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Complex Topological K-Theory (Cambridge Studies in Advanced Mathematics)
Complex Topological K-Theory (Cambridge Studies in Advanced Mathematics)
by Efton Park
Edition: Hardcover
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5.0 out of 5 stars Perfect for self-study, March 28, 2014
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K-theory, whether it be algebraic or topological has many uses and applications in mathematics and high energy physics, and this entails that its understanding is crucial if one is to enter into those areas which it has found applicability. As in almost all areas of modern mathematics, the intuitive explanations of the ideas and concepts behind them are usually lacking, with emphasis placed on formal constructions and proofs. The latter of course is what makes mathematics what it is, but for those who thirst for a real understanding of a particular subject area, such as K-theory, the formal style of writing in modern mathematical texts and monographs will not quench this thirst.

This book is different in that it offers such an understanding, but without sacrificing the rigor that is expected in mathematics. Students of K-theory, or those who want to understand its applications, will therefore benefit greatly from the study of this book, and definitely take away an appreciation of the context in which K-theory arose historically. This is especially the case in the manner in which the author discusses the needed mathematical tools in the first chapter of the book. Indeed, the notion of an idempotent is clearly understandable as being a generalization of an ordinary projection operator in Euclidean space. Readers will learn the enormously important role that idempotents play in K-theory, and good examples of them occur throughout the book.

That one can treat vector bundles and idempotents as groups, even though they are not, is one of the unique features of K-theory, and being able to add and subtract vector bundles and idempotents comes from taking what is called the Grothendieck completion of these objects. The author shows in detail that when this is done vector bundles and idempotents become naturally isomorphic. This isomorphism between classes of idempotents and classes of ranges of these idempotents makes it crystal clear why idempotents and be viewed as generalizations of the projection operators in ordinary vector space theory. The Grothendieck completion of the class of vector bundles (or idempotents) of a compact Hausdorff space X is the zeroth K-group of of X.

Just as in the ordinary theory of vector spaces, where one can study subspaces of the vector space at hand, K-theory can be done for closed subspaces of a compact Hausdorff space. This goes by the name of ‘relative K-theory’ and the author gives a good motivation from a geometric point of view in the book. Of particular importance in the study of relative K-theory is the construction of a ‘one-point compactification’, since in later developments and applications of K-theory to areas such as homotopy theory and the theory of spectra it is used quite extensively, along with its generalization called the ‘suspension.’ The one-point compactification is also used in the book to prove the famous Bott periodicity theorem, and in the proof of the latter the author is kind to the reader in discussing the general structure of the reader before jumping immediately into its details. Subtracting trivial vector bundles (which have a zeroth K-group isomorphic to the integers) from non-trivial vector bundles is the topic of ‘reduced K-theory’, where the intent is to concentrate the effort on the non-trivial part of the vector bundle. As the author shows, this is accomplished by using pointed spaces, which should be very familiar to the reader acquainted with homotopy theory.

Readers familiar with differential topology will appreciate the discussion and the proof of the Thom isomorphism, due in part to the use of exterior calculus and the ball and sphere bundles. It is somewhat surprising to learn that the K-theory of a vector bundle V over a compact Hausdorff space X and the K-theory of X are in fact isomorphic, and readers who go through the proof of the Thom isomorphism and who are familiar with the suspension of a space will see it generalized to the case of vector bundles.

An Introduction to Galois Cohomology and its Applications (London Mathematical Society Lecture Note Series)
An Introduction to Galois Cohomology and its Applications (London Mathematical Society Lecture Note Series)
by Grégory Berhuy
Edition: Paperback
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5.0 out of 5 stars Excellent, February 16, 2014
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Given a field k, how does one classify mathematical structures defined over k which become isomorphic over a finite Galois extension or over an arbitrary Galois extension? This question is answered with great clarity in this book, and the explanations and motivations given in it have to rank its didactic quality as being one of the best in the mathematical literature. And most importantly, it does so without sacrificing mathematical rigor, which proves that the latter and intuitive understanding are not inversely related. Readers will walk away with an appreciation of Galois cohomology that might be difficult to attain by the study of other books or research papers.

Readers who intend to study this book will know that they must have a thorough grounding in abstract algebra, and be familiar with field extensions and Galois theory. The author motivates the subject of Galois cohomology by examining the simple case of the descent problem for matrices, namely the problem of determining whether two matrices are conjugate over a (finite) Galois extension of a field k remains so over the k itself. The author shows how this problem involves finding an obstruction to descent, and this obstruction is essentially a map that measures how far a matrix is from being conjugate to another by an element of the matrices over k. Infinite Galois extensions are dealt with by using pro finiteness and then the challenge is to patch the obstructions together. Finding such obstructions in contexts more general than matrices is the subject of Galois cohomology.

A context in modern mathematical terms is of course a category and to expose the generality of Galois cohomology the author gives a short review of category theory in the book. One category of particular interest in the book is the category whose objects are field extensions of a field and whose morphisms are morphisms of extensions of this field extension. Also of interest is a covariant functor from this category into the category of sets. For a field extension O, the elements of the Galois group GalO of O gives rise eventually to a continuous action of GalO on the category of sets and a representable functor from the category whose objects are associative unital commutative k-algebras and whose morphisms are k-algebra morphisms to the category whose objects are groups and whose morphisms are group morphisms.

This representable functor is known as a group scheme and it is in this context that the author formulates and solves the descent problem using Galois cohomology. As the author shows, group schemes allow one to understand the action of a Galois group on a group, and this allows the definition of cohomology sets of the (pro finite) Galois group GalO. This depends on finding groups on which GalO acts by group automorphism, and this can be accomplished by considering O-points of group-valued functors. Such a strategy allows the definition of the nth Galois cohomology set and the author shows how to obtain the Galois cohomology of GalO to the Galois cohomology of its finite Galois sub-extensions.

Towards the goal of formulating a general Galois descent problem, it is advantageous to define an equivalence relation on the category of sets for every field extension K of k. This equivalence relation identifies elements that are in the same G(K)-orbit, where G is the group-valued functor acting on the functor F from field extensions to sets. The natural question here is whether two elements that are equivalent in O are also equivalent in k. An answer to this question involves the notion of a twisted element of F, which is an element of F that is equivalent to a fixed element over O. An element a' is defined to be a twisted K-form of a if a is equivalent to a' over O.

For the element a, the author then defines a collection of equivalence classes [a']. This collection, denoted Fa, formulates the Galois descent problem in terms of twisted forms, namely that of showing that Fa is in fact equal to {[a]}. The author then goes on to describe Fa in terms of the Galois cohomology of a group scheme associated to a. The Galois descent condition comes down to showing that every element of F(O) on which GalO acts trivially comes from an element of the value of F on K. Examples of Galois descent for vector spaces and central simple k-algebras are given.
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$J$-Holomorphic Curves and Quantum Cohomology (University Lecture Series)
$J$-Holomorphic Curves and Quantum Cohomology (University Lecture Series)
by Dusa McDuff
Edition: Paperback
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3.0 out of 5 stars Needs to be supplemented by considerable outside reading., February 7, 2014
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As the authors of this book explain, J-holomorphic curves are a generalization of holomorphic curves the latter of which solve the Cauchy-Riemann equations. The Cauchy-Riemann equations are replaced by an expression involving the differentials of a map of a Riemann surface into a closed symplectic manifold M and what is called an `almost complex structure' J on M, which has the property that J^2 = -1. If J is chosen to be `compatible' with the symplectic structure w on M, then this allows the use of hermitian geometry and then one can show that the area of a J-holomorphic curve is a symplectic invariant for M. Such a strategy for finding an invariant of a symplectic manifold follows upon that of using ordinary holomorphic curves to study symplectic topology in four dimensions. If one can find a single holomorphic curve with the right local properties, then the manifold in which it is embedded can in a sense be determined by the holomorphic curve.

But the applications of J-holomorphic curves to symplectic geometry is much more involved than some of these relatively elementary constructions, but in order to appreciate these applications readers of this book will have to pay close attention to the details in these constructions. The authors do not always give the necessary insight to understand them, and so some outside reading will be required in order to gain this insight. For example, it is advantageous when reading this book to stand back from the formalism from time to time and think clearly about what kind of geometric consequences come from some of stated conditions. One example is to think of the closure condition on a symplectic form as representing the fact that the symplectic area of a surface with boundary does not change as the surface moves, as long as the boundary is held fixed. Another example is to view a J-holomorphic curve as giving a method by which one can cut a cylinder into 2-dimensional slices of area pi r^2. Still another example would be to view J as essentially being a rotation by a quarter turn, and that an almost complex structure is a collection of such rotations, one for each point, that varies smoothly as a function of the points. And, as contrasted with the case of complex structures, almost complex structures have no local symmetries. Locally though, J-holomorphic curves behave like holomorphic curves, but J is not integrable in general.

Also important for readers is to have an understanding of the images of J-holomorphic maps in the target manifold, and so for this purpose it might be necessary to review the connection between embeddings and compactness. Along these lines, it is helpful to note that the image of a J-holomorphic map is not necessarily an embedding, and in addition J-holomorphic curves are parametrized and are only "approximately holomorphic" in the sense that they are not obtained from zero sets of sections of line bundles. In the book it is shown a J-holomorphic map is simple (not a multiple cover of any other curve) and has at most finitely many self-intersections and critical points.

One of the most important discussions in this book though has to be on the topic of the compactification of the moduli space of J-holomorphic curves and its relation to the interesting phenomenon of "bubbling". It might be a struggle for the reader to visualize what is going on with bubbling, since examples in the book are lacking. But since the energy of a non-constant J-holomorphic curve cannot be arbitrarily small, bubbling can only occur near finitely many points, and the "energy density" is concentrated at isolated points. Readers can find other examples of bubbling in the mathematical literature, such as in the Yamabe equation and the Yang-Mills equation, which are elliptic equations that have nonlinear terms that do not satisfy the Sobelev inequalities. If readers are willing to consult outside sources, they will find that many of the examples of bubbling take place in the context of maps between 1-dimensional and n-dimensional complex projective space. These examples are fairly clear if readers are familiar with the Fubini-Study metric on n-dimensional complex projective space. The most important thing to learn from these examples is that there can be several different bubbles for a curve, depending on the scaling and the parametrization, but the genus of the curve is to be unchanged in the limiting process. The preservation of the genus is the origin of the interest in the study of `stable maps'.

The compactification of the moduli space makes use of what are called `cusp curves' in the book, which are essentially unions of J-holomorphic spheres, the latter of which can be parametrized by a smooth non-constant J-holomorphic map from one-dimensional complex projective space into the symplectic manifold of interest. Cusp curves can represent a homology class A, and along with an `evaluation map', are used to construct "strata" which are essentially images of evaluation maps on space spaces of simple cusp curves. If W is the moduli space of evaluation maps, its closure will be described by a stratified space. An evaluation map is used to obtain a `pseudo-cycle' for a generic almost complex structure. The image of this map can be compactified by adding pieces of codimension greater than or equal to 2, and it carries a `fundamental class' that is independent of J. The strategy of the proof of compactification is to choose a regular path of almost complex structures, to arrive at a cobordism between the endpoints of these almost complex structures. The proof of compactification and its use of evaluation maps motivated the construction of the famous Gromov-Witten invariants for symplectic manifolds, which are defined as the number of isolated curves which intersect specified homology cycles in the symplectic manifold. The authors show that there are two different special of looking at these invariants, one where the curve can intersect cycles anywhere, and one where the intersection points are fixed.

Of course the most important part of the book is the discussion on quantum cohomology, and readers with a background in quantum physics/quantum field theory will no doubt immediately raise the question as to why the adjective "quantum" is used to describe this cohomology theory. In the opinion of the reviewer, the closest justification is in the context of the Floer theory wherein two curves are said to intersect if there is a J-holomorphic curve connecting them, thus making the intersection "uncertain" in some sense. Or, one could view the intersection from the standpoint of how the ordinary cup products in cohomology are "deformed" by "quantum" corrections. These corrections however are merely the result of taking the tensor product of the ordinary cohomology groups with a coefficient ring, the latter of which in the book is taken to be the collection of Laurent polynomials in variables of a chosen degree. There is nothing really "quantum" about this.

The best way to view quantum cohomology however is to forget about any "quantum" interpretation and view it as a method of doing intersection theory, as of course it was designed to do. If one consults the research literature not referenced in this book, one will find that the variable q in the coefficient ring raised to a power d has a natural interpretation in terms of complex projective spaces, wherein one is interested in the intersection of hyperplanes. A projective hyperplane can be represented by a generator "p" in the second (ordinary) cohomology group of CP(n), whereas the intersection of two hyperplanes can be represented by squaring p, which is an element of the fourth (ordinary) cohomology group. If one continues to do this, namely if one takes the intersection of n generic hyperplanes, then this can be represented by p^n, which is an element of the 2n (ordinary) cohomology group of CP(n). If another intersection is attempted, then the empty set will result, and so one could view the ordinary cohomology of CP(n) as represented by Q[p]/p^(n+1). The "quantum" cup product will come into play when taking ordinary cup products of the generators p raised to some powers. Defining p^(n+1) = q, if one cups p^k with p^l and takes the quantum cup product with p^m, then one will obtain various powers of q depending on how k, l, m are related to n. For example, if k + l + m = n, then one will obtain q^0, which is viewed as degree 0 holomorphic spheres passing through the cycles p^k, p^l, and p^m. If k + l + m = 2n + 1, then the quantum cup product gives q^1, which because of the exponent being equal to 1 is viewed as all lines connecting p^k and p^m. These lines form a projective subspace of dimension l which meets p^l in one point. In general then the element q^d represents the contributions of the holomorphic spheres of degree d.

All of these considerations about the element q^d are formalized when the the authors show how to prove associativity of the quantum cup product and the connection to the famous Novikov ring, which is taken to be the coefficient ring for the case of a closed symplectic manifold. The Novikov ring is the completion of the group ring of the second homology group of the symplectic manifold, and as such it allows the interpretation of quantum cohomology as being the encoding of information about the second homology of the symplectic manifold. This reflects the strategy of counting J-holomorphic curves in a given homology class. When the group ring has dimension 1 and the symplectic manifold is monotone, the Novikov ring will be the Laurent power series ring in q and q^(-1).

Lectures on the Theory of Pure Motives (University Lecture Series)
Lectures on the Theory of Pure Motives (University Lecture Series)
by Jacob P. Murre
Edition: Paperback
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4.0 out of 5 stars Good introduction to the subject, December 29, 2013
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The theory of motives originated with the mathematician Alexander Grothendieck and can loosely be described as an attempt to find a 'universal cohomology theory' for algebraic varieties. In this book, attention is focused for the most part on the theory of pure motives, which are those motives that are related to smooth projective varieties. Every such variety X is viewed as a motive, which is viewed as a "piece" of X that accounts for the geometric and arithmetic properties of X. This view is inspired from algebraic geometry, wherein for an algebraic curve X, the "essential" part of X is determined by the Jacobian variety of X, and every abelian variety is an abelian subvariety of a Jacobian. This fact motivates the search for finding a Jacobian variety for any arbitrary variety. Loosely speaking, a motive could then be viewed as an analog in higher dimensions of the Jacobian of a curve.

Crucial to the understanding of pure motives as outlined in this book is the concept of a correspondence. Given smooth projective varieties X and Y, correspondences are special types of maps between X and Y which form an abelian group. Compositions of correspondences are defined using the fiber product and involve a complex set of operations using projections on factors. Algebraic cycles are then taken to be formal linear combinations of correspondences. Correspondences reflect the idea that in general, there do not exist regular maps from one algebraic variety to a second one, and hence correspondences are the "many-valued maps" that reflect the absence of regularity.

More specifically, a correspondence from one algebraic variety X to another algebraic variety Y is a cycle in the product X x Y. If Z is such a correspondence and if T is a cycle in X of codimension i with d = dim(X), then Z will "push forward" T to a cycle Y of codimension i + t - d where t = dim(X x Y). If it happens that t = d, then Z is said to preserve the codimension of the cycle T. The fact that Z can raise the dimension in this way is a sign that correspondences are not so simple as one might imagine at first glance. This complexity is responsible for some of the nagging issues that must be settled in order to get a reasonable notion of intersection of cycles and definitions of push forward and pullback maps. As in most areas of mathematics, the strategy for dealing with such complexity is define an equivalence relation on cycles, in order that the operations of intersection, pullback, etc can be defined.

The goal therefore is to find an "adequate" equivalence relation, and a few proposals have been made, going by the names of rational, algebraic, numerical, and homological equivalence. iRational equivalence is the easiest to understand, being that it is a generalization of the classical notion of linear equivalence of divisors. The original definition of rational equivalence involved the notion of 'specialization', which in turn relied on the notion of an 'associated form.' A more specialized notion of rational equivalence is that of 'smash-nilpotent' equivalence, which means that some integer power of the cycle is rationally equivalent to the zero cycle. Algebraic equivalence is somewhat similar to rational equivalence, with the difference being that one can find a smooth irreducible curve that in a sense that can be rigorously defined serves as an interpolation between the cycle and the zero cycle. Readers will find a fairly detailed discussion of the conjectured equivalences between some of these notions of equivalence, such as that of homological and numerical equivalence etc.

If readers look into the history of the notion of a correspondence, they will find that correspondences were widely used in "classical" algebraic geometry. This is readily apparent when considering them as examples of a graph of a morphism or the closure of such a graph. Correspondences have a product operation and there exists homomorphisms that generalize the notion of composition, push-forward, and pull-back for morphisms. In addition, one can view obtain a motivating example of the intersection theory of correspondences by remembering that the Lefschetz fixed point formula allows one to study intersections of an object with the "diagonal". If X and Y are both n-dimensional and T is an n-dimensional correspondence then the degree of the intersection of T with the diagonal will give the number of virtual fixed points of T. The classical theory of correspondences is very rich and touches on many "modern" topics such as enumerative geometry. Correspondences have been shown to have a connection with the theory of Hecke operators, but this connection is not discussed in this book.

An immediate question concerns the issue of whether a motive is finite-dimensional in some sense. This issue is discussed in this book using the theory of group representations. Finite-dimensionality of motives is defined in terms of what happens to a product motive under the action of the symmetric group. One says a motive is 'evenly finite dimensional' if there exists a positive integer n such that the nth exterior product is zero. A motive is 'oddly finite dimensional' if there exists a positive integer n such that the nth symmetric product of M is zero. A motive is then said to have finite dimension if it can be written as the direct sum of evenly finite dimensional and oddly finite dimensional motives. The dimension of the motive is then the sum of the dimensions of the summands. If a motive is both evenly and oddly finitely dimensional then the motive is identically zero. Dimension is preserved under surjective morphisms of motives, and so it is important to pinpoint the cases where a morphism between motives is surjective. It is an open question as to whether every Chow motive is finite dimensional. One can show that the dimension of objects of a full tensor pseudo abelian subcategory in the category of Chow motives generated by motives of smooth projective curves is finite.

Also of importance in this discussion of finite-dimensionality of motives is the notion of a 'phantom motive' which becomes zero after passing to numerical equivalence. Phantom motives arise from the fact that the forget functor from motives under rational equivalence to motives under homological equivalence is not faithful. A Chow motive is a phantom motive if it is not zero but equal to zero under homological equivalence. Phantom motives do not arise for motives of finite dimension.

The underlying need for discussing the finite-dimensionality of motives is that there exists a big difference between the theory of divisors and the theory of algebraic cycles of codimension greater than one. For example, for divisors, the Chow group is nothing other than the Picard group which is finitely-generated, as is the Neron-Severi group, which is the Chow group modulo algebraic equivalence.

As the authors show, one can take the direct sum of pure motives, along with their tensor product, and one has a notion of 'unit' motive, which is the identity for the tensor product. There is also a notion of a 'Lefschetz motive', which should be viewed as one of the "elementary" motives, in the sense that any motive can be expressed as a direct factor of a power of the Lefschetz motive. Interestingly, this result alleviates the need to deal with motives based on varieties with components of different dimensions. The dimensionality of direct sums of motives is straightforward to define, whereas for tensor products it is somewhat more involved. It is an open question as to whether every Chow motive is finite-dimensional.

Motives (Proceedings of Symposia in Pure Mathematics) (Pt. 2)
Motives (Proceedings of Symposia in Pure Mathematics) (Pt. 2)
by Uwe Jannsen
Edition: Paperback
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5.0 out of 5 stars Good articles on the theory of pure and mixed motives, November 28, 2013
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Review of the article 'A Summary of Mixed Hodge Theory' by J.H.M. Steenbrink:

The article on mixed Hodge structures could be approached historically, by remembering that the origin of the concept of a Hodge structure arose in complex analysis, where it was found that the real components of a holomorphic function satisfy the Cauchy-Riemann equations, and this loosely speaking allowed one to study the properties of such a function by breaking it up into its holomorphic and anti-holomorphic parts. Such a viewpoint carries over to complex differential forms, as long as one is dealing with a compact 'Kahler' manifold, which is a complex manifold with a very special metric that is the "best" analog of the Euclidean metric in real manifolds. Complex differential forms can be written in local holomorphic coordinates as linear combinations of p holomorphic differentials and q anti-holomorphic differentials and are then said to have 'type (p, q)'. A very fundamental result in later later generalizations to the theory of mixed Hodge structures is that of 'Hodge's theorem' which states that the complex de Rham cohomology of a compact Kahler manifold X can be written as the direct sum of cohomology classes Hp,q(X) representable by closed differential forms of type (p, q). For a general complex manifold, the differential forms decompose into types but the complex cohomology does not decompose into a direct sum of subspaces Hp,q(X), which is another way of saying that in the general context differentials are not compatible with types.

The general theory of Hodge structures is an abstraction of this situation on compact Kahler manifolds. In fact, Hodge structures form a category, and in this category one can define a Hodge decomposition as a formal structure in linear algebra, namely a Hodge structure of weight n is a finitely generated abelian group HZ, where Z is the integers. Taking the tensor product of HZ with the complex numbers gives an object HC that can be decomposed into complex subspaces Hp,q, where p and q are integers. This implies there are linear operations on Hodge structures and using them one can define vector spaces over Q called 'rational Hodge structures' of integer weight m.

Pursuing this line of reasoning, mathematicians interested in the theory of motives, such as Peter Deligne, defined the notion of a 'mixed Hodge structure', which can be viewed as representing the "analytic side" of the theory of motives. In this viewpoint, the nth cohomology group of a complex algebraic variety with complex coefficients carries two finite filtrations by complex sub vector spaces, one is called a 'weight filtration' W and the other is called a 'Hodge filtration' F. The weight filtration is trivial for a smooth compact variety and the subspaces are rationally defined. These subspaces are generated by rational elements in Hn(X, Q) and taking the homogeneous part results in a graded vector space associated to W. For the Hodge filtration F, F and its conjugate give the Hodge decomposition with respect to rational cohomology. For the case of linear algebra defined on a complex vector space this gives the ordinary Hodge structure. F and its conjugate also induce a Hodge decomposition of the W-induced grading. This gives a mixed Hodge structure on the cohomology. Interestingly, it turns out that a geometric interpretation can be given to F and W, namely that they reflect properties of the normal crossing divisor at infinity of a completion of a smooth variety. Hodge filtrations can be used to study the variations of Hodge structures, and they vary holomorphically with respect to certain parameters. A Hodge decomposition can be recovered from the Hodge filtration.

Examples of Hodge structures include the Tate Hodge structure Z(1) which has weight -2, m-twists of Hodge structures of weight n, which have weight n - 2m, and the pth exterior power of a Hodge structure, which has weight pn. Additional examples include the Hodge structure on the homomorphisms of Hodge structures H and H', which have weight n' - n, and the Hodge structure of a dual of a Hodge structure of weight n, which has weight -n. Using the trace map on 2n-dimensional complex cohomology and Poincare duality, one can also induce a Hodge structure on homology. Mixed Hodge structures can also be constructed with the observation that if the integers Z or the rational numbers Q are tensored with Q, the result is Q; whereas if the real numbers R are tensored with Q, the result is R. If B stands for Q, R, or Z then one can construct mixed Hodge structures with respect to B. This B-mixed Hodge structure has both a weight filtration W and a Hodge filtration F associated with it, and a grading consisting of a Hodge structure of weight n with respect to the tensor product of B with Q. It can be shown that this grading has a Hodge decomposition.

Hodge structures form an abelian category whose objects are Hodge structures and whose morphisms are morphisms of Hodge structures. The kernels and cokernels of a morphism f between two Hodge structures H and H' are induced by the Hodge decomposition of H. More generally, if B is an object of an abelian category, then there are increasing and decreasing filtrations F of B, and one can view the pair (B, F) as a filtered object. Filtered objects form an additive category, but since the image and co-image are not necessarily filtered-isomorphic, these objects do not form an abelian category. There are graded objects associated to (B, F), and two types of filtrations on sub objects of B, namely the induced filtration and the quotient filtration. These can be used to define a cohomology on the filtered objects, and using a notion of strictness between morphisms of filtered objects one can make the category of filtered objects abelian. One can show that the condition for being strict is equivalent to an exact sequence involving the gradings of the kernels and cokernels, and the resulting cohomology respects the grading. If there are two filtrations on an object of an abelian category, then they induce a bigraded object. In addition, there is a notion of opposite filtrations, that is motivated by the consideration of Hodge filtrations on Hodge structures, namely that a Hodge structure F of weight n is n-opposite to conjugate(F).

The Making of the Atomic Bomb: 25th Anniversary Edition
The Making of the Atomic Bomb: 25th Anniversary Edition
by Richard Rhodes
Edition: Paperback
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1 of 1 people found the following review helpful
5.0 out of 5 stars One of the best histories ever written., October 14, 2013
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After reading this book one carries away a deep appreciation of what it takes to produce works of such high quality. The writing style and historical overview are superb, and even though the topic of the book will instigate feelings of sadness because of the dangerous forces that a large group of individuals released upon innocent people, it also is an example of how a government/military program can succeed if its participants remain focused and respect scientific protocol.

This book not only gives insight into the physics and technology behind nuclear weapons, it also assists in the understanding of the personalities idolized (and ridiculed) by popular culture and Hollywood that were involved in the Manhattan project. One should always be cautious in imputing too much to the individuals that found their way to popularity, either through their own intent or via the eagerness of the press to inflate a story. One must also not forget the contributions of those individuals who worked in the trenches to make the Manhattan project successful. Their contributions have not been popularized, but they are real and the project would not have been able to be completed without them.

Thankfully the author was not content to just give an historical account of the actual beginning of the Manhattan project, but also the decades that preceded it with emphasis on the physics of radioactivity and the early speculations on atomic fission that will be familiar to those readers, such as the reviewer, who have a strong physics background. Such readers can't help be feel excited when reading of the methodologies and patterns of thought that went into developing the first particle accelerator and fission pile. Monte Carlo simulations, now done on a massive scale worldwide in financial engineering, radiology, and many other areas, have their origin in the Manhattan project with the goal of understanding neutron diffusion.

At the same time, one feels a moral ambivalence about these developments and the eventual Trinity event, due to the frightening consequences that resulted in the years after it. And it is apparent that many of the participants of the Manhattan project had themselves these feelings of moral ambivalence and doubt, even during the early stages of the project. It is interesting to read for example that Edward Teller, who Hollywood has ridiculed beyond measurable levels, expressed reservations about working on weapon's research. One can respect his moral ambiguity since his life was disrupted twice by European conflicts, and this no doubt played a role in his decision to continue with the project, and with the eventual development of thermonuclear weapons.

Readers can also walk away with an appreciation of the fact the physicists are fundamentally just a human as everyone else, and have interests beyond just science. It is surprising to learn for example of the soccer playing interests of Niels Bohr, and the everyday amusements of the scientists as they tried to lower their stress levels in Los Alamos. Perhaps one can conclude from these anecdotes that one should not describe people as scientists except from a purely statistical perspective. Sometimes they engage in the rational behavior that scientific research requires, and sometimes they do not. There is ample evidence for such a conclusion in this book.

It is amazing that the individuals who participated in the Manhattan project were able to complete this mission in so short a time. That brings out what in the reviewers opinion is the most important conclusion to be drawn from the study of this book, namely that if a group of people remain focused on the science and the problem solving that it requires, they will achieve success, in spite of the inevitable clashing of personalities and conflicts of personal interest.

New and Collected Poems: 1931-2001
New and Collected Poems: 1931-2001
by Czeslaw Milosz
Edition: Paperback
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2 of 2 people found the following review helpful
5.0 out of 5 stars Incites a neuronal riot, October 5, 2013
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Poetry is inspired by and appreciated by every human emotion, even negative ones like anger and resentment, but whatever the causes they are not always manifestly apparent in the spontaneity of words that this book abundantly exemplifies. Whether Czeslaw Milosz was inspired by suffering or by the celebration of its alleviation, or a superposition of both, this collection represents his best, and some of the best of poems now in print. And by the best in poetry one means those poems that provoke the strongest and greatest variety of emotions. Laurel leaves therefore go to those poets whose poems allow a hyper-proliferation of interpretations; whose poems strongly perturb readers out of their conceptual and emotional equilibrium.

A reader could perhaps extract a slice of twentieth-century history form the poems of Milosz, but he cautions poets to not yield to the temptation to become reports. Milosz wants poets to "contemplate the things of the world as they are without illusion', but whether he did so in his poetry is to a large degree irrelevant. All that matters is the beauty of his poetry and its propensity to cause a neuronal riot in the minds of its readers.

And indeed, this collection of poems just precisely that, for Milosz speaks of the earth's happiness as being terrible. He speaks for the need of living more than one life in order to decipher its sufferings and probe its laws. He scolds the mob for having laid ashes to Giordano Bruno, and those who give up hope. He exalts men who are small but who produce great works.

And Milosz is not hesitant to receive messages from "a world that is bright, beautiful, warm and free", and to speak of the artistry of the sun and a poem called the Earth. He cautions against becoming a "ritual mourner", of "poetry which does not save", and in this regard promises "no wizardry of words." Milosz gives a characterization of European society and culture that is characteristically Stendahlian, and he shows no mercy to its "malignant wisdom". He delightfully pokes fun at the Hegelian justification of power structures and scientific fakery. Stendahl's glorification of obsequious flatteries and praising of power supported by ambiguity of communication is admonished with subtle literary skill.

Milosz wants to perform combat with prose. There is ample proof of his battle campaigns between the covers of this book.

Multivariate Survival Analysis and Competing Risks (Chapman & Hall/CRC Texts in Statistical Science)
Multivariate Survival Analysis and Competing Risks (Chapman & Hall/CRC Texts in Statistical Science)
by M. J. Crowder
Edition: Hardcover
Price: $88.46
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1 of 2 people found the following review helpful
4.0 out of 5 stars Very helpful, September 10, 2013
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Survival theory and the theory of competing risks have been around for quite some time now, due to intense interest from the insurance industry and the medical field. The reviewer has found application of the theory of competing risks to financial modeling in the form of mortgage analytics, and in the modeling of information networks with particular emphasis on calculating the capacity of wireless networks. The part of the book on competing risks, which was the only one studied by the reviewer, should be read in conjunction with some of the vast references that it quotes, is a good overview of the theory, and contains some real world examples that illustrate some of the main points. The author has chosen the `R" statistics package to illustrate these examples, so readers such as the reviewer who are not acquainted with `R" will probably not appreciate these examples as much as readers who do.

The theory of competing risks will be helpful to those analysts who need a notion of causality built in to their models that is not available for example from Monte Carlo studies or other frameworks where randomization is built-in and only correlations, and not causation, can be discussed meaningfully. It is also very useful in that it takes into account data that will inevitably be missing in a real-world study, due to the constraints on time during which the study is being conducted. This omission of data is called "censoring", and is the rule rather than the exception in conducting studies that begin at a certain time and end after a chosen time interval. As examples, one can quote studies where patients suffering from a particular disease are given treatment and monitored in a clinic for a given length of time. Patients sometimes leave the study, either because of relocation or some other reasons. Also, data on patients who survive over the duration of the study will be "right-censored" and the estimates on the efficacy of the treatment will have to deal with the right-censored data. Another example of right-censoring occurs in the context of using competing risk analysis for the study of information networks, where typically some sort of packet capture device is placed on a network for a certain duration in order to count the number of packets that are dropped because of adverse conditions on the network. Packets that have not been delivered to their destination before the time at which the capture device is removed will thus be right-censored, and once again the estimates and predictive analytics of packet drop will be affected by the data that is right-censored.

The author gives many other examples that will be of interest to the reader working in a wide range of fields, and also the theory behind the analysis of these examples. A portion of this theory, especially the kind that deals with more advanced mathematical topics such as the theory of stochastic processes, is delegated to the references, so readers who need this level of discussion will have to study these references in conjunction with the book. There are some very important discussions of properties in the theory of competing risks that at first seem counterintuitive, such as the "Tsiatis non-identifiability theorem", which may disappoint those analysts who are approaching the theory of competing risks with the expectation that correlations among risk factors can be determined unambiguously from survival data. The Tsiatis result indicates that they cannot, and along these lines the author discusses some "independent-risk" parametric models that illustrates the non-identifiability behavior. Even more interesting is the theorem (proof left to the references) that illustrates that even dependent-risks models will suffer from the same problem of non-identifiability. The issue of non-identifiability problem and ensuing issues of unaccounted for risk correlations has lead to something called "false protectivity", which has recently even made national headlines in claim that cigarette smoking can thwart the onset of Alzheimer's disease. The reviewer did not find an explicit discussion of false protectivity in the chapters studied, but this seeming omission does not detract from the quality of the material that is presented in this very helpful book.

Note: This review is based solely on a reading of Chapters 12-17 of the book.

Trust: A New Beginning (Volume 1)
Trust: A New Beginning (Volume 1)
by Cristiane Serruya
Edition: Paperback
Price: $17.81
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2 of 3 people found the following review helpful
4.0 out of 5 stars Beautiful people in beautiful clothes, and hot steamy sex., August 4, 2013
The challenge of writing erotica is to offer a good story that is not merely a "filler" between the sex scenes. And the sex, hot and steamy as it needs to be, should be intertwined with the storyline, and not serve as a distraction for the reader who is bored with the characters or is merely seeking titillation. In addition, there is some value in erotic literature for making it gender-specific, in that it might be worthwhile for the authors of such literature to target the female or male readership exclusively. One can easily find many examples of this strategy in the voluminous erotic literature in this venue and others, with only a small sample that has as its goal the reading pleasure of both men and women.

With an effective and plausible plot, this book is an example of the latter, in that both male and female can find commonality in outlooks between its covers, and easily imagine themselves between their own bed covers, finding robust approximations to the physical entanglements of the characters, who in this book know how to make love as much as they know how to ****, and they know there is a difference. The clothes the author decorates them in are superb, and for that rare male reader who loves seeing women in beautiful clothes, a strong identification will take place with the character of Ethan. What a lot of women don't really know is that there are men in the world who like Ethan absolutely adore women who are dressed in nice clothes, and further, they have strong knowledge of how to dress women in these clothes, and frequently want to do so at the early stages in their relationships with them. The author definitely has an appreciation for such men, as clearly brought out in the Ethan's character, and in the tasteful clothes that she overlays on the character Sophia, who is the prime mover of the book, and a woman who most men could easily find themselves obsessed with.

The author also deploys some literary devices, such as the use of italics, that are effective in revealing the character's introspection. The latter seems to decrease in frequency as the characters indulge themselves and each other physically. Whether this was the intent of the author or not, it is noticeable, and serves as an example of the diminution of self-awareness as bodies move closer to orgasm. All of the character's encounters are a high-temperature illustration of the geometry of sex, namely that whenever a cylinder encounters a triangle, the results should always be totally explosive and totally uninhibited, regardless of how the former fits into the latter, along with being coupled with the intense anticipation of splashing themselves with the finality of passion.

The Conceptual Framework of Quantum Field Theory
The Conceptual Framework of Quantum Field Theory
by (Professor of physics) Anthony Duncan
Edition: Hardcover
Price: $101.61
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15 of 16 people found the following review helpful
5.0 out of 5 stars Excellent, June 29, 2013
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When one engages in the learning of quantum field theory it is easy to be amazed by what typically can be a collection of mathematical tricks can result in such unprecedented agreement with experiments. At the same time, there are many aspects of the subject that can be troubling to both the physicist and the mathematician alike. Physicists can be troubled by the typically abstract nature of quantum field theory while mathematicians can be annoyed by the constructions that are not defined rigorously. These concerns have instigated an enormous amount of research and many questions still remain to be answered in quantum field theory. It is a very active field of research and no doubt this will be the case for years to come.

Some of the questions that are frequently asked by those first learning quantum field theory and even experienced researchers include:

1. What is really the meaning of Haag's theorem in terms of the interaction picture in quantum field theory?

2. Why is cluster decomposition important and should it be considered on the same level as for example the requirement of Lorentz invariance? Can cluster decomposition be established without using the creation-annihilation operator formalism (so-called "second quantization")? The author views cluster decomposition as being responsible for most of the interesting phenomena in quantum field theory, and Lorentz invariance as being a kind of incidental side constraint. This is an interesting comment and many who have learned quantum field theory (such as the reviewer) have not found this emphasis on cluster decomposition in other texts and monographs on quantum field theory. Along these same lines the author views cluster decomposition as a kind of search engine for finding sensible quantum field theories, in that those theories that incorporate it will have long-distance behavior that is more in line with physical reality.

3. Why is it so difficult to establish the existence of a bound state in quantum field theory? Should quantum field theory be viewed as a formalism to be used strictly for scattering phenomena with bound state calculations done using an extension of quantum field theory? Will insisting on the formation of bound states in quantum field theory entail a radical revision of the formalism? It would seem at first glance that the 'clustering principle' that the author views as being absolutely fundamental is really only of relevance for scattering phenomena. What is its applicability for lets say a proton and an electron who come close enough together but do not scatter, but instead form a bound state (a hydrogen atom)? The author gives a fine discussion of various Hamiltonians that lead to clustering theories. Can the same be done for Hamiltonians that lead to "bound-state theories" or is this not a meaningful question to ask within the scope of quantum field theory? Are quantum field theories that allow bound states 'ultra local' in the sense that the author uses this term in the book, or will they violate ultra locality? Do the 'particle-field duality axioms', which are deemed to be essential to the development of a "comprehensible" theory of particle scattering in quantum field theory have analogs in showing how particles (fields) come together to form bound states?

4. Do ideas from the renormalization group only make sense for quantum field theories that are renormalizable? Along these same lines, can a non-renormalizable quantum field theory have any kind of "predictive power"?

5. What is the nature of number-phase complimentarity for fields?

6. Did the founders of quantum field theory find it difficult intuitively to accept the concepts behind quantum field theory? One might ask this question in modern terms as wondering whether there was any kind of cognitive dissonance among the early practitioners of quantum field theory?

7. What is the micro causality principle in quantum field theory and why is it important? Can it be dispensed with without affecting the successful predictions of quantum field theory?

8. The "action-at-a-distance" effects that the author speaks of in the book are to be distinguished he says from the "entangled" states of EPR fame. But do these effects really predict phenomena that are of interest observationally or are they too small to be relevant from the standpoint of what is measurable from a technological perspective? Should this issue be viewed then in the context of "imperfect resolution" of measuring devices? If so, this might allow a resolution of the 'infrared problem' by approximating massless photons by massive photons and concentrating only on 'inclusive cross sections'.

9. What is really the mechanism behind the appearance of inequivalent representations of the canonical commutation relations in quantum field theory?

10. Can quantum fields "restore" broken symmetries at the classical level? For example, are there quantum field theories that when viewed from the standpoint of perturbation theory will recover momentum conservation when the classical counterpart manifestly breaks symmetry translation? This would imply that momentum convservation, although broken classically, could be recovered in quantum field theory (at least perturbatively).

11. What conditions guarantee that the vacuum state is cyclic for products of quantum field operators localized in a bounded Euclidean spacetime region at positive time?

12. Can inhomogeneous local gauge transformations be used to generate massive gauge bosons, and consequently avoid the need for the Higgs mechanism?

Not all of these questions are answered in this book, but it has enough discussion of elementary principles and concepts to allow the reader who is genuinely interested in understanding quantum field theory ample food for thought. In this regard, some of the highlights of the book include:

1. The historical introduction and the insight it offers on how quantum field theory arose as a methodology: (a) The role played by Pascual Jordan on showing that the view of Einstein who held that energy fluctuations of electromagnetic radiation has two "structurally independent" causes is not necessary if one uses an electromagnetic field description.The work of Jordan has recently been formalized in the framework of algebraic quantum field theory using a concept called 'modular localization' and is a highly interesting development in the foundations of quantum field theory. (b) The contrast between today's emphasis on the use of functional methods in quantum field theory versus what was the preferred method in the early years of quantum field theory, namely the interaction-picture methods of Julian Schwinger. It was the contributions of Freeman Dyson that apparently convinced many to switch over to the Feynman paradigm of path integrals, with a few physicists however hanging on to the Schwinger methods well into the 1970's.

2. The many discussions throughout the book on how to implement classical symmetries in quantum field theory and problems faced when attempting to do this (the famous anomalies). The author gives an interesting example in quantum mechanics involving (canonical) point transformations to motivate the occurrence of anomalies.

3. That the author believes that quantum field theory is manifested most powerfully as a scattering theory is readily apparent throughout the book. His discussion of the need for requiring more than just Lorentz invariance in building a scattering theory in relativistic quantum mechanics is very interesting and gives good motivation for the consequent discussion of cluster decomposition.

4. The treatment of the Majorana field, which lately has become very important from an experimental point of view in condensed matter physics.

5. The chapter on the classical limit of quantum fields. Those who are interested in whether or not superconductivity for example is a "macroscopic manifestation of quantum phenomena" as is claimed many times should find this chapter of great interest. In addition, the discussion of the anti cyclic permutation operator as being the quantum analog of the complex classical phase is very helpful from the standpoint of the number-phase uncertainty principle.

6. The author is not shy about discussing the bound state problem, and this is refreshing considering the extremely difficult nature of this problem. Even outside of the context of his discussion on bound states, the author gives information that indicates that many of the results in quantum field theory cannot be derived if bound states are present. One example of this is the discussion on the principle of asymptotic completeness. Another example is the discussion of the Heisenberg field, where certain 'auxiliary' fields are introduced to characterize the kinematical structure of asymptotic states. What is not so clear is whether or not these asymptotic states can indeed be bound states. i.e. can the "out-states" be bound states, and will the Heisenberg field be of assistance in constructing these states, conforming to the claim that "knowledge of the Heisenberg field" will always allow a construction of the out-states. The author is well-aware of the fact that the S-matrix philosophy concentrates on what happens in the distant past and future, and ignores what happens "in-between" to use his terminology.

The only major omission that the reviewer is aware of is that the author does not give an in-depth discussion of the Reeh-Schlieder theorem, but merely refers to this interesting result only in a footnote.

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