5 comments about Yet Another Introduction to Analysis.

1. Bryant's book on analysis is a great illustration of what a textbook should be. He takes what many upper level college mathematics students consider to be the most tedious and boring topic - analysis- and presents it in a clear, interesting and effective way. Calculus at the college undergraduate level is usually taught in 3 semester long classes where integrals and derivatives are seen as tools for finding areas under curves, or volumes of objects, etc. which is the way engineers are made to view Calculus during their scholastic careers. Past that introductory level, in their junior years students of pure mathematics must be reintroduced to Calculus in a rigorous proof-driven way - thus enters the dreaded subject of "analysis", also sometimes called "advanced calculus" and thankfully, thus also enters this book. Bryant starts off assuming that rational numbers behave as we know from elementary school and then constructs the real numbers by adding a completeness axiom. From there he introduces the concept of limits and also the epsilon-delta technique in an accessible way before going on to the topics of differentiation and integration. Even though this is a mathematically rigorous book, the author manages to keep things interesting by introducing topics and theorems in bite-sized chunks. Basically, the book doesn't go beyond the analysis of calculus normally taught at the undergraduate level, but rather reintroduces it properly and puts it on the rigorous plane with which all graduate mathematics students shall become familiar. Along with all that, there's an excellent selection of interesting exercises with solutions at the back. These exercises range from the rather simple to the very tricky. If you are a mathematics major, you will probably not be lucky enough to have this as your textbook in analysis class, but you should buy a copy and read it before and during the class so that you know what is really trying to be conveyed.



2. Look inside the book! This book has a table of content, with only 5 entries. You you want to look up a thema to repeat it, or to learn a special thing you are interested in, you have to read 50 pages. This is realy stupid!



3. Just right!
   
   This is the real analysis book for all us Goldie Locks out there.



4. Victor Bryant's informal, conversational text, Yet Another Introduction to Analysis, offers an engaging, well-motivated introduction to real analysis, but it is not a full substitute for a more formal, more axiomatically structured approach. However, Bryant's text is a great companion text, and is especially suitable for self-tutoring purposes, or as pre-read prior to taking that first rigorous analysis class. The reader need only be familiar with first year calculus.
   
   As is so often said, mathematics is not a spectator sport, and Bryant clearly expects his readers to work the problem sets; the text frequently makes direct use of the results of previous problems. Bryant provides full solutions to nearly every problem, another reason why this book is so good for self-study. (The solutions section is 67 pages.) Bryant's problems were rarely difficult or overly time consuming, and are most notable for clarifying key points in the text.
   
   Bryant begins with a brief examination of real numbers, looking at why the irrational numbers so out number the rational ones. (The completeness axiom is introduced in the short first chapter.) I particularly enjoyed the next section, Bryant's examination of whether a series converges or not and ways to determine the sum of an infinite series. (I had not previously been all that interested in the study of series, but Bryant's approach peaked my interest. I have now purchased a more advanced Dover reprint, Infinite Series by James M. Hyslop, for follow-up reading.)
   
   A longer section examines the familiar concept of a function from various perspectives, using the inverse relationship between exp and the log as one of the key examples. The final two chapters focus on a primary topic of analysis, the basic theorems of differentiation and integration. Familiarity with partial differentiation and multiple integration is not needed.
   
   Some readers may find Bryant's conversational approach to be too wordy and occasionally digressive, but I personally enjoy his leisurely style. I also recommend Bryant's short text titled Metric Spaces, Iteration and Application, published by Cambridge University Press.
   
   Another good choice is Maxwell Rosenlicht's Introduction to Analysis, available in an inexpensive Dover edition. It offers a more traditional, structured approach to analysis that is suitable either as follow-up to Yet Another Introduction to Analysis, or as a stand-alone self-tutorial text. Although Rosenlicht's text emphasizes generality and abstraction to a greater extent, it is still more concrete and less terse than many standard texts on real analysis.



5. I found this book an excellent introduction to real analysis. The math courses I took during my US undergraduate engineering degree (your standard Calc I - Calc III) focused more on computation than theory. This book gave me a deeper understanding of the real number line, sequences and series, functions, differentiation, and integration, as well as some much-needed practice in writing proofs.
   
   I was a bit worried starting the book that it would be too difficult, but fortunately, the book started at just the right level for me and continued at a good pace. The book is written in a friendly and conversational style and all the concepts are well-explained, with lots of graphs to make things clear.
   
   The exercises often have you proving some key theory that is referred to later on, which gives a strong motivation to work through all the exercises. For someone with little experience writing proofs like myself, the exercises were not overly difficult, but provided a good challenge. The book provides full, worked-out solutions to all the exercises, which makes it great for self-study (I used the book to get some background on analysis over summer before I started my master's).
   
   Overall, I found this to be an excellent book. I highly recommend it for self-study or as a supplement to a course.

4 comments about Geometry Civilized: History, Culture, and Technique.

1. All books are unique, as George Orwell might have said, but some are more unique than others. And Heilbron's "Geometry Civilized" may be the most unique of all. It is, on the one hand, a coffee table book, in size and presentation, with beautiful illustations. On the other hand, it is a serious geometry text with full proofs of many theorems in Euclidean geometry, and plenty of interesting exercises for the reader. But perhaps most of all, it is a fascinating ramble through a wide range of topics, written by a leading historian of science with a strong esthetic sense and equally strong views on math and science education. He is, in the words of W.S. Gilbert, "Teeming with a lot o' news", including "Many cheerful facts about the square of the hypotenuse" -- the title of his chapter on the Pythagorean Theorem. Another chapter, "From Polygons to Pi," includes the exact geometry of a Gothic arch and much of the accompanying ornamentation, as well as other topics ranging from Stonehenge to the Pentagon building, and from the idea behind burning mirrors attributed to Archimedes and actually constructed by Lavoisier and others, to the octagonal room designed by Thomas Jefferson. Anybody who enjoyed geometry in high school should love this book, and many people who feared or hated high school geometry may discover what they missed by not having a John Heilbron to show them the wonderful richness and flavor of what, presented badly, can appear a dry and useless subject.



2. The rise of geometry was simultaneous with the rise of civilization. When people are aggregated into permanent population centers, it becomes necessary to precisely measure areas so that the proper taxes can be collected and also measure in the third dimension so that magnificent buildings can be constructed. While both require knowledge of geometry, they are experiential in nature.
   The true rise of Western civilization as we know it took place in Greece, and the most permanent feature of that culture was the development of abstract mathematics. From this point one, mathematics was an endeavor that involved objects whose true nature was only in the mind. Any geometric diagram could only at best be a crude approximation of the true situation. It is hard to underestimate how much of a change this was from earlier forms of reasoning. I for one, firmly believe that all of the other ideas of democratic government, ethics, logic and philosophy that arose at the same time and place were a consequence of the new, more abstract and theoretical thinking that was taking place in geometry.
   This book is a combination of history and geometry, showing how intertwined the two are. It is also one of the more extensive collections of solved geometry problems that exists today. Heilbron poses many problems, solving nearly all of them immediately after they are posed. As you step through the solutions, it is with a sense of wonderment as the steps are so direct, sequential and easy to understand, which is the hallmark of good geometric proofs.
   Packed with figures, this book is suitable as a text for courses in geometry that the students will love, and I will point it out to anyone who claims that geometry is dull. With supplements, it could also be used as a text for courses in the history of mathematics.



3. The rise of geometry was simultaneous with the rise of civilization. When people are aggregated into permanent population centers, it becomes necessary to precisely measure areas so that the proper taxes can be collected and also measure in the third dimension so that magnificent buildings can be constructed. While both require knowledge of geometry, they are experiential in nature.
   The true rise of Western civilization as we know it took place in Greece, and the most permanent feature of that culture was the development of abstract mathematics. From this point one, mathematics was an endeavor that involved objects whose true nature was only in the mind. Any geometric diagram could only at best be a crude approximation of the true situation. It is hard to underestimate how much of a change this was from earlier forms of reasoning. I for one, firmly believe that all of the other ideas of democratic government, ethics, logic and philosophy that arose at the same time and place were a consequence of the new, more abstract and theoretical thinking that was taking place in geometry.
   This book is a combination of history and geometry, showing how intertwined the two are. It is also one of the more extensive collections of solved geometry problems that exists today. Heilbron poses many problems, solving nearly all of them immediately after they are posed. As you step through the solutions, it is with a sense of wonderment as the steps are so direct, sequential and easy to understand, which is the hallmark of good geometric proofs.
   Packed with figures, this book is suitable as a text for courses in geometry that the students will love, and I will point it out to anyone who claims that geometry is dull. With supplements, it could also be used as a text for courses in the history of mathematics.



4. Heilbron's greatest accomplishment in this work is the very thorough cutting and pasting that brings us many pretty pictures, especially from the worlds of art and architecture and old textbooks. Other than that there is little of value. The bulk of the book is the same old terse Euclidean geometry that you can find in just about any geometry book. You might as well read Euclid because Heilbron adds basically nothing in terms of insight and readability when it comes to the geometry itself. In fact, he repeatedly manages to create technical obstacles even in clear terrain; see for example what must surely be the most incomprehensible introduction ever of radian angle measure on page 278. Still, the book also discusses many diverse applications which perhaps makes it worthwhile? Unfortunately, no. First of all there are some horrendously formulated statements, such as the claim that pi "cannot be expressed as a number, even an irrational one" (p. 241) and the implicit claim that three points need not lie in a plane: "Assuming, what is more or less true, that Rhodes and Alexandria lie on the same noon circle or meridian (that is, that Rhodes, Alexandria, and the centre of the earth lie in the same plane), ..." (p. 66). One wonders how such things survived into the "corrected" paperback edition. More seriously, Heilbron frequently breaks the rule that in science and mathematics everything should be explained and nothing should be pulled out of a hat. He is forced to do so because he doesn't have very many interesting applications of Euclidean geometry to offer and so has to discuss applications that are thoroughly incompatible with the mathematics covered. This is completely unnecessary since Euclidean geometry has many wonderful applications, but Heilbron simply ignores them: remarkably, conic sections, for example, are never mentioned even though there is a section on burning mirrors (!?), where we are told in a parenthesis that "a slightly different surface, whose intersection with the plane of the paper makes a parabola, gives a more intense focus" than a spherical mirror (p. 282, this is the only occurrence of the word parabola in the book). Instead, for example, we learn that "a Dutch geometer named Willebrord Snel" simply "proposed" the law of refraction (p. 107), apparently on a whim, and there is no indication of why nature would choose to obey this curious law. Later this law is fundamental when we study the rainbow, yet another topic that our methods are completely incapable of handling. Lacking calculus, we resort to the use of a table of values and read off that "evidently" the properties of refraction in raindrops are such-and-such (p. 197). If this is a valid method then why did we bother toiling with proofs of the Pythagorean theorem, for example? We could have just thrown up a bunch of numerical calculations and said that "evidently" the theorem is true. Geometry is not "civilized" by betraying the very soul of rational inquiry.

No comments about Dynamical Systems and Ergodic Theory (London Mathematical Society Student Texts).

1 comments about Unsolved Problems in Geometry (Problem Books in Mathematics / Unsolved Problems in Intuitive Mathematics).

1. After the counting numbers, geometry is the oldest branch of mathematics and no doubt the first one that required abstract thinking. Even so, there is always a certain "concreteness" about it in the sense that diagrams can almost always be constructed. The range of problems that fall under the geometric umbrella is extremely wide and some even have practical uses.
   This book is a testament to the wide range of problems that are geometric in nature. One of my favorites is known as the "worm problem." To be more precise, the question is, "find the convex set of least area where any continuous curve of length one can be placed in it." This type of problem has ramifications in optimal packings, where a single type of container needs to be constructed for all possible ways an object can fold. Other problems such as tiling and dissection; packing and covering and combinatorial geometry are also covered.
   However, the best part of the book may be the extensive references. Every problem is followed by a list of references, so if you wish to take a crack at it, you will have little difficulty in locating the work done to the date of publication.
   This is one of those books that always seems to beckon me when it lies on my bookshelf. Every once in awhile I pull it off and browse through it, admiring the skill and breadth of mathematicians in their pursuit of truth. It should be in every academic library.
   
   Published in Journal of Recreational Mathematics, reprinted with permission

1 comments about Matrix Groups: An Introduction to Lie Group Theory.

1. I've been been reading it for only half an hour so I can't say much about the content of this book,however one thing is clear this book's life is going to be very very very short,in fact I don't think that tomorrow will be in one piece,I'm very disappointed to say the least,I don't understand how springer can produce such a poor quality book
   now I could try to glue it wich is a mess or collect the pages as they fall apart and staple them,unfortunately I have to say that this is not a very rare case, cause many Springer paperbacks suffer from similar lack of binding quality

No comments about Algebra and Trigonometry with Analytic Geometry (with CengageNOW Printed Access Card).

3 comments about Catastrophe Theory.

1. Catastrophe theory was the first mathematical "advance" to receive extensive coverage in the popular press since quantum mechanics. The interest and adulation afforded to chaos theory and nonlinear dynamical systems research barely immitated the enthusiasm and later scorn generated by Thom and his followers. In retrospect it is hard to see what all the excitement was about. Perhaps that's exactly why this book deserves serious attention.

   While the doctoral trained mathematician may find more faults here than the rest of us, this book provides sufficient complexity for the professional, and at times is "gentle" enough for the nonmathematician. It will probably find its most useful audience among people with "semi-mathematical" trainings, scientists in life and behavioral or social sciences. For these people, who may just want to find out why Thom's initial theorems generated so much excitement and controversey, this book will be a readable delight.




2. Catastrophe theory was the first mathematical "advance" to receive extensive coverage in the popular press since quantum mechanics. The interest and adulation afforded to chaos theory and nonlinear dynamical systems research barely immitated the enthusiasm and later scorn generated by Thom and his followers. In retrospect it is hard to see what all the excitement was about. Perhaps that's exactly why this book deserves serious attention.

   While the doctoral trained mathematician may find more faults here than the rest of us, this book provides sufficient complexity for the professional, and at times is "gentle" enough for the nonmathematician. It will probably find its most useful audience among people with "semi-mathematical" trainings, scientists in life and behavioral or social sciences. For these people, who may just want to find out why Thom's initial theorems generated so much excitement and controversey, this book will be a readable delight.

   This book should be mandatory reading for anyone claiming an interest in chaos theory but who does not understand how some of the overzealousness of a new paradigm can have devestating consequences. Needless to say, those unfamiliar with contemporary mathematical advances and behavioral/life/physical applications in European literarature will find this book invaluable.




3. Catastrophe theory is introduced as a sort of merger of Whitney's theory of singularities of mappings and Poincaré's qualitative theory of dynamical systems. First Whitney. A surface is projected onto a plane. Somewhere the surface is folded, so that the inverse of the projection is multi-valued. Now, the plane may represent the possible values of the control parameters of a dynamic system, and the surface the possible states of the system. Moving continuously in the plane across the boundary between a single-valued and a multi-valued region may cause a jump on the surface to one of the other sheets--i.e. a small external change causes the system's state to change drastically: a "catastrophe". Poincaré's bifurcation theory of dynamical systems may now be perceived similarly on a metalevel where the systems themselves are points in a space--again an infinitesimal move in the system space may cause drastic changes of the system's equilibria. This type of geometric thinking may then be used in applications--but only sober ones, mind you: elasticity, optics, etc. Back in the old days Thom quite successfully pushed his catastrophe theory on gullible non-mathematicians. Arnol'd states his own view on those matters clearly and repeatedly throughout the book. From the preface: "Neither in 1965 nor later was I ever able to understand a word of Thom's own talks on catastrophes. He once described them to me as 'bla, bla, bla'". Arnol'd instead prefers the mathematical meat and potatoes of catastrophe theory: the theory of singularities. If only one were as enthusiastic as Arnol'd about singularity classification theorems then this would be very interesting indeed.

No comments about Geometry of Quantum States: An Introduction to Quantum Entanglement.

1 comments about Noncommutative Geometry, Quantum Fields and Motives (Colloquium Publications (Amer Mathematical Soc)).

1. Mathematics and physics has had an intense symbiotic relationship in the last few hundred years, but even more so in the last three decades, due in part to the willingness of physicists to learn highly sophisticated "pure" mathematics, but also the willingness of mathematicians to learn some physics. This symbiosis has resulted in brilliant developments in fields of physics and mathematics such as quantum field theory, superstring theory, and differential and algebraic geometry. This research is fascinating but it has put those entering these areas into somewhat of a quandary, in that they must master both the physics and the mathematics in order to be able to get to the frontiers of research. This is complicated by the relative lack of good books on the subjects that go beyond the mere formalism and strive instead to give insight and needed intuition.
   
   In this regard this book does not attempt to be instructional and it presupposes that the reader has already mastered perturbative quantum field theory as well as the theory of mixed motives from algebraic geometry. To obtain mastery of both of these areas is a formidable undertaking, and the authors' brief overviews of them in the book will not suffice to give an in-depth understanding. In addition, the book does not make an attempt to reach into the area of nonperturbative quantum field theory with all its difficulties and unknowns. Indeed, research into this area has been extremely slow and this shows no sign of improvement.
   
   The authors want to view a renormalizable perturbative quantum field theory T in terms of an affine group scheme that is associated with the subcategory of flat equisingular vector bundles arising from T. This group scheme as it turns out can be viewed in terms of Galois theory, which is a very curious development and which the authors spend the first eight sections of the book discussing in some detail. The quantum field theory T will be finite if its Galois group is trivial and super-renormalizable if its Galois group is finite-dimensional. Even though expressing quantum field theory in this manner will not facilitate computations such as the calculation of cross-sections, it does offer hope that the renormalization procedure does have some mathematical rigor to it. Many mathematicians have been skeptical of quantum field theory because of this lack of rigor.
   
   Their discussion begins with the `Connes-Kreimer' theory of renormalization which expresses the Bogoliubov-Parsiuk-Hepp-Zimmermann (BPHZ) procedure in terms of a Birkhoff factorization of loops in a particular Lie group associated to a commutative Hopf algebra. To the physicist reader who has worked in the theory of integrable systems in statistical mechanics, Hopf algebras are very familiar, in that they arise in the factorization conditions for the partition function. In the mathematical literature they are now usually discussed under the topic of `quantum groups', but historically they arose in studying the singular cohomology of locally finite topological spaces with a base point. If one applies the singular homology functor to the diagonal mapping of a pointed space, then this induces a homomorphism on the homology called a `comultiplication.' If the homology functor is applied to the basepoint mapping this induces a homomorphism on the homology called the `unit.' Abstracting from these considerations to the case where the homology is a graded vector space over a field gives a `comultiplication' as mapping from the homology to the tensor product of the homology. One can then define a notion of coassociativity for this comultiplication, a `counit', the latter being a mapping of the homology to the field over which the homology is defined that factors through the tensor product in a natural way. These mappings define a `coalgebra' structure for the singular homology of a locally finite space. If now one does essentially the same thing with the cohomology functor, then one obtains the cup product and a unit which are the "dual" notions to the coalgebra case. But if then one wants to be concerned with the homotopies of these maps, and how to multiply them and so on, one obtains the notion of an `H-space'. The homology of an H-space has a product called the Pontryagin product, and if one dualizes this algebra one obtains a comultiplication. A Hopf algebra has both of these products satisfying the conditions as delineated by the authors.
   
   The authors are not concerned with how Hopf algebras are used in topology but instead are interested in their relation to `affine group schemes.' These objects may sound exotic at first but they just a formalized methodology for constructing groups from other algebraic objects. For example, if R is any commutative ring with identity then the 2 x 2 matrices with entries in R having determinant 1 form a group under matrix multiplication, called SL2(R) in the literature and which is an affine group scheme. Many other such elementary examples of affine group schemes are available, and the procedure by which the groups are constructed can be abstracted into the notion of a `representable functor'. For an algebra R defined over a field k (a k-algebra) one wants a method by which a group G(R) can be obtained from R. The functor obtains G(R) by finding the solutions in R of some family of polynomial equations defined over k. The trick is to find the most general solution of these equations, and to do this one considers a polynomial ring over k with indeterminates the variables in the equations. One then factors out the ideal generated by the relations expressed by the equations to obtain a quotient algebra A. If G(R) is the solutions of the equations in R, then a homomorphism from A to R will take the general solution to a solution in R that corresponds to an element of G(R). For such an A, this gives rise to a correspondence between G(R) and Hom(A, R; k), and in general one can show that if G is a functor from k-algebras to sets, and if G(R) is the set of solutions in R of a family of equations, then there is a k-algebra A and a correspondence between G(R) and Hom(A, R; k). A is then said to represent F and an affine group scheme over k is a representable functor from k-algebras to groups.
   
   The authors are focused on affine group schemes where A is a commutative Hopf algebra and R is a unital k-algebra, and the Lie algebras of affine group schemes. Their strategy is to assign a Hopf algebra over the complex numbers to a given renormalizable quantum field theory in such a way that the coproduct of this Hopf algebra respects the operations in the BPHZ formalism. Thus they construct a "Hopf algebra of Feynman graphs", by first concentrating on the graphs responsible for divergences and then later on the full collection of Feynman graphs by the inclusion of arbitrary one-particle irreducible graphs. Interestingly, the Hopf algebra they construct has a grading by both the loop and line number of the Feynman graphs, with the Hopf algebra being connected in the second case by not in the first. For the second case, this allows them to show, although they do not do so explicitly, that the affine group scheme associated to this Hopf algebra is a projective limit of linear algebraic groups. For the general case where external lines are included, the full Hopf algebra is a symmetric algebra on the linear space of distributions, with the latter being necessary to model the incoming momenta.
   
   The crucial result that allows the authors to get a handle on the BPHZ formalism is the Birkhoff factorization of loops. Those familiar with the inverse scattering method, the theory of complex vector bundles, or the factorization conditions in the theory of integrable systems should have no problem understanding Birkhoff factorization. The authors assume prior familiarity with it since they do not motivate it intuitively or historically in the book. The authors use the language of affine group schemes in order to obtain a recursive formula for the factorization. For the case of the Hopf algebra of Feynman graphs, they show that the recursive formulae are identical to the BPHZ renormalization. This is a very interesting result, and should placate critics who have been skeptical of the non-rigorous nature of the renormalization procedure in quantum field theory. Even more interesting is that the authors show what role the renormalization group and the beta function play in this new formalism. The renormalization group appears as a 1-parameter subgroup of the affine group scheme associated with a positively graded connected commutative Hopf algebra and the beta function is the infinitesimal generator of the renormalization group.
   
   It is after the discussion of renormalization that the level of mathematics may become very challenging to some readers, especially those from the physicist community. For it is here that the authors bring in some heavy machinery from algebraic geometry. Indeed, even specialists in this field may find the reading difficult, as the authors only summarize the main results from what is now called the theory of `motives.' Loosely speaking, motives allow one to do algebraic topology over algebraic varieties and schemes. Before getting into the theory of motives and the advantages they bring to their program, the authors prove that the counterterms in perturbative renormalization depend only on the beta function. In the interim they give a rigorous formulation of the `time-ordered exponential' in quantum field theory, which is a bread-and-butter calculation tool in quantum field theory and its mathematical formulation done rather sloppily in most textbooks and treatises on quantum field theory. There have been attempts in the literature to express the time-ordered exponential in the context of C*-algebras, but here the authors formulate it in the language of affine group schemes. This approach allows them to give a geometric interpretation of the divergences in renormalizable quantum field theory in terms of `flat equisingular connections.' This unexpected and very interesting strategy involves the use of a principal fiber bundle where the choice of base point is essentially fixing the value of Planck's constant, and the fibers are possible values of the mass parameter scaled by Planck's constant. When a particular class of connections, the equisingular flat connections are introduced in this bundle, this allows the authors to interpret the negative part of the Birkhoff factorization for loops in terms of equivalence classes of these connections. Interestingly, the compactification of this bundle corresponds to the classical limit where Planck's constant is zero.
   
   The reformulation of quantum field theory in terms of equisingular flat connections allows the authors to use the Riemann-Hilbert correspondence to view them using the language of representation theory, a familiar strategy in physics. This therefore entails a kind of linearization, but it is done in this book in the context of affine group schemes. The authors motivate this more in a more general context of what is called a `neutral Tannakian category.' Loosely speaking, this is a category that can be shown to be equivalent to the category of finite-dimensional linear representations of an affine group scheme. To construct such a category they first consider `tensor categories', which have the familiar properties the linear transformations over ordinary vector spaces have and satisfy a coherence condition with respect to the tensor product. A particular type of tensor category, called `rigid', has a duality operation that allows one to have a well-defined notion of dimension, i.e. that the dimension be non-negative. The astute reader will note that this requirement is not as trivial as it appears at first glance, since if the dimension of an object in a category is defined as the Euler characteristic, it can be negative (take for example a curve of genus g). A Tannakian category is obtained if one can construct a `fiber functor' from this tensor category into the category of vector spaces defined over an extension of the field over which the tensor category is defined. The Tannakian category is `neutral' if this extension field is equal to the original field. The neutral Tannakian category of interest is the category of finite-dimensional linear representations of affine group schemes.
   
   At this point the authors want to make the Tannakian formalism more concrete by applying it to the category of `differential systems.' This entails they delve a little deeper into the Riemann-Hilbert correspondence and how to use it to obtain a Galois theory of differential equations. As expected if compared to the familiar Galois theory of algebraic equations, this theory looks at the symmetries of the possible solutions of a (linear, meromorphic) differential equation to obtain its `differential Galois group'. Such a differential equation of course may have singularities, but if these singularities are `regular' they can studied by the well-known Riemann-Hilbert correspondence. The latter examines the behavior of the differential equation in the neighborhood of a point, and the regularity allows one to obtain an analytical continuation of the solution along any path that does not encounter any singularities. Deforming the path continuously does not alter this continuation, and so one obtains the action of the homotopy group of loops on the finite-dimensional space of local solutions at this point. The representation of this action in the general linear group over this space is the familiar `monodromy' group of the differential equation, and this has been generalized recently for the case of irregular singular differential equations by the use of what is called the `wild fundamental group.' The authors do not go into these constructions in any kind of detail, but instead refer the reader to the references (most of which are written in French). The differential Galois group is the Zariski closure of the wild fundamental group, and the authors briefly show how to construct the Riemann-Hilbert correspondence for this more general case. This is done in the context of the category of differential modules, and the differential Galois group of a differential equation is obtained by using the affine group scheme associated to the Tannakian subcategory generated by the differential module associated to the equation.
   
   Along these lines, the main interest of the authors though is in using the Riemann-Hilbert correspondence for the class of flat equisingular connections constructed in earlier sections. In order to carry this out, they must construct the category of flat equisingular vector bundles, which they do by utilizing filtered vector bundles. For those readers who have been introduced to mixed Hodge structures, filtered vector bundles should be very familiar, for mixed Hodge structures can be straightforwardly defined using them (and those readers who are familiar with mixed Hodge structures will find the discussion on motives later on much more palatable). The category of flat equisingular vector bundles that they construct is shown to be a neutral Tannakian functor, and also to be equivalent to the category of representations of a particular affine group scheme. The latter is constructed by making a connection with time-ordered products and `universal counterterms', thus solidifying the connection with perturbative quantum field theory. This connection is made even more concrete in that they show that this affine group scheme acts on the coupling constants of quantum field theories and can be used to define a `Galois group' for such theories. A renormalizable quantum field theory is shown to be finite if and only if its Galois group is trivial, and super-renormalizable if its Galois group is finite-dimensional.
   
   It is at this point that the authors make a giant leap into an area that might strain the mathematical background of the physicist reader. Using what they have done so far, the authors want to express perturbative quantum field theory in terms of the theory of motives. The latter is a very esoteric branch of algebraic geometry with some parts of it being ill defined at the time of publication. The authors only give a short summary of this field, and speak of it as the goal of finding a "universal" cohomology theory for algebraic varieties over a semi-simple monoidal category. It can also be viewed as an attempt to generalize the K-theory of vector bundles to the context of algebraic cycles or `correspondences'. A correspondence is essentially a linear combination of irreducible algebraic subvarieties of a product of smooth projective varieties. The collection of correspondences forms an abelian group and since they are defined in a product it makes sense to speak of their intersection number with respect to a subvariety of complementary dimension. One then defines an equivalence relation called `numerical effective equivalence' on the cycles by saying two are equivalent if they have the same intersection number with this subvariety. Readers with a strong background in algebraic geometry will realize that if this equivalence relation is replaced with `rational equivalence' one obtains the Chow groups, which are widely studied and which are discussed briefly by the authors. If one instead fixes the Weil cohomology and defines an algebraic cycle to be trivial if its image under the cycle map is trivial, then one obtains a notion of `homological equivalence' of cycles.
   
   As the authors review (very briefly) there are many different versions of cohomology for algebraic varieties, depending on the characteristic of the field over which they are defined. For smooth projective algebraic varieties over fields with separable closure, one can define the `etale cohomology'; for fields of characteristic zero, one has the `de Rham cohomology' which is very familiar to physicists, and when this field can be embedded in the complex numbers the `Betti cohomology'. When the field has positive characteristic, there is another cohomology, called the `crystalline cohomology' that is probably the one unknown to most readers and somewhat mysterious because of its connection to what are called the ring of `Witt vectors' of the field. Crystalline cohomology was invented to fulfill the Grothendieck dream of integrating number theory with algebraic geometry, and the role of Witt vectors comes in as sort of methodology for lifting the Frobenious endomorphism from a context of prime characteristic to one where the characteristic is zero.
   
   As stated before, these different cohomology theories can be related to each other, but mathematicians naturally are lead to ask whether they can be viewed as different manifestations of an underlying "universal" cohomology theory that has all the properties that a suitable cohomology should have. This strategy is manifested in the concept of a `Weil cohomology', and showing that the various cohomology theories are actually Weil cohomologies has resulted in a huge amount of mathematical research, marked by brilliant developments. The authors mention some of these developments but leave the details to the references. The idea of a motive is to encapsulate the notion of a universal cohomology theory for algebraic varieties but whose coefficient space is not the graded vector space of Weil cohomology but rather a semi-simple monoidal category.
   
   The authors give some of the details behind how to construct a motive beginning with category of algebraic cycles under the numerical equivalence relation. They show that this category can be made into a semi-simple Abelian category with a rigid tensor structure. The latter structure needs to have a notion of duality, and this is obtained by adding the famous `Tate motives' to this category. This could be made into a Tannakian category if one had a suitable fiber functor. A restriction on the positivity of the trace for Tannakian categories forces the authors to modify the tensor product structure on their category of algebraic cycles in order for it to be a Tannakian category. A neutral Tannakian category is obtained if one assumes the "standard conjectures" of Grothendick. The authors identify various subcategories of this category, such as the subcategory of Tate motives and the subcategory of Artin motives, and the motivic Galois group of the latter is a generalization of the ordinary Galois group.
   
   
   Note: This review is based on a reading of sections 1 - 8.2 of the book.

No comments about Plane Algebraic Curves (Student Mathematical Library, V. 15).