1.1. From the Publisher.

We present a self-contained account of measure theory and integration in a Banach space. We give a detailed analysis of the weak Baire probabilities on a Banach space E, and on its second dual. Scalarly (= weak) measurable functions valued in E are studied via their image measure and it is shown how to regularise them using lifting. General criteria are given to ensure that they are Pettis integrable. This study relies on tools from topological and abstract measure theory.

1.2. From the Abstract.

We present a self-contained account of measure theory and integration in a Banach space. We give a detailed analysis of the weak Baire probabilities on a Banach space E and on its second dual. Scalarly (= weak) measurable functions valued in E are studied via their image measure and it is shown how to regularise them using liftings. General criteria are given to ensure that they are Pettis integrable; when it is the case, we study when the corresponding vector measure has a compact range. When E is a dual space, an almost complete description is given. Under very mild conditions, functions have versions which are Pettis integrable, and conditional expectation exist. We also investigate the converse problem of the Weak RNP, i.e., when bounded variation measures have a Pettis density, and related properties. This study relies on tools from topological and abstract measure theory. Most of the striking results rely on properties of pointwise compact sets of measurable functions. The theory of these sets is developed in great detail, but unfortunately the techniques available make use of (mild) set theoretic assumptions. Application of the measure-theory techniques is given to a non-standard characterisation of Riemann-measurable functions on the torus.

1.3. From the Introduction.

The theory of Pettis integration and more generally of measure theory in a Banach space was started in the thirties by Pettis, Dunford and others. It made only slow progress for a long time, until the 1977 paper of G Edgar brought renewed attention to the topic. The remarkable advances of Edgar's paper are permitted by the use of elaborate tools from topological measure theory and abstract measure theory. Since Edgar's paper, the pace of development was much faster, but the progress still came from applications of increasingly more sophisticated measure theoretic results. This line of approach has been so successful (modulo, a set-theoretical restriction to be explained later) that all the problems which were considered as important a few years ago have been solved. The theory is now in search of new directions, as well as of applications of its many powerful results. At this stage, there is a need for a self-contained unified approach of the theory and its tools.

The main results of this work should be accessible with only a working knowledge of abstract measure theory, and elementary knowledge of functional analysis and topological measure theory. No previous knowledge of Pettis integration is required and the reader should never have to look for a reference in specialised papers but rather only in basic books.

The book is organised in three parts. Chapters 1 through 7 deal with Pettis integration and topological measure theoretic tools. Pettis integration also needs abstract measure theoretic tools. These tools are used in Chapters 2, 6 and 7 but since they are more technical, their study is delayed until Chapters 8 through 14. Chapters 15 and 16 deal with applications and more specialised questions.
...

I was introduced to this kind of measure theory by D H Fremlin back in 1976. This work contains many of his results and is considerably influenced by his powerful ideas.

Part of the research presented here, as well as the actual writing of the book was done while I visited the Department of Mathematics at The Ohio State University. I am grateful to Professors W Davis and W Johnson for making this visit possible. The excellent atmosphere and working conditions were of great help. Special thanks are due to G Edgar, from whom I learned a lot during many conversations, and who suggested many improvements.

I am very grateful to the Department of Mathematics of The Ohio State University for generously supporting the typing of this work; this actually made possible its publication under the present form. Expert typing was done by Barbara Pletz, Donna Cotton and Elaine Bolton whom went through the various versions of the work with exceptional care and patience.

1.4. Review by: J J Uhl, Jr.
Mathematical Reviews MR0756174 (86j:46042).

The Pettis integral for vector-valued functions was introduced by B J Pettisin 1938 and has remained a mystery ever since. In his original paper, Pettis made his contribution to the Orlicz-Pettis theorem for the purpose of proving that the Pettis integral is countably additive. This subtle fact was the only nontrivial property of the Pettis integral for the better part of thirty years. In retrospect, this is not so surprising because it is now clear that the kind of measure theory needed to deal with the Pettis integral did not even begin to arrive on the scene until 1975 with the publication of Fremlin's subsequence theorem. This theorem says that under the assumption of a perfect measure space, a sequence of measurable functions either has an a.e. convergent subsequence or a non-measurable pointwise cluster point! The reviewer believes that this theorem and its offspring have completely changed measure theory. "Halmos-type" measure theory is fine, but it is not enough to deal with many problems of today; in particular it is not enough to handle the Pettis integral.

When Fremlin's theorem appeared it was seized on by Stegall, Geitz and others who used it skilfully to begin to understand the Pettis integral. Tools like Fremlin's subsequence theorem become extremely potent in the hands of a measure-theorist like Talagrand who also has a trick or two of his own up his sleeve.

Now something about the memoir under review. What the author has done is to build measure theory to the point at which he can make a good run at the Pettis integral. The memoir is pleasing to read because it begins with elementary material and becomes increasingly deeper and deeper, thus allowing a reader to complete it or to bail out at the right time.
...

This impressive memoir will leave its imprint on measure theory for a long time.

2.1. From the Publisher.

Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.

2.2. From the Introduction.

Probability in Banach spaces is a branch of modern mathematics that emphasizes the geometric and functional analytic aspects of probability theory. Its probabilistic sources may be found in the study of regularity of random processes (especially Gaussian processes) and Banach-space-valued random variables and their limit properties, whose functional developments revealed and tied strong and fruitful connections with classical Banach spaces and their geometry.
...

This book emphasises the recent use of isoperimetric inequalities and related concentration of measure phenomena, and of modern random process techniques in probability on Banach spaces. The two parts are introduced by chapters on isoperimetric background and generalities on vector valued random variables.
...

In the first part we study vector-valued random variables, their integrability and tail behaviour properties and strong limit theorems for sums of independent random variables. Successively, vector-valued Gaussian variables, Rademacher series, stable variables and sums of independent random variables are investigated using recent isoperimetric tools. The strong law of large numbers and the law of the iterated logarithm, for which the almost sure statement is shown to reduce to the statement in probability, complete this first part with extensions to infinite-dimensional Banach-space-valued random variables of some classical real limit theorems. In the second part, tightness of sums of independent random variables and regularity properties of random processes are presented. The link to geometry of Banach spaces through type and cotype is developed with applications in particular to the central limit theorem. General random processes are then investigated and regularity of Gaussian processes characterised via majorizing measures with applications to random Fourier series. The book is completed with an account on empirical process methods and with several applications, especially to local theory of Banach spaces, of the probabilistic ideas presented in this work.
...

We would like to mention that the topics of probability in Banach spaces selected in this book are not exhaustive and actually only reflect the tastes and interests of the authors.
...

On our exposition itself, we took the point of view that completeness is sometimes prejudicial to clarity. With a few exceptions, this work is however self-contained. Actually, many of our choices, reductions or simplifications were motivated only by our lack of resistance. We did not try to avoid repetitions and we use from time to tie results that are proved or stated only further in the exposition.

2.3. Review by: Evarist Giné.
Mathematical Reviews MR1102015 (93c:60001).

This book gives an excellent, almost complete account of the whole subject of probability in Banach spaces, a branch of probability theory that has undergone vigorous development during the last thirty years. There is no doubt in the reviewer's mind that this book will become a classic.
...

Regarding the first subject, Talagrand completely characterised sample boundedness and continuity of Gaussian processes by means of their covariances (or, equivalently, by means of the geometric properties of their index sets). In connection with the second, a unifying theme, at least for almost sure behaviour of sums of independent random vectors, is an isoperimetric inequality of Talagrand for product measures. Isoperimetric inequalities (on the sphere and in Gauss space) had already been shown to be very useful in the study of Gaussian processes by C Borell and others, and also in limit theorems via randomisation. The extensive use of techniques from these two subjects gives elegance and unity to the exposition and at the same time leads to best, definitive accounts of many topics. The book is divided into a preamble, or Part 0, "Isoperimetric background and generalities", and two parts: Part I, "Banach space random variables and their strong limit properties", and Part II, "Tightness of vector valued random variables and regularity of random processes".
...

As warned by the authors, there are a few interesting topics of probability in Banach spaces not covered by this book, in particular, Banach-space-valued martingales and their relation to geometry, many aspects of empirical process theory, large deviations in the context of Banach spaces, infinitely divisible distributions and the general central limit theorem in Banach spaces, and rates of convergence in the central limit theorem (but there are books on the last two subjects).

The notes and references at the end of every chapter provide exceptionally complete and accurate accounts of the history of the subject. The bibliography is complete.

Although this is mainly a reference book for researchers and teachers of probability and linear analysis, it can also be used as an advanced graduate text.

2.4. Review by: Marjorie G Hahn.
The Annals of Probability 22 (2) (1994), 1115-1120.

This is a book about the importance of inequalities and the power of isoperimetric methods. The vehicle chosen to illustrate these concepts is the theory of probability in Banach spaces.
...

Talagrand's recent paper on "Concentration of measure and isoperimetric inequalities in product spaces," presented in March 1993 as an IMS Special Invited Lecture at the Second International Symposium on Probability and Its Applications in Bloomington, Indiana, shows that isoperimetry and the associated concentration of measure phenomena have substantial implications for other areas of probability as well. These are typically areas which have problems involving a large number (perhaps infinitely many) of random variables (independent or not). The areas in which profound applications have already been established include stochastic models in physics (such as percolation) and computer science (bin packing, assignment problem, geometric probability). The Ledoux-Talagrand book serves both as a good, convincing illustration of the power of the isoperimetric methods and as good preparation for the further developments. Consequently, the book should prove useful to readers from many diverse areas in pure and applied probability, as well as in functional analysis and the geometry of Banach spaces.
...

The book is divided into three parts. Part 0 concerns the isoperimetric background and generalities on vector-valued random variables. Part I is devoted to Banach-space-valued random variables and their strong limiting properties. Part II focuses on tightness of vector-valued random variables and regularity of random processes.
...

In summary, the power and beauty of this book is that it represents a shift in perspective in probability in Banach spaces: Inequalities are of fundamental importance and isoperimetric methods are a powerful means of establishing them. These insights and the results they have yielded thus far are likely to stimulate more research in many different directions.

3.1. From the Publisher.

In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called "spin glasses". These models are simple and rather canonical random structures, that physicists studied by non-rigorous methods. They predicted spectacular behaviours, previously unknown in probability theory. They believe these behaviours occur in many models of considerable interest for several branches of science (statistical physics, neural networks and computer science).

This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics, and contains proofs in complete detail of much of what is rigorously known on spin glasses at the time of writing.

3.2. From the Introduction.

Consider a large finite collection $(X_{k})_{k≤M}$ of random variables. What can we say about the largest of them? More generally, what can we say about the "few largest" of them? When the variables $X_{k}$ are probabilistically independent, everything is rather easy. This is no longer the case when the variables are correlated. Even when the variables are identically distributed, the answer depends very much on their correlation structure. What are the correlation structures of interest? Most of the familiar correlation structures in probability are low-dimensional, or even "one-dimensional". This is because they model random phenomena indexed by time, or, equivalently, by the real line, a one-dimensional object. In contrast with these familiar situations, the correlation structures considered here will be "high-dimensional" - in a sense that will soon become clear and will create new and truly remarkable phenomena.
...

The mathematical objects studied in this book are of a rather fundamental nature. Yet they have been discovered by physicists. At the beginning of an already long story are "real" spin glasses, alloys with strange magnetic properties, which are of considerable interest, both experimentally and theoretically. It is believed that their remarkable properties arise from a kind of disorder among the interactions of magnetic impurities. To explain (at least qualitatively) the behaviour of real spin glasses, theoretical physicists have invented a number of models. They fall into two broad categories: the "realistic" models, where the interacting atoms are located at the vertices of a lattice, and where the strength of the interaction between two atoms decreases when their distance increases; and the "mean field" models, where the geometric location of the atoms in space is forgotten, and where each atom interacts with all the others. The mean-field models are of special interest to mathematicians because they are very basic mathematical objects. Yet some physicists predict extremely intricate structures arising from these objects. (As for the "realistic" models, they appear to be intractable at the moment.) Moreover, they believe that their ideas are applicable in a wide range of situations. ... The methods used by the physicists are probably best described here as a combination of heuristic arguments and numerical simulation. They are probably reliable, but they have no claim to rigour, and it is often not even clear how to give a precise mathematical formulation to some of the central predictions.

It is rather paradoxical for a mathematician like the author to see simple, basic mathematical objects being studied by the methods of theoretical physics. It is also very surprising, given the obvious importance of what the physicists have done, and the breadth of the paths they have opened, that mathematicians have not tried earlier to prove their conjectures.

This book will study through mathematical methods some of these fascinating mathematical objects discovered by the physicists. The reader has already been explained all the statistical mechanics (s)he needs to know. No knowledge whatsoever of physics is required or probably even useful to read it. The vast body of knowledge and techniques from statistical mechanics has little relevance because the overwhelming feature is the randomness of the interactions. (This randomness seems in particular to preclude the existence of "infinite-volume limiting objects".) Most of the book is devoted to finding ways to estimate apparently hopelessly complicated random quantities.

The main purpose of the book is to bring this wonderful new area of mathematics discovered by the physicists to the attention of the mathematical community.
...

It is customary for authors, at the end of an introduction, to warmly thank their spouse for having granted them the peaceful time needed to complete their work. I find that these thanks are far too universal and overly enthusiastic to be believable. Yet, I must say that in the present case even what would sound for the reader as exaggerated thanks would not truly reflect the extraordinary privileges I have enjoyed. Be jealous, reader, for I yet have to hear the words I dread the most: "Now is not the time to work."

3.3. Review by: Anton Bovier.
Mathematical Reviews MR1993891 (2005m:82074).

Mathematical models for spin glasses were proposed in the 1970s: one considers a graph $\Gamma$ and places spin variables $\sigma_{i}$ on each vertex $i$. Each $(ij)$ is equipped with a random variable $J_{ij}$ that represents the interaction between spins. ...

The Sherrington-Kirkpatrick model corresponds to the case when $\Gamma$ is the complete graph on $N$ vertices, a so-called mean field model, and where $J_{ij}$ are centred Gaussian random variables of variance $\large\frac{1}{N}\normalsize$. This model and a number of more or less closely related models are the subject of the book under review.

From a purely mathematical point of view, the SK model can be described as a Gaussian process on the hypercube $\{−1, 1\}^{N}$. The author, who has in the past pioneered research on Gaussian processes on high-dimensional (and Banach) spaces, was intrigued to find out that the theoretical physics community had devised methods to compute fine properties of these systems that were inaccessible to the conventional tools of mathematical analysis. The book gives a comprehensive account of the endeavour by its author, as well as some selected works of other people in the same field, to turn these heuristic results into rigorous mathematics.

The importance of the subject is highlighted by the fact that the mathematics involved is not limited to the SK model, nor to Gaussian processes on the hypercube. In fact, it had been long noticed that a great number of other problems that are highly relevant in diverse application areas fall into the same category. These include, but are not limited to, models of neural networks, such as perceptron models and the Hopfield model, and the K-SAT and other problems from computer science, which are all covered here.

The approach presented here circles around the so-called cavity method. The cavity method essentially consists in the attempt to compute key quantities of interest (functionals of the random process at hand) by inductions over the number N (the volume). That is, given a function $F_{N}$of the process, one tries to derive a recursion relation for this function in the variable $N$. Usually, in this attempt it turns out that no closed form can be achieved, and a number of new functions have to be introduced. The goal is to show that it is enough to introduce a finite set of such functions, while all further terms produced in the program can be treated as error terms.
...

This book will remain a comprehensive reference on the subject of mean field spin glasses for years to come.

3.4. Review by: David Aldous.
SIAM Review 47 (1) (2005), 168-170.

Part of my standard advice to graduate students seeking a career in mathematics research is that, sometime during their second or third year, they should spend six months coming to grips with the core technical details of some currently active area of research. This is most conveniently done by reading every line of a recent monograph. I speak from personal experience: reading, 30 years ago, Billingsley's monograph [Convergence of Probability Measures] on weak convergence has served me well ever since. The monograph under review is ideally suited for this purpose, not only by virtue of its content and style being "core technical details of some currently active area of research," but also because the author poses open problems, from Level 1 (the author feels he could do them, if he tried) to Level 3 (touching essential issues, with currently no way of telling how difficult they might be).
...

This book seeks to start from basics (assuming little more than undergraduate probability and analysis, and in particular assuming no knowledge of statistical physics) and to develop the rigorous theory, in almost complete detail, as far as possible. By focusing on a small set of models, the book manages to get close to the frontier of what is known rigorously about these models, though this often falls short of what has been discussed non-rigorously in the physics literature. Much of this rigorous work is due to Talagrand himself, though due credit is paid to others, Guerra in particular.
...

In summary, this is a book of ideas and calculations which build upon each other to create a rich and informative theory. Not a book for browsing, but one which demands concentrated study. Having these basis rigorous results in one place, with clear exposition of the techniques, will surely catalyse research into rigorous theory over the next few years.

It is intriguing to draw parallels between current work on spin glass models and the previous generation's work on percolation models, which also were studied in non-rigorous detail by physicists before attracting attention as rigorous mathematics. The basic percolation models, like the SK model, are highly oversimplified as models of real-world phenomena, but have turned out to be conceptually fundamental and useful for comparison purposes (e.g., "percolation substructures") in the study of more general real world-inspired stochastic spatial models. It is reasonable to guess that these fundamental spin glass models and their analysis techniques will in future turn out to be equally pervasive.

4.1. From the Publisher.

What is the maximum level a certain river is likely to reach over the next 25 years? (Having experienced three times a few feet of water in my house, I feel a keen personal interest in this question.) There are many questions of the same nature: what is the likely magnitude of the strongest earthquake to occur during the life of a planned building, or the speed of the strongest wind a suspension bridge will have to stand? All these situations can be modelled in the same manner. The value $X_{t}$ of the quantity of interest (be it water level or speed of wind) at time $t$ is a random variable. What can be said about the maximum value of $X_{t}$over a certain range of $t$?

4.2. From the Introduction.

In 2000, while discussing one of the open problems of this book with K Ball (be he blessed for his interest in it) I discovered that one could replace majorizing measures by a suitable variation on the usual chaining arguments, a variation that is moreover totally natural. That this was not discovered much earlier is a striking illustration of the inefficiency of the human brain (and of mine in particular). This new approach not only removes the psychological obstacle of having to understand the somewhat disturbing idea of majorizing measures, it also removes a number of technicalities, and allows one to give significantly shorter proofs. I thus felt the time had come to make a new exposition of my body of work on lower and upper bounds for stochastic processes. The feeling that, this time, the approach was possibly (and even probably) the correct one gave me the energy to rework all the proofs. For several of the most striking results, such as Shor's matching theorem, the decomposition theorem for infinitely divisible processes, and Bourgains solution of the $\Lambda _{p}$ problem, the proofs given here are at least three times shorter than the previously published proofs.

Beside enjoying myself immensely and giving others a chance to understand the results presented here (and even possibly to get excited about them) a main objective of this book is to point out several problems that remain open. Of course opinions differ as to what constitutes an important problem, but I like those presented here. One of them deals with the geometry of Hilbert space, a topic that can hardly be dismissed as exotic. I stated only the problems that I find really interesting. Possibly they are challenging. At least, I made every effort to make progress on them. A significant part of the material of the book was discovered while trying to solve the "Bernoulli problem" of Chapter 4. I have spent many years thinking to that problem, and will be glad to offer a prize of $ 5000 for a positive solution of it. A smaller prize of $ 1000 is offered for a positive solution of the possibly even more important problem raised at the end of chapter 5. The smaller amount simply reflects the fact that I have spent less time on this question than on the Bernoulli problem. It is of course advisable to claim these prizes before I am too senile to understand the solution, for there will be no guarantee of payment afterwards. (Cash awards will also be given for a negative solution of any of these two problems, the amount depending on the beauty of the solution.)
...

In conclusion, a bit of wisdom. I think that I finally discovered a foolproof way to ensure that the writing of a book of this size be a delightful and easy experience. Just write a 600 page book first!

4.3. Review by: Werner Linde.
Mathematical Reviews MR2133757 (2006b:60006).

Unfortunately, the concept of majorizing measures is not so easy to understand and, moreover, it is very complicated to work with those measures. For example, in general it is very difficult to construct a concrete majorizing measure for a given bounded Gaussian process. In 2000 the author of the present book discovered that majorizing measures may be replaced by a suitable variation of the classical chaining arguments. Thus the main purpose of this book is to present this new technique and to show how it applies in various situations.
...

Summing up, in the reviewer's opinion this is a very important book about concepts developed during the investigation of stochastic processes, yet with far reaching consequences also for difficult problems in other fields of mathematics. The book is very well written; all results are proved in detail. Unfortunately, by a mishap, the announced appendix is missing. In that appendix the new approach is related to the old one using majorizing measures (they do not appear in the book at all). Fortunately, this appendix is available on the web.

4.4. Review by: Radu Zaharopol.
SIAM Review 49 (2) (2007), 363-365.

One of the topics that are at the same time very challenging and of significant importance in probability theory, analysis, and ergodic theory is the study of inequalities involving suprema of families of random variables (or, more generally, real-valued measurable functions.
...

Even though the book under review appeared in 2005, people have already started to use the monograph's approach to the generic chaining.

The results discussed in the book involve a tremendous amount of creativity, and the author has made every possible effort to explain how he arrived at the results he deals with.

I believe that the monograph is of interest to anyone who works even remotely with indexed families of maps (be those maps continuous functions, measurable functions, or measures) because, while discussing specific results, the author also offers a wealth of general principles for dealing with such families, principles that could be used in related approaches but in very different settings. Thus, in my opinion, the book is of interest to almost everyone who works in probability, analysis, or ergodic theory even if she/he has not done any research involving the topics covered by the volume under review. Actually, I agreed to review the work, even though my research interests are very far from the results discussed in the monograph, exactly because I wanted to emphasise the fact that the book is of interest to a large audience, and not only to people who publish in the area presented in the work.

As we all know, there is a perception in the mathematical community (especially among the people who have written at least one book) that one cannot find flawless monographs in mathematics. In the work under review, the author refers repeatedly to an appendix that is assumed to be part of the book, and the bad news is that there is no appendix in the volume. However, the good news is that the appendix can be downloaded from the web.

5.1. From the Publisher.

This is a new, completely revised, updated and enlarged edition of the author's Spin Glasses: A Challenge for Mathematicians. This new edition will appear in two volumes, the present first volume presents the basic results and methods, the second volume is expected to appear in 2011. In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called "spin glasses". These models are simple and rather canonical random structures, of considerable interest for several branches of science (statistical physics, neural networks and computer science). The physicists studied them by non-rigorous methods and predicted spectacular behaviours. This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics. The first volume of this new and completely rewritten edition presents six fundamental models and the basic techniques to study them.

5.2. From the Introduction.

... this is a book of probability theory (mostly). Attempting first a description at a "philosophical" level, a fundamental problem is as follows. Consider a large finite collection $(X_{k})_{k≤M}$of random variables. What can we say about the largest of them? More generally, what can we say about the "few largest" of them? When the variables $X_{k}$are probabilistically independent, everything is rather easy. This is no longer the case when the variables are correlated. Even when the variables are identically distributed, the answer depends very much on their correlation structure. What are the correlation structures of interest? Most of the familiar correlation structures in Probability are low-dimensional, or even "one-dimensional". This is because they model random phenomena indexed by time, or, equivalently, by the real line, a one-dimensional object. In contrast with these familiar situations, the correlation structures considered here will be "high-dimensional" in a sense that will soon become clear and will create new and truly remarkable phenomena. This is a direction of probability theory that has not yet received the attention it deserves.
...

It was rather paradoxical for a mathematician like the author to see simple, basic mathematical objects being studied by the methods of theoretical physics. It was also very surprising, given the obvious importance of what the physicists have done, and the breadth of the paths they have opened, that mathematicians had not succeeded yet in proving any of their conjectures.

Despite considerable efforts in recent years, the program of giving a sound mathematical basis to the physicists' work is still in its infancy. We already have tried to make the case that in essence this program represents a new direction of probability theory. It is hence not surprising that, as of today, one has not yet been able to find anywhere in mathematics an already developed set of tools that would bear on these questions. Most of the methods used in this book belong in spirit to the area loosely known as "high-dimensional probability", but they are developed here from first principles. In fact, for much of the book, the most advanced tool that is not proved in complete detail is Hölder's inequality. The book is long because it attempts to fulfil several goals (that will be described below) but reading the first two chapters should be sufficient to get a very good idea of what spin glasses are about, as far as rigorous results are concerned.

The author believes that the present area has a tremendous long-term potential to yield incredibly beautiful results. There is of course no way of telling when progress will be made on the really difficult questions, but to provide an immediate incitement to seriously learn this topic, the author has stated as research problems a number of interesting questions (the solution of which would likely deserve to be published) that he believes are within the reach of the already established methods, but that he purposely did not, and will not, try to solve. (On the other hand, there is ample warning about the potentially truly difficult problems.)

This book, together with a forthcoming second volume, forms a second edition of our previous work, "Spin Glasses, a Challenge for Mathematicians". One of the goals in writing "Spin Glasses" was to increase the chance of significant progress by making sure that no stone was left unturned. This strategy greatly helped the author to obtain the solution of what was arguably at the time the most important problem about mean-field spin glasses, the validity of the "Parisi Formula". This advance occurred a few weeks before "Spin Glasses" appeared, and therefore could not be included there. Explaining this result in the appropriate context is a main motivation for this second edition, which also provides an opportunity to reorganise and rewrite with considerably more details all the material of the first edition.

The programs of conferences on spin glasses include many topics that are not touched here. This book is not an encyclopaedia, but represents the coherent development of a line of thought. The author feels that the real challenge is the study of spin systems, and, among those, considers only pure mean-field models from the "statics" point of view. A popular topic is the study of "dynamics". In principle this topic also bears on mean-field models for spin glasses, but in practice it is as of today entirely disjoint from what we do here.

5.3. Review by: Francis Comets.
Mathematical Reviews MR2731561 (2012c:82036).

This book, together with the forthcoming second volume, forms a second edition of the influential Spin glasses: a challenge for mathematicians. It is primarily motivated to include some of the many results which have been obtained since the publication of that book, including the celebrated Parisi formula. This new edition has been considerably expanded and fully rewritten, and as a result stands as a real novelty.

Mean-field spin glasses originated in theoretical physics in the 1970's as solvable models of glasses. The archetype is the Sherrington-Kirkpatrick model (SK), which is central in this book.
...

From the mathematical perspective, the aim is the probabilistic analysis of nonlinear functions of a Gaussian process indexed by a product space, where the covariance has a special structure (it is invariant under permutations of the coordinates). Being an expert and successful explorer of such processes on high-dimensional spaces, the author brings to bear the appropriate fundamental tools in the context of disordered systems, and he adapts fine techniques and develops them, ending with powerful weapons tailored to the main questions raised. Concentration of measure, interpolation techniques and sharp convexity estimates are among the main tools.
A central technique is interpolation between the model of interest and a simpler one, a method the author calls the "smart path method" to emphasise that choosing the proper path is essential. The key to the method is to control the derivatives along the interpolation path, usually after an integration by parts. In the case of the SK model, the smart path method coincides with Gaussian interpolation, and the idea can be traced back at least to Kahane's proof of Slepian's lemma for Gaussian processes. It was a fundamental discovery of F Guerra that, for the SK model and some variations, the sign of the derivatives along a certain smart path can be controlled, a fact that is at the root of most of today's rigorous low-temperature results. Unfortunately this does not seem possible for many models of interest. For these models, the author uses the smart path method as a basis to obtain a rigorous version of the cavity method introduced by physicists as an alternative approach to the replica method and which was at the core of the above-mentioned book by the author. One starts to express a given function of the $(N+1)$-spins system in terms of functions of the system with N spins, and one is left with the problem of closing the system of equations generated in this way. In some circumstances - the so-called "replica-symmetric regime" - only a few functions are necessary, while all other terms that are produced vanish as $N$ tends to infinity. This latter fact, however, has to be proven in turn by fine induction, using some sort of high-temperature assumption (and so one does not know how to control these models without such an assumption).
...

The chapters are organised as independently as possible, one separate from another, which allows the reading of only a few chapters to give the full story on a particular model. An appendix recalls many basic facts from probability theory.

Though very complete, the book is not an encyclopaedia. Rather, it represents the development of a coherent line of thought. It builds stone by stone a remarkable construction, revealing at each step a new piece of the fascinating picture of mean-field spin glass models. The author has obviously put extreme care and effort into the writing, preserving the reader's attention for the mathematics. To help the reader, exercises and research problems ranked by level of difficulty are scattered all through the text. Enlightening explanations, written with strong commitment, help the reader to grasp the general line of thought before entering into the computations, which range from elementary considerations that motivate the questions or the models, to the most subtle or technical ones. As a result, the book is totally self-contained with respect to both its mathematics and its modelling aspects. Only a solid mathematical temperament is required of the reader in order to benefit from the wonders presented in this book.

With its depth and proficiency of important techniques, this book opens new perspectives on probability theory and the statistical mechanics of disordered systems. It will be a source of inspiration for generations of mathematicians.

6.1. From the Publisher.

This is a new, completely revised, updated and enlarged edition of the author's Spin Glasses: A Challenge for Mathematicians in two volumes (this is the 2nd volume). In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called "spin glasses". These models are simple and rather canonical random structures, of considerable interest for several branches of science (statistical physics, neural networks and computer science). The physicists studied them by non-rigorous methods and predicted spectacular behaviours. This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics. The present Volume II contains a considerable amount of new material, in particular all the fundamental low-temperature results obtained after the publication of the first edition.

6.2. From the Introduction.

Welcome to the second volume of the treatise "Mean fields models for spin glasses". You certainly do not need to have read all of Volume 1 to enjoy the present work. For the low-temperature results of Part II, starting with Chapter 12, only (the beginning of) Chapter 1 is really needed. This is also true for Chapter 11.

In the first part of this volume we continue, at a deeper level, the study of four of the models that were introduced in Volume I. Chapter 8 continues the study of the Shcherbina-Tirozzi model of Chapter 3; Chapter 9 continues the study of the Perceptron model of Chapter 2. Both chapters culminate in the proof of the "Gardner formula" which computes the proportion of the sphere (respectively the discrete cube) that belongs to the intersection of many random half-spaces. Chapters 8 and 9 are somewhat connected. They could in principle be read with only the previous understanding of the corresponding chapter of Volume 1, although we feel that it should help to have also read at least a part of each of Chapters 2 to Chapter 4, where the basic techniques are presented.

Chapter 10 continues and deepens the study of the Hopfield model of Chapter 4. We achieve a good understanding for a larger region of parameters than in Chapter 4 and this understanding is better, as we reach the correct rates of convergence in $\large\frac{1}{N}\normalsize$. This chapter can be read independently of Chapters 8 and 9, and in principle with only the knowledge of some of the material of Chapter 4.

Chapter 11 provides an in-depth study of the Sherrington-Kirkpatrick model at high temperature and without external field. As this is a somewhat simpler case than the other models considered in this work, we can look deeper into it. Only (the beginning of) Chapter 1 is a prerequisite from this point on.

In my lecture in the International Congress of Mathematicians in Berlin, 1998, I presented (an earlier form of) some of the results explained here. At the end of the lecture, while I was still panting under the effort, a man (whose name I have mercifully forgotten) came to me, and handed me one of his papers with the following comment "you should read this instead of doing this trivial replica-symmetric stuff". To him I dedicate these four chapters.

The second part of this volume explores genuine low-temperature results. In Chapter 12 we describe the Ghirlanda-Guerra identities and some rather striking consequences. This chapter can be read without any detailed knowledge of any other material presented so far.

In Chapters 13 and 14 we learn how to prove a celebrated formula of G Parisi which gives the value of the "limiting free energy" at any temperature for the Sherrington-Kirkpatrick model. A very special case of this formula determines that high-temperature region of this model. We present first this special case in Chapter 13. This seems to require all the important ideas, and these are better explained in this technically simpler setting. Parisi's formula is believed to be only a small part of a very beautiful structure that we call the Parisi Solution. We attempt to describe this structure in Chapter 15 where we also prove as many parts of it as is currently possible. Chapter 15 can be read without having read the details of the (difficult) proof of Parisi formula in Chapter 14, and is probably the highlight of this entire work. We also explain what are the remaining (fundamental) questions to be answered before we reach a really satisfactory understanding.

In the final Chapter 16 we study the p-spin interaction model, in a case not covered by the theory of Chapter 14. The approach is based on a clear physical picture of what happens in the phase of "one step of replica-symmetry breaking" and new aspects of the cavity method.

6.3. Review by: Francis Comets.
Mathematical Reviews MR3024566.

This is the second volume of the author's treatise Mean field models for spin glasses.

The first part of this volume, "Advanced replica-symmetry", is a continuation, at a deeper level, of the analysis of the four models introduced in Volume I: Shcherbina-Tirozzi model, perceptron, Hopfield model, Sherrington-Kirkpatrick (SK) model. The gain in accuracy is quite impressive in Chapter 10, where new subtle ideas and fine arguments are added.

The second part of this volume, "Low temperature", deals with replica-symmetry breaking. An ultimate goal is a proof of the celebrated formula of G. Parisi, which gives the value of the limiting free energy of the SK model. The formula, derived by Parisi in 1980 using a non-rigorous replica method, is a true mathematical challenge; it was proved by the author in 2006. In Chapter 13 the main ideas are explained in the simplest case. The heart of the matter is the extended Chapter 14, where serious technicalities come after presenting the central line of the proof. Chapter 15 can be read without mastering all these technicalities. It covers many fascinating features of the Parisi solution, including ultrametricity, Ghirlanda-Guerra identities, Poisson-Dirichlet cascades, and random overlap structures. The volume ends with the p-spin model.

Not only does this book lead the reader towards the proof of the Parisi formula, but it also reveals the beautiful structure which is embedded in the model and the "new territories" mentioned in the dedication. With its depth and profusion of important techniques, this book, together with Volume I, opens new perspectives in probability theory and statistical mechanics of disordered systems. It will be a source of inspiration for generations of mathematicians.

7.1. From the Publisher.

The book develops modern methods and in particular the "generic chaining" to bound stochastic processes. This methods allows in particular to get optimal bounds for Gaussian and Bernoulli processes. Applications are given to stable processes, infinitely divisible processes, matching theorems, the convergence of random Fourier series, of orthogonal series, and to functional analysis. The complete solution of a number of classical problems is given in complete detail, and an ambitious program for future research is laid out.

7.2. From the Introduction.

What is the maximum level a certain river is likely to reach over the next 25 years? What is the likely magnitude of the strongest earthquake to occur during the life of a planned nuclear plant, or the speed of the strongest wind a suspension bridge will have to stand? The present book does not deal with such fundamental practical questions, but rather with some (arguably also fundamental) mathematics which have emerged from the consideration of these questions. All these situations can be modelled in the same manner. The value $X_{t}$ of the quantity of interest (be it water level or speed of wind) at time $t$ is a random variable. What can be said about the maximum value of $X_{t}$ over a certain range of $t$? In particular, how can we guarantee that, with probability close to one, this maximum will not exceed a given threshold?

A collection of random variables $(X_{t})_{t\in T}$, where t belongs to a certain index set $T$, is called a stochastic process, and the topic of this book is the study of the supremum of certain stochastic processes, and more precisely to find upper and lower bounds for these suprema. The key word of the book is

INEQUALITIES.

It is not required that $T$ be a subset of the real line, and large parts of the book do not deal directly with the "classical theory of processes" which is mostly concerned with this situation. The book is by no means a complete treatment of the hugely important question of bounding stochastic processes, in particular because it does not really expand on the most basic and robust results which are the most important for the "classical theory of processes". Rather, its specific goal is to demonstrate the impact and the range of modern abstract methods, in particular through their treatment of several classical questions which are not accessible to "classical methods".

The most important idea about bounding stochastic processes is called "chaining", and was invented by A Kolmogorov. This method is wonderfully efficient. With little effort it suffices to answer a number of natural questions. It is however not a panacea, and in a number of natural situations it fails to provide a complete understanding. This is best discussed in the case of Gaussian processes, that is processes for which the family (Xt)t∈T consists of jointly Gaussian random variables. These are arguably the most important of all. ...

Probably the single most important conceptual progress about Gaussian processes was the gradual realisation that the metric space $(T, d)$ is the key object to understand them, irrelevant of the other possible structures of the index set. This led R Dudley to develop in 1967 an abstract version of Kolmgorov's chaining argument adapted to this situation. This provides a very efficient bound for Gaussian processes. Unfortunately, there are natural situations where this bound is not tight. Roughly speaking, one might say that "there sometimes remains a parasitic logarithmic factor in the estimates".

The discovery around 1985 (by X Fernique and the author) of a precise (and in a sense, exact) relationship between the "size" of a Gaussian process and the "size" of this metric space provided the missing understanding in the case of these processes. Attempts to extend this result to other processes spanned a body of work which forms the core of this book.

A significant part of this book is devoted to situations where one has to use some skills to "remove the last parasitic logarithm in the estimates." These situations occur with unexpected frequency in all kinds of problems.
...

Even though the book is largely self-contained, it mostly deals with rather subtle questions. It also devotes considerable energy to the problem of finding lower bounds for certain processes, a topic considerably more difficult and less developed than the search for upper bounds. Therefore it should probably be considered as an advanced text, even though I hope that eventually the main ideas of at least Chapter 2 will become part of every probabilist's tool kit. In a sense this book is a second edition (or, rather, a continuation) of the monograph [M Ledoux and M Talagrand, Probability in a Banach Space], or at least of the part of that work which was devoted to the present topic. I made no attempt to cover again all the relevant material of [M Ledoux and M Talagrand, Probability in a Banach Space]. Familiarity with [M Ledoux and M Talagrand, Probability in a Banach Space] is certainly not a prerequisite, and maybe not even helpful, because the way certain results are presented there is arguably obsolete. The present book incorporates (with much detail added) the material of a previous (and, in retrospect, far too timid) attempt [M Talagrand, The Generic Chaining] in the same direction, but its goal is much broader. I am really trying here to communicate as much as possible of my experience working in the area of boundedness of stochastic processes, and consequently I have in particular covered most of the subjects related to this area on which I ever worked, and I have included all my pet results, whether or not they have yet generated activity. I have also included a number of recent results by others in the same general direction. I find that these results are deep and very beautiful. They are also sometimes rather difficult to access for the non-specialist (or even for the specialists themselves). I hope that explaining them here in a unified (and often simplified) presentation will serve a useful purpose. Bitter experience has taught me that I should not attempt to write about anything on which I have not meditated enough to make it part of my flesh and blood (and that even this is very risky). Consequently this book covers only topics and examples about which I have at least the illusion that I might write as well as anybody else, a severe limitation. I can only hope that it still covers the state-of-art knowledge about sufficiently many fundamental questions to be useful, and that it contains sufficiently many deep results to be of lasting interest.

A number of seemingly important questions remain open, and one of my main goals is to popularise these. Of course opinions differ as to what constitutes a really important problem, but I like those I explain in the present book. Several of them were raised a generation ago in [M Ledoux and M Talagrand, Probability in a Banach Space], but have seen little progress since. One deals with the geometry of Hilbert space, a topic that can hardly be dismissed as being exotic. These problems might be challenging. At least, I made every effort to make some progress on them. The great news is that when this book was nearly complete, Witold Bednorz and Rafal Latala solved the Bernoulli Conjecture on which I worked for years in the early nineties (Theorem 5.1.5). In my opinion this is the most important result in abstract probability for at least a generation. I offered a prize of $ 5000 for the solution to this problem, and any reader understanding this amazing solution will agree that after all this was not such a generous award (specially since she did not have to sign this check). But solving the Bernoulli Conjecture is only the first step of a vast (and potentially very difficult) research program, which is the object of Chapter 12. I now offer a prize of $ 1000 for a positive solution of the possibly even more important problem raised at the end of Chapter 12. The smaller amount reflects both the fact that I am getting wiser and my belief that a positive solution to this question would revolutionise our understanding of fundamentally important structures (so that anybody making this advance will not care about money anyway). I of course advise to claim this prize before I am too senile to understand the solution, for there can be no guarantee of payment afterwards.

7.3. Review by: Sasha Sodin.
Mathematical Reviews MR3184689.

The reader will find in the book a complete solution for the case of Gaussian processes (due to Fernique and the author), for the case of Bernoulli processes (due to Bednorz and Latała), and additional results, as well as numerous applications. All the material from [M Talagrand, The generic chaining, 2005] as well as a significant part of [M Ledoux and M Talagrand, Probability in Banach spaces, 1991] is incorporated.

7.4. Review by: René L Schilling.
The Mathematical Gazette 100 (547) (2016), 180-181.

Bounds for Gaussian processes is one of the recurrent themes in the oeuvre of Michel Talagrand. The monograph under review - which may well be his magnum opus on bounds for stochastic processes - is entirely devoted to the study of the supremum of a stochastic process $(X_{t})_{t\in T}$ indexed by a general index set $( T, d)$ (usually seen as a finite or infinite metric space). The aim is to present a unified approach to the problem of boundedness (and the then (!) 'simple matter to study continuity' [Chapter 1, p. 12]) for Gaussian and many other classes of mostly non-homogenous stochastic processes, e.g. Bernoulli processes, random Fourier series, p-stable processes and infinitely divisible processes. The principal tool, Kolmogorov's chaining idea, which has been refined and used by Dudley for his maximal 'entropic' estimate for Gaussian processes, has been earlier ascribed by Talagrand in The Generic Chaining, but the present approach is much more general and far-reaching. It includes most of the material of the predecessor book and takes into account a wealth of material from Talagrand's other classic (jointly with M Ledoux) Probability in Banach Spaces. The reader is assumed to be familiar with classical methods from the theory of processes (e.g. Kolmogorov-Chentsov criteria, the Garsia-Rodemich-Rumsey lemma or martingale methods), but as far as new ideas and developments since the1970s are concerned (many of them pioneered by Talagrand), the present volume is self-contained.

Rather than trying to do the impossible in this brief review (i.e. to do justice to this work, giving a true picture of the content or at least an overview of the material covered in the 600-odd pages) let me point out that this is a most scholarly account written by a master of the field. Each chapter begins with an explanation of the overall philosophy and gives a brief survey of the things to come. This makes a rewarding read for everyone with a sound probability background, and for the specialist it is even entertaining and most inspiring. Apart from the chapters devoted to special classes of processes (Chapters 3-11, 14), the first two introductory chapters (pp. 1-74) are a wonderful introduction into the topics and methods of this book, while Chapters 12 and 13 and 14 through 16 provide (sometimes unexpected) applications of the bounds and inequalities for stochastic processes. There are many areas where Talagrand has a very personal view of things which, I am sure, will be food for thought for future generations of probabilists.

7.5. Review by: Antonio Auffinger.
Bulletin of the American Mathematical Society 53 (1) (2016), 173-177.

I was at the beginning of my graduate studies at the Courant Institute when one of my professors, in his typical calm voice, advised in class: "Probability is all about inequalities. Knowing how to do upper bounds is essential, but the true art lies in handling the lower bounds." Somehow, I kept these words in the back of my mind. Now, they truly came alive while reading and reviewing this wonderful book by M Talagrand. The quote certainly does not do full justice to my field, but it is too tempting not to recall.
...

The examples and methods of the "classical theory of stochastic processes" have been widely used and generalised in several directions. They belong to the toolbox of almost every probabilist. Due to their importance, they are included in any classical first-year graduate course in probability. They appear in several textbooks in probability, combinatorics, statistics. ... But not in Talagrand's book.

Although he writes a book about inequalities of stochastic processes, Talagrand focuses on modern abstract methods, completely abdicating the "classical approach". He describes problems on which the above strategies would not work and develops an abstract methodology to deal with some of these situations. The methods described in his book, many of them developed by its author, are much inspired by the idea of "chaining" that goes back to Kolmogorov.
...

Talagrand's goal in this book is, without any doubt, very ambitious. It is not an introductory volume, but it contains marvellous ideas that should very likely be in the toolbox of anyone dealing with stochastic processes. It was my companion during many long (and happy) days of last summer. And I still feel I barely scratched its surface.

8.1. From the Publisher.

This book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results.

The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist's toolkit. The effectiveness of the scheme is demonstrated by the characterisation of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures.

A large part of this unique book is devoted to the author's influential work. While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers.

8.2. From the Introduction.

This book had a previous edition [M Talagrand, Upper and lower bounds for stochastic processes (2014)]. The changes between the two editions are not only cosmetic or pedagogical, and the degree of improvement in the mathematics themselves is almost embarrassing at times. Besides significant simplifications in the arguments, several of the main conjectures of [M Talagrand, Upper and lower bounds for stochastic processes (2014)] have been solved and a new direction came to fruition. It would have been more appropriate to publish this text as a brand new book, but the improvements occurred gradually and the bureaucratic constraints of the editor did not allow a change at a late stage without further delay and uncertainty.
...

One of my main goals is to communicate as much as possible of my experience from working on stochastic processes, and I have covered most of my results in this area. A number of these results were proved many years ago. I still like them, but some seem to be waiting for their first reader. The odds of these results meeting this first reader while staying buried in the original papers seemed nil, but they might increase in the present book form. In order to present a somewhat coherent body of work, I have also included rather recent results by others in the same general direction. I find these results deep and very beautiful. They are sometimes difficult to access for the non-specialist. Explaining them here in a unified and often simplified presentation could serve a useful purpose. Still, the choice of topics is highly personal and does not represent a systematic effort to cover all the important directions. I can only hope that the book contains enough state-of-art knowledge about sufficiently many fundamental questions to be useful.

8.3. Review by: Erick Trevino-Aguilar.
Mathematical Reviews MR4381414.

Studying and solving a problem for a mathematical object through the properties of another associated object is of fundamental importance in mathematics, and it is a characteristic in the theory developed in the reviewed book.
...

As often is the case in probability theory, there are real life problems that motivate the interest in the expected supremum $S$. In the book, motivating questions include: What is the maximum level a certain river is likely to reach over the next 25 years? What is the likely magnitude of the strongest earthquake to occur during the life of a planned nuclear plant?

The book presents advanced and specialised material with a systematic treatment. Hence, for a first reading, it helps to keep in mind the aforementioned questions, and that in their mathematical formulation such questions lead to the problem of studying the expected supremum of the stochastic process $X$, to be estimated through quantities derived from the metric space $(T, d)$.

The reason why a metric structure appears is that the estimation of $S$ builds upon the concept of chaining, which the author attributes to Andrey Kolmogorov. It is not an overstatement to describe chaining as the key concept bonding the chapters together and to claim that it is the main method which allows for a successful solution to the estimation of $S$ for Gaussian processes. A stylised formulation on generic chaining leads to the association of the metric space $(T, d)$. Preserving it as an effective tool to estimate S for other classes of processes is the subject of many of the chapters in the book.
...

The book is organised into three parts. In the first, the focus is on Gaussian processes, and in Chapter 2, the Majorizing Measure Theorem 2.10.1 is proved. Chapter 3 is dedicated to trees on metric spaces. Chapter 4 presents an application of generic chaining to matching problems, which is continued in Chapters 17 and 18. The second part starts in Chapter 5 and finishes in Chapter 13. It is dedicated to generic chaining for other classes of processes. The goal is the estimation of S, for which generic chaining continues to be effective in combination with decomposition theorems. These are: Theorem 6.2.8 for Bernoulli processes, Theorem 7.5.14 for trigonometric sums, Theorem 11.1.1 for empirical processes, Theorem 11.10.3 for random series, and Theorem 12.3.5 for infinite divisible processes. 'A road map' for decomposition theorems is presented in Section 7.8.3. The third part of the book presents matching theorems and outlines further directions, particularly applications of the theory to Banach spaces.

The book includes a rich collection of exercises that will allow the reader to gain skills for a better understanding. The book is then suitable as a textbook for an advanced course. The expertise of the author as a leading researcher in the field is reflected in the formulation of another class of exercises labelled 'research problems', enhancing and inspiring new investigations. The systematic and deep treatment of the subject under study makes the book a good reference for the specialist.

9.1. From the Publisher.

Quantum field theory (QFT) is one of the great achievements of physics, of profound interest to mathematicians. Most pedagogical texts on QFT are geared toward budding professional physicists, however, whereas mathematical accounts are abstract and difficult to relate to the physics. This book bridges the gap. While the treatment is rigorous whenever possible, the accent is not on formality but on explaining what the physicists do and why, using precise mathematical language. In particular, it covers in detail the mysterious procedure of renormalisation. Written for readers with a mathematical background but no previous knowledge of physics and largely self-contained, it presents both basic physical ideas from special relativity and quantum mechanics and advanced mathematical concepts in complete detail. It will be of interest to mathematicians wanting to learn about QFT and, with nearly 300 exercises, also to physics students seeking greater rigour than they typically find in their courses.

9.2. From the Introduction.

As a teenager in the sixties reading scientific magazines, countless articles alerted me to "the infinities plaguing the theory of Quantum Mechanics". Reaching 60 after a busy mathematician's life, I decided that it was now or never for me to really understand the subject. The project started for my own enjoyment, before turning into the hardest of my scientific life. I faced many difficulties, the most important being the lack of a suitable introductory text. These notes try to mend that issue.

I knew no physics to speak of, but it was not particularly difficult to get a basic grasp of topics such as Classical Mechanics, Electromagnetism, Special and even General Relativity. They are friendly for mathematicians as they can be made rigorous to our liking.

Quantum Mechanics was a different challenge. Quite naturally, I looked first for books written by mathematicians for mathematicians. By a stroke of bad luck, the first book I tried was to me a shining example of everything one should not do when writing a book. Being a mathematician does not mean that I absolutely need to hear about "a one-dimensional central extension of $V$ by a Lie algebra 2-cocycle" just to learn the Heisenberg commutation relations. Moreover, while there is no question that mastery of some high level form of Classical Mechanics will help reaching a deeper understanding of Quantum Mechanics, Poisson Manifolds and Symplectic Geometry are not absolute prerequisites to get started. Other books written by mathematicians are less misguided, but seem to cover mostly topics which barely overlap those in physicists' textbooks with similar titles. To top it all, I was buried by the worst advice I ever received, to learn the topic from Dirac's book itself! The well-known obstacle of the difference of language and culture between mathematics and physics is all too real. To a mathematician's eye, some physics textbooks are chock-full of somewhat imprecise statements made about rather ill-defined quantities. It is not rare that these statements are simply untrue if taken at face value. Moreover arguments full of implicit assumptions are presented in the most authoritative manner. Looking at elementary textbooks can be an even harder challenge. These often use simple-minded approaches which need not be fully correct, or try to help the reader with analogies which might be superficial and misleading.

Luckily, in 2012 I ran into the preliminary version of Brian Hall's Quantum Theory for Mathematicians, which made me feel proud again for mathematicians. I learnt from this book many times faster than from any other place. The "magic recipe" for this was so obvious that it sounds trivial when you spell it out: explain the theory in complete detail, starting from the very basics, and in a language the reader can understand. Simple enough, but very difficult to put into practice, as it requires a lot of humility from the author, and humility is not the most common quality among mathematicians. I don't pretend to be able to emulate Brian's style, but I really tried and his book has had a considerable influence on mine.

After getting some (still very limited) understanding of Quantum Mechanics came the real challenge: Quantum Field Theory. I first looked at books written by mathematicians for mathematicians. These were obviously not designed as first texts or for ease of reading. Moreover, they focus on building rigorous theories. As of today these attempts seem to be of modest interest for most physicists. More promising seemed studying Gerald Folland's heroic attempt to make Quantum Field Theory accessible. His invaluable contribution is to explain what the physicists do rather than limiting the topic to the (rather small) mathematically sound part of the theory. His book is packed with an unbelievable amount of information, and, if you are stuck with minimum luggage on a desert island, this is a fantastic value. Unfortunately, as a consequence of its neutron-star density, I found it also much harder to read than I would have liked, even in sections dealing with well-established or, worse, elementary mathematics. Sadly, this book has no real competitors and cannot be dispensed with, except by readers able to understand physics textbooks. No doubt my difficulties are due to my own shortcomings, but still, it was while reading Weinberg's treatise that I finally understood what induced representations are, and this is not the way it should have been. So, as the days labouring through Folland's book turned into weeks, into many months, I felt the need to explain his material to myself, and to write the text from which I would have liked to learn the first steps in this area.

In the rest of the introduction I describe what I attempt to do and why. I try to provide an easily accessible introduction to some ideas of Quantum Field Theory for a reader well versed in undergraduate mathematics, but not necessarily knowing any physics beyond the high-school level or any graduate mathematics.

I must be clear about a fundamental point. A striking feature of Quantum Field theory is that it is not mathematically complete. This is what makes it so challenging for mathematicians. Numerous bright people have tried for a long time to make this topic rigorous and have yet to fully succeed. I have nothing new to offer in this direction. This book contains statements that nobody knows how to mathematically prove. Still, I try to explain some basic facts using mathematical language. I acknowledge right away that familiarity with this language and the suffering I underwent to understand the present material are my only credentials for this task.

My main concern has been to spare the reader some of the difficulties from which I have very much suffered reading others' books (while of course I fear introducing new ones), and I will comment on some of these.
...

9.3. Review by: Marek Nowakowski.
Mathematical reviews MR4367366.

Relativistic Quantum Field Theory (QFT) is the place in physics where quantum mechanics meets successfully the special theory of relativity, not only to give rise to a full covariance of the formalism, but also to realise explicitly the energy-mass equivalence in reactions. This local formalism is the pillar of the Standard Model of particle physics, i.e., the theory of three fundamental interactions in nature. Apart from that it is widely used in effective theories. All predictions so far derived from QFT on the basis of perturbation theory seem to agree with experiments in a rather spectacular way. The points above are the ones which led to its importance. It is therefore understandable that many scientists have tried to put QFT on a rigorous mathematical ground, be it by axiomatising it, by using a geometric quantisation or a quantisation based on deformation of Poisson structures, by invoking von Neumann algebra or by trying to give a rigorous definition of products of generalised functions (distributions). The book under review chooses a middle way - in spite of the title's promise that it is meant for mathematicians - suitable for mathematicians and physicists willing to see their subject from a different angle. This makes it exceptional and exceptionally nice among the books on the market. Whatever way one looks at it, from the mathematics or from the physics perspective, it has a pedagogical easy-going style. Indeed, the future reader gets a glimpse of this nice feature when he/she notices that in 740 pages the book covers QFT in operator language (linear operators on a Hilbert space) up to renormalisation in higher order perturbation theory trying the best possible mathematical rigour. It necessarily leaves out non-abelian gauge theories and path integral quantisation (I fear this would add another 300 pages to the text). It is obvious from the style and content that the book wishes to thoroughly explain the topics it chooses to cover and not rush by trying to cover them all. To appreciate this point I mention that the book starts in the first hundred pages with an introduction to non-relativistic quantum mechanics and non-relativistic quantum fields. What other book does this for the reader? In this way the book does not follow any other book, but many others might follow its style later.

After the introductory material it continues with a chapter on the Poincaré and Lorentz groups and the first glimpses of a quantisation of a scalar field. The next part, entitled "Spin", is devoted to the representations of these groups and making the quantisation of free fields more precise and deeper. Up to this point the book follows a rigorous mathematical style with definitions, lemmas, theorems and proofs. All can be understood by both mathematicians and physicists, but the author makes it a point to distinguish between the two "schools" and give a dictionary translating their respective ways. From here on, i.e., when the interaction is switched on, the mathematical rigour is reduced and more hand waving arguments find their way into the explanations. I suspect that this might have to do with the fact that the interaction picture might, strictly speaking, not exist. Although it is understood from the table of contents that the book is not meant for applications of QFT, Chapters 12 and 13 discuss the main observable in QFT, namely the cross-section ending with an example of the first divergency. The next chapter picks up the interaction and tries to put back some mathematical sanity, as far as it is possible, into it. But again the theorems offered should be understood rather as supporting arguments, as the author himself points out. Here the reader will find many formal aspects of QFT, like the Gell-Mann-Low formula, the LSZ reduction formalism and the Källén-Lehmann representation. The book ends with the most difficult subject: renormalisation. For some it is an art to sweep divergencies under the carpet, for others it is the pragmatism of a physical theory in which one has to identify the real direct observables (e.g., the charge is not such an observable) and needs the skill to bring the higher order results into a canonical form. The author manages to formulate some exact theorems in this area of QFT. The Bogoliubov-Parasiuk-Hepp-Zimmermann renormalisation scheme is discussed in more detail. It could even be that here the mathematical sanity came back to life.

The text has many exercises and sixteen (!) appendices from which one can learn quite a bit. This shows the dedication of the author to the subject and his wish to share his knowledge with others. The book hits the point between mathematics and physics where the first is not too abstract and the second not too phenomenological. My personal wish for the second edition is to mention that the perturbation theory in QFT is not convergent, but asymptotic which, however, does not harm its predictions.

In short, the book is exceptional and might set standards.

9.4. Extracts of reviews chosen by the Publisher.

1. From Sourav Chatterjee, Stanford University.
   
   This book accomplishes the following impossible task. It explains to a mathematician, in a language that a mathematician can understand, what is meant by a quantum field theory from a physicist's point of view. The author is completely and brutally honest in his goal to truly explain the physics rather than filtering out only the mathematics, but is at the same time as mathematically lucid as one can be with this topic. It is a great book by a great mathematician.
   
   
2. From Brian Hall, University of Notre Dame.
   
   Talagrand has done an admirable job of making the difficult subject of quantum field theory as concrete and understandable as possible. The book progresses slowly and carefully but still covers an enormous amount of material, culminating in a detailed treatment of renormalisation. Although no one can make the subject truly easy, Talagrand has made every effort to assist the reader on a rewarding journey though the world of quantum fields.
   
   
3. from Shahar Mendelson, Australian National University.
   
   A presentation of the fundamental ideas of QFT in a manner that is both accessible and mathematically accurate seems like an impossible dream. Well, not anymore! This book goes from basic notions to advanced topics with patience and care. It is an absolute delight to anyone looking for a friendly introduction to the beauty of QFT and its mysteries.
   
   
4. From Ellen Powell, Durham University.
   
   I have been motivated to try and learn about quantum field theories for some time, but struggled to find a presentation in a language that I as a mathematician could understand. This book was perfect for me: I was able to make progress without any initial preparation, and felt very comfortable and reassured by the style of exposition.
   
   
5. From Philippe Sosoe, Cornell University.
   
   In addition to its success as a physical theory, Quantum Field Theory (QFT) has been a continuous source of inspiration for mathematics. However, mathematicians trying to understand QFT must contend with the fact that some of the most important computations in the theory have no rigorous justification. This has been a considerable obstacle to communication between mathematicians and physicists. It is why despite many fruitful interactions, only very few people would claim to be well versed in both disciplines at the highest level. There have been many attempts to bridge this gap, each emphasising different aspects of QFT. Treatments aimed at a mathematical audience often deploy sophisticated mathematics. Michel Talagrand takes a decidedly elementary approach to answering the question in the title of his monograph, assuming little more than basic analysis. In addition to learning what QFT is, the reader will encounter in this book beautiful mathematics that is hard to find anywhere else in such clear pedagogical form, notably the discussion of representations of the Poincaré group and the BPHZ Theorem. The book is especially timely given the recent resurgence of ideas from QFT in probability and partial differential equations. It is sure to remain a reference for many decades.