry Tao starts with the following words:

[see Preface above]

Indeed, expander graphs have emerged as the area with the most fruitful interactions between computer science and pure mathematics. In computer science, expanders appear everywhere as basic building blocks of (communication) networks, in algorithms, derandomization, error correcting codes, and much more. The reader is referred to [Shlomo Hoory, Nathan Linial, and Avi Wigderson, Expander graphs and their applications (2006)] for an excellent survey (though in the last decade since it was written, so much more has been done that an updated version will be a welcome addition to the literature). The current book gives the story from a different angle: the importance of expander graphs in pure mathematics and the use of pure mathematics to further advance the theory of expanders. In this sense this book follows [Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures (1994)], but so much has been done in the last twenty-five years, and the theory went to some totally unexpected directions, that except for some similarity in the early chapters, the books are very different.

Reviewing this book gives an opportunity to describe the fascinating development this area has made in the last decades. In spite of (or maybe because) I am personally involved in this process, going over this book was, for me, a wonderful journey in a beautiful interdisciplinary mathematics.
...
In summary, this wonderful book describes, in a readable fashion, an active area of mathematics whose beauty is also due to the fact that it is very interdisciplinary. The author dedicates almost half of the 300 pages to six appendices on various subjects, where each one is a lovely, short course in itself which helps in reading the main part of the book.

I must end with a personal remark. The book appeared in the series Graduate Studies in Mathematics of the AMS, and as such it is indeed suitable for a pretty advanced graduate course. I have been involved in this area for the last thirty years, still I am learning a lot from this book. This is an outstanding exposition by one of the most outstanding mathematicians of our generation, which can be read at several different levels.

22.1. From the Publisher.

This is part one of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning - the construction of the number systems and set theory - then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each.

The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

The fourth edition incorporates a large number of corrections reported since the release of the third edition, as well as some new exercises.

Note.

The Publisher gives the same information as for Analysis I.