This book, while by no means inclusive of the whole spectrum of mathematics (albeit encyclopedic after a fashion) is quite a demanding read, not the least reason being that to best understand math, one must "do" math, and even the most talented mathematicians recognize that sometimes to progress in math one is faced with the brutal harshness of learning math.  However, mathematics is an enormous field, and one can only choose to study, in one's lifetime, just a few areas in depth.  Therefore, a relatively serious, broad-based book like this, given a non-ideal world in which we individually simply cannot comprehend the field as a whole in the very finite, and short time we have available in our lives for such "cultural" reading, is a good book to acquire some of the flavor and perspective of mathematics on the large scale.  Even so, I find reading the articles, which for the most part are well-written, organized and edited, is a struggle and demanding (partly in that I cannot give each article the attention it deserves).  I find I am probably going to return to read some articles again, and possibly even third and fourth times, and do some of the calculations and proofs suggested in the articles.  The intellectual level varies somewhat, but my impression is that the authors tried to write to the educated audience of those with basic mathematical knowledge and skills in undergraduate-level mathematics.  (Some of the writing is extremely difficult for me to understand, requiring both concentrated attention and advanced knowledge.)  My impression is that this would be a fairly good book for graduate students, having passed Ph.D. quals, to get an idea of what might be interesting thesis areas, in addition to areas that appeal to the student from the basic studies in the first couple of years at graduate school.  Many students can be somewhat isolated (despite attending colloquia and seminars) and naive, and a book like this can awaken them to a fuller appreciation of mathematics.  I certainly would have appreciated such a book when I was a graduate student, but I cannot claim to have been well-prepared for graduate school in math:  It was a real struggle for me, often almost overwhelming.  It is important to bear in mind, too, that there are true conceptual and intellectual positions in mathematics to assess beyond the "definition-theorem-proof" diet that is fed the student of mathematics.  These conceptual issues are not necessarily trivial to grow in mathematical "maturity".  Therefore, a book like this can probably be worthwhile for any mathematician.  There may be some tendency to view the short articles as a mere dilettante's perspective on math.  As a companion, the articles are intended to enhance, and encourage additional exploration of the fields it addresses.<br><br>I have read the entire companion, commenting on some of it below, and have found that it has helped me to understand a little better some of the important aspects of modern mathematics.  It gives a huge perspective on twentieth century math, mostly focused on pure math, but with some articles that reach broadly into applied math, and is a very ambitious work.  One cannot expect a great improvement in knowledge from such reading, but I would say that each time I spend an hour or two with the book, I derive at least as much as I would from a colloquium.  I find that I am able to derive something of interest from even the most opaque articles after 30 or 45 minutes, and reading an article twice can often lead to substantial clarification (particularly if one consults some outside resources).  Sometimes the topics are well-known to me and I derive little, but a fair amount of the reading so far has been enlightening.  I have recommended the Princeton companion to others, because it has been quite exciting and exhilarating to experience some of these lucid presentations.  The Companion has been carefully edited and focused, which is often substantially important.  The Princeton Companion, overall, seems to be mainly directed toward well-prepared readers and to appeal to the young mathematician, but I think there is something of the young mathematician in all mathematicians.  There are many theorems and basic elements of vocabulary, which while not difficult, would amount to a very unusually well-prepared high school student.  I do not mean to suggest that high school students, or first or second year college students cannot benefit from the perspective one obtains through reading this book.<br><br>In reading the section on mathematical concepts, I find that it has been particularly helpful, with each concept, to give at least one concrete example.  In a way, this loses some of the sense of power entailed in the more abstract concepts, such as "K-Theory", but it can be a great aid to the understanding when being introduced, or perhaps re-introduced, to a concept.  For some of the more esoteric concepts, such as "Knot Polynomials", it is very helpful to discuss their importance, and context of their use.  Sometimes this more or less degenerates to "name-dropping", which I think in very short discussions is somewhat unavoidable.  I personally, also appreciated it when something of the historical context was presented, to give a larger perspective on the significance of a concept.  There are so many "mathematical concepts", it is not easy to give a short list, but I feel that the list provided in the companion is pretty representative, as evidenced by the numerous cross-references, that show that these concepts have lots of connections to many areas of mathematics.  The centrality of a few concepts, such as "group" or "invariant", which appear again and again in the discussion of each concept suggest that it would have been helpful to have added a bit to the list, but, of course, with "big" concepts like "matrix" or "group" or "invariant", it is difficult to confine one's remarks meaningfully to a few paragraphs or at most to a few pages.  Also, there is a sense that, to be representative, one must include a spectrum of concepts, with most of the concepts such that they display some of the richness of mathematics in a short discussion.  On a personal note, I was very pleased to see the Ising model included, as it is certainly one of my favorite among all the concepts, and I have somewhat specialized in this very simple, yet deceptive, concept that appears superficial, and yet turns out to be broad and deep.  It was a bit of a shock for me to see it included.  My other real favorite (I admit, I'm pretty specialized) is the Pade approximant, which I feel has been somewhat under-appreciated.  This latter concept was not included in the list of math concepts discussed.  Overall, I think the editors have handled a huge task of developing the section on mathematical concepts in a pretty optimal way, given the space limitations of the book.  There were very few discussions that I have read so far on concepts that have been excessively opaque.  I might mention that, although I derived some insight from the article on quantum groups, I found that article to be almost impenetrable, but the article itself, did not seem to be poorly written.  Some of the articles, such as that on special functions, focus on concrete examples, and do not require so much technical knowledge or vocabulary as the article on quantum groups.  I found the article on special functions to be pretty insightful, and not require much more than basic vocabulary and knowledge one would expect from an undergraduate major in mathematics.<br><br>In the section on Branches of Mathematics, the article by Prof. Mazur on algebraic numbers is the first one encounters.  It really boosted my insight into algebraic numbers, and was, indeed, a good companion.  There are numerous facets to algebraic numbers, and many rather basic aspects of them which one covers in undergraduate and graduate courses on algebra, and I was pleased that he did not just regurgitate standard textbook material, but of course, this makes his article somewhat difficult.  I liked the level that he tried to achieve, which I feel was that of concrete computation, accessible to the motivated high school student.  It is not so much concreteness that helps, but the careful examination of simple examples that are illustrative but may be somewhat abstract.  For example, he does discuss rings, which are fairly abstract, but in the context of unique factorization, which is simple and focused.  It was rather difficult for me to follow a few areas of the article, motivated as they were by getting the reader to make computations.  I have to say, Prof. Mazur's article was very inspiring, and led me to consult more material on the internet from YouTube in this area.  My time is very limited, due to a very tight work schedule, and as an amateur mathematician, I carry out my own researches "after hours".  But I did feel that I would like to spend more time on this subject after reading his article.  As a result of this article, I definitely plan to work more at this field, because one certainly gains a sense of its appeal and importance from the article.  If I had written the article, I would have stayed focused on certain limited specific examples.  I felt that his article lost some focus by trying to touch on too many different ideas, but, of course, this is a minor criticism, since he is addressing a huge field in just a few pages.<br><br>The next article, by Prof. Granville, concerns analytic number theory.  The article was written at a level of about a bright high school student, but I still had a fair amount of difficulty understanding it in its totality.  Actually, it would probably take me two or three readings to really catch on, and I would probably have to consult outside sources.  Nevertheless, it was a fascinating story about how people have tried, at least over the past 150 years or so, to understand prime numbers.  When one considers that these numbers are probably distributed quite randomly, as suggested by the prime number theorem, a fair amount of interesting work has been done.  The Riemann hypothesis is a highlight, of course, which the article discusses in a very illuminating way.  There is evidently also some connection to physics, which was suggested originally by Freeman Dyson.  Actually, I have been impressed in reading the Companion, how much of the mathematics discussed, which is presumably "pure", seems to be connected to physics, other sciences and engineering.  (However, I believe many mathematicians have emphasized that a substantial fraction of mathematics derives from "inspiration" out of the applied areas.)  If I were a high school student, I would really be inspired to study number theory in depth from this article.  I do not know, despite the incredible fascination of the subject, if I will ever return to read the article again.  Unfortunately, one finds with us old codgers, that we have highly prioritized "to-do" lists, and there are already too many things to do at the top of the list, plus, I cannot afford to put analytic number theory toward the top.  There are just too many other more important demands on my time that must be addressed first.  Still, even though I am old, I could sense the power and attraction of this field.<br><br>The article on computational number theory proved to be quite a strain, at least for a first reading.  Although perhaps not as deep as the other two number theory articles, computational number theory relies on many detailed computational tricks and heuristics.  Overall, it leaves an impression of great achievement in the face of daunting uncertainties, but the field is one that seems to require of one calculations, and not just casual reading of a single article.  As with the other articles, I definitely felt that my understanding would improve from multiple readings and consulting outside sources, in addition to working at some of the computations.  This field is certainly not well-suited for mere spectators.<br><br>The next two articles, on algebraic geometry and concerning the relationship between polynomials and geometry, describe a field, that while concerning rather simple objects, viz. polynomials, is remarkable in its technical subtlety.  It is this field that led to what is arguably the most important breakthrough of the twentieth century in mathematics, the proof to Fermat's last theorem.  Therefore, despite its technical depth and abstraction, because it concerns some of the most important mathematics of our time, it is certainly worth continued study.  I was helped (a little) in my understanding, especially through discussions of varieties and schemes in the second article, which seemed to be more didactic than the first.  One cannot help but feel, too, that this important area of mathematics, is worth continued effort to study.  Unfortunately for the initiate, this is a deeply difficult subject.  That usually does not discourage the motivated youngster.<br><br>The article on algebraic topology was intriguing.  While a graduate student, I just studied algebraic topology somewhat superficially, but I have an appreciation for its depths.  The article started with discussions of material with which I have some familiarity, but toward the end, I was having a great deal of difficulty fitting everything together:  There was simply too many new concepts for me.  Some concepts, like fiber bundles, are not too difficult to grasp, and I had encountered them before.  I am thinking that, since algebraic topology is rather basic, I may get a text and pursue improving my knowledge in this area somewhat.<br><br>Prof. Taubes wrote the article on differential topology, which starts with a basic introduction to smooth manifolds, and to the characterization of manifolds.  The article has a good discussion of the geometrization conjecture, proved by Perelman, and the associated Poincare conjecture.  I thought the discussions of dimension 4 or greater manifolds were pretty interesting and illuminating.  I found myself getting a bit lost, especially in the latter part of the article, discussing Perelman's ideas more carefully.  Overall, it was an article that helped pique my interest, and I would like to study these matters in more depth, as Perelman's result was certainly among the most important of recent years.  I'm glad many of the authors include references and further reading suggestions, as one would like, ideally (that is, given the time and motivation), to study many of these topics in more depth.<br><br>The article on moduli spaces by Prof. Ben-Zvi is very well written and informative.  He first discusses lines in the plane, considering how they vary continuously as families.  From there he heads into Riemann surfaces, the moduli of curves and Teichmuller spaces.  This provides some concrete basis for analogies with moduli spaces.  The article develops material that I found too difficult and abstract in addressing moduli spaces of genus of curves, with general genus.  However, thanks to the careful structuring of the article and the development of analogies, I do feel I gained a little from having read the article.  This article evidently is related to some areas in physics such as quantum field theory and string theory.  I think it represents, as all the articles have to this point, a concentration and focus that goes far beyond what I have been able to do in math, but it really seems to invite one to learn more and progress in this area.  It makes me wish I were young, and that I had more time to work and study at this.  This article really made me feel humble before the profoundness of mathematical science.  It was an excellent article, despite my lack of comprehension of the more difficult parts.<br><br>I read the article by Ian Grojnowski on representation theory twice.  I felt that I definitely understood the article better the second time.  He has done a very nice job at presenting some of the basic perspective on this theory, starting with simple considerations of finite groups, character tables and Fourier series, and attempting to describe how one can gradually generalize these notions to situations that are much more complex.  He ends with a brief description of the Langlands program.  I feel that a couple more readings would improve my understanding, but, of course, with such technical, and to a certain extent deep issues, one must go beyond a simple introductory article like this.  He does give some suggested references.<br><br>My impression of the fairly long article on group theory (finitely "presented" groups) by Martin Bridson is that it is well-worth reading a few times, as his discussion of free groups seems to really indicate that this is not only interesting mathematics, but highly versatile and tied into useful "universal" structures.  The title "Geometric and Combinatorial Group Theory", in itself, seems to suggest the field is very broad, as well as deep.  I got overwhelmed at the end of the article by just too much, but even there I found little aspects, as in his discussion of tree graphs, that seemed somewhat intelligible and well worth coming back to study again.  For now, however, I continued on (without even a second reading, which I think would improve my understanding), because I had so much of this book to read yet.  I intend to get the book co-authored by Prof. Bridson that he lists for further reading, and another book, as well on this subject.<br><br>The article by Terence Tao "Harmonic Analysis" was a beautiful piece of didactic explanatory work.  One feels, after having read this, that one has been very privileged, as Prof. Tao comes across as a superb teacher, someone the likes of which one rarely encounters.  In the references that he gives for further reading, he does not list any of his own work, but I suspect it would be very worthwhile to read.  He cuts through a lot of the "fog" that often makes studying mathematics doubly hard:  Not only can math be very difficult, but all of the specialist jargon can amplify the difficulties, and even notations can be complicated and impede understanding.  All of the articles I have read so far in the section "Branches of Mathematics" have been very interesting and certainly warrant further exploration, but his is a little gem.<br><br>Since my PhD is in physics, in the area of statistical mechanics, I have a fair familiarity with PDEs and found Prof. Klainerman's article on partial differential equations to be quite understandable, and interesting.  My background provides me with a perspective on much that he discussed.  He referred to a longer version of the article on the internet, which I certainly think would be wise to consult, but I am not going to take time to read the extended article.  His discussion centers around a "general theory" for PDEs, even though, as he clearly demarcates, such a theory is not possible, at least not at this time.  Taking this larger view, however, is very nice because it brings together physical and mathematical ideas in something of a synthesis.  He focuses on a discussion of elliptic, hyperbolic, parabolic and dispersive categories, and I was especially drawn to the idea of "bootstrapping", which he just discussed very briefly, but sounds intriguing.<br><br>I found the discussion by Prof. Dafermos in the article "General Relativity and the Einstein Equations" to be very interesting, because it indicates how closely coupled considerations of pure mathematics can be to the science of physics.  General relativity is conceptually very difficult for me, partly because the methodical aspect of the mathematics, which can get very profound, is sometimes also very confusing to me, and in general relativity seems to be closely tied to the pragmatic aspects of the science.  One often feels, in physics, that sheer method (to which so many pure mathematicians are dedicated) can lead one astray, as one must also pay attention to the world, as it is.  One recognizes that this is a limitation for many mathematicians, but in general relativity it seems to be an advantage, as the large steps in progress there seem to have been made by pure mathematicians.  I found the article, like others in this book, to be illuminating and certainly deserving a second reading.  Nevertheless, I came away still fairly mystified about general relativity, in a way that I am not about quantum mechanics, despite the strangeness of the latter subject.  (Certainly, von Neumann's brilliant insights there have better opened the door of mathematics.)  Partly, one simply realizes that my confusions result from the fact that I have never focused, in my studies, on general relativity, and a short article like this is far too short to adequately discuss general relativity.  General relativity is physics, and I feel more uncertain about my understanding of this subject than about the material I read on PDEs in this book.<br><br>The article "Dynamics" by Prof. Banner discussed the concept of Julia set in the context of some simple examples, and then more deeply discussed the Mandelbrot set and Bottcher maps.  She largely restricted her discussion to rational functions, in the context of being holomorphic.  Much of what she discussed was new to me, but I have studied chaotic dynamics a little, so I had some background to draw on.  Her writing is very clear and didactic, and yet, as with the other areas addressed in the book, there is much that demands extended study, and I found I was lost at several points in the article, despite the clarity of her exposition.  The juxtaposition of this article with the previous one on general relativity brings the Bottcher maps, as something fairly closely related to the maps of manifolds associated with general relativity, into a context that I had never thought about before.  It left one with the impression that Prof. Banner's article is one of the more important articles in the book.<br><br>I found I had to read the article "Operator Algebra", by Profs. Higson and Roe twice to derive enough meaning from it.  I felt that the article was pretty helpful in its discussions of group representations and C*-algebras (despite the fact that the authors do not define some things very clearly, including C*-algebra).  I noticed a definite improvement in my level of understanding when they got around to discussing K-theory a bit.  Despite the fairly substantial connections to physics, I thought the article was heavy going, but I also learned a fair amount from reading, with a number of interesting topics discussed, including Fredholm theory.  This develops a very nice perspective without overwhelming the reader, but I had to work!  It's nice to see that I connected a few things together that I have previously read about in the book.<br><br>The article "Mirror Symmetry" by Prof. Zaslow reports on the intriguing "duality" that has been used by the string theorists to simplify some of their problems. One is bound to make comparisons with general relativity and quantum field theory that preceding articles discuss.  Because string theory was developed much later in the twentieth century than either of these two latter theories, and was, indeed, intended as a unifying perspective on both theories, it places larger intellectual demands on us, as a theory that remains under development.  Nevertheless, the notion of mirror symmetry is very impressive in indicating that the symmetric "image" of a theory does not provide us with exactly the same point of view as the original theory.  Quantum field theory is a huge theory in physics, and one cannot expect Prof. Zaslow to provide an adequate treatment in just a short introduction.  However, he definitely whets one's appetite to look further by giving us some insight (and insight that is not that available to people working outside his specialty).  Furthermore, one realizes that while the physics seems to pose insurmountable problems at this time, the mathematics of conformal field theory connects with many other areas of mathematics, and seems to provide an avenue for fruitful research.<br><br>A related article to the preceding is "Vertex Operator Algebras", by Prof. Gannon.  This article discusses conformal field theory further, and describes some apparently important ideas about the vertex operator algebras, but I was not able to understand well-enough to evaluate this for myself.  Clearly, these algebras are regarded as extremely important, not only as a result of a number of interesting conjectures, called Monstrous Moonshine, but also as a result of the award of the Fields Medal to one of the seminal workers, R. E. Borcherds.  I got the idea that quite a lot is involved, even apart from the physics of string theory, because two important proofs in the field are described as "extremely long" and very "lengthy".  This seems to be a very promising area in which to work, because it is connected with a vast reach of mathematics, and appears to be developing into something of central importance, whatever the fate of string theory in physics.<br><br>The article "Enumerative and Algebraic Combinatorics" by Prof. Zeilberger discussed several very interesting counting problems, and considered in fair depth the complicated world of enumeration, which can be addressed with a certain set of tools, especially generating functions (using formal convergence of series).  I have studies counting problems to some extent, because they naturally arise in statistical mechanics, but this article probed deeper than the level of my rather superficial knowledge, and would be worth more extended study at some later time.  There is also a larger theoretical structure surrounding the field that would be interesting to explore, but computer computations seem to be a formidable driving factor in this field at the present.  One is impressed with the complications that can be handled sometimes rather elegantly in these problems, but I have to say that there also seems to be quite a playful side to Prof. Zeilberger's view on this field.  Sometimes the deepest (or at least deep) math arises from people who like to play and explore, just for the pure exuberant joy of working with mathematics.<br><br>Profs. Alon and Krivelevich are responsible for the article "Extremal and Probabilistic Combinatorics".  This is a tremendously rich area of mathematics, addressing a great multitude of often difficult problems in graph theory or combinatorics.  An extremely interesting feature of the work is the application of probabilistic arguments to establish results outside of the realm of probability theory.  I definitely felt that their article warranted a second more careful reading.  This seems to be an extremely important area of mathematics, with numerous connections to other areas of mathematics and applications, to computer science, physics and other areas.  Many seminal pieces of work, such as in random graph theory, in this area seem to be associated with Paul Erdos.<br><br>The article on computational complexity by Oded Goldreich and Avi Wigderson was one of the longer articles, and I think, also, one of the more informative and better written.  I thought the long discuss on P vs. NP was very enlightening.  The probabilistic methods, of course, seem intriguing and the discussion of pseudorandomness was very good.  The authors emphasize that, although their article is relatively long for this book, it is barely scratching the surface of a truly fascinating subject at the interface between interesting theory and important applications in computer science.<br><br>The article on numerical analysis by Lloyd Trefethen gave an excellent perspective on the development and evolution of numerical analysis.  It was shorter than some of the other articles, but I think that given its multidisciplinary nature, and the extent of its importance in computation, a really deep, full account would have taken one pretty far afield from focusing just on pure mathematics.  I enjoyed particularly the discussion of numerical linear algebra, which seemed to be a bit deeper than Prof. Trefethen's discussions in other areas.<br><br>Because mathematics is so deep, and vast in extent, the value of a companion, as this is, even just for pure mathematics, may be questioned.  Certainly, once one has penetrated the fog and obfuscation of some of the specialist jargon and confusing notations, and is able to read profitably good expert accounts of mathematics, say as in Rudin's Real and Complex Analysis, one can see a great merit to a companion, in say, real analysis or functional analysis, because at the expert level, mathematics and philosophy are very comparably deep subjects, and companions in philosophical fields are well-known to be extremely helpful.  However, most serious mathematicians do not share their insights as readily as philosophers, and so there is little awareness among mathematicians that companions would be useful.  This book, despite its fine articles, is not likely to change that situation.  One must simply struggle with mathematics by working problems, proving theorems, doing computations, etc. which gives mathematics inevitably a somewhat different character than philosophy, where open debate can often be beneficial, and makes one skeptical as to the merit of companions for mathematics, or beyond that, for physics.  These are subjects where, to paraphrase Newton, one often feels like a child on a beach, fascinated by the shells and rocks, while the great ocean of truth lies all unknown before one.  In one's lifetime, one can just explore and journey out into that ocean a little.  I find that it often takes me years to see the point of something, or why a certain strategy was adopted.  However, as I have read this book, I certainly think it has been a good companion.<br><br>I feel that studying the companion is akin more to a coffee shop conversation with someone who has dedicated his or her life to a subject, namely pure mathematics, and in discussion has much that is pertinent and wise to offer someone who takes more than a casual interest in math.  Most people in math who are professionals are specialists, and are little better able to offer insights into math in a broad perspective than others like graduate students who study math seriously.  So, the companion is really quite different, in outlook, than discussing something with a particular professional mathematician.  One is likely, in one's life, to encounter very few people in any field that could offer the broad, compelling perspective of the companion.  I am not claiming that there are not mathematicians who are not capable or who do not possess this far-reaching view of mathematics, just that one is not likely to encounter such a person, and if one did, an extended encounter, to transmit the level of information of the companion, seems very unlikely, indeed.  I strongly recommend reading the companion completely at least once:  It will serve one better as a reference by doing this.