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An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves (Progress in Mathematics)
An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves (Progress in Mathematics)
by Joachim Kock
Edition: Hardcover
Price: $44.67
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5.0 out of 5 stars A fine overview with helpful, pictorial examples, September 20, 2014
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The Kontsevich combinatorial formula of stable algebraic curves can be loosely described as being a generalization of what is done for Grassmann varieties in the context of vector bundles. A Grassmann variety Gr(k, n) is a collection L of k-dimensional linear subspaces of a complex n-dimensional vector space. The geometry of Gr(k, n) can be viewed as a kind of measure of how complicated things can get if L is permitted to vary in families. A family can be viewed as a collection of linear spaces parametrized by points of a base space B, and this leads naturally to the concept of a locally trivial vector bundle over B. One can then obtain a ‘tautological’ vector bundle Ltaut over Gr(k, n) consisting merely of pairs (L, v) where v is an element of L. Forgetting v gives a map from Ltaut to Gk(r, n) with fiber L. Given a map from B into Gr(k, n) there is a ‘pull-back’ of Ltaut which happens to be a vector bundle over B of rank k. It turns out that this procedure for n arbitrarily large and for B compact is gives a lot of information and is “universal” in the sense that there is a bijection between homotopy classes of maps from B to Gr(k, n) and the set of isomorphism classes of rank n vector bundles on B.

In the context of algebraic geometry a natural question to ask is whether this “universality” can be repeated when the families of linear spaces are replaced by families of curves of genus g. In other words, given a family F of smooth algebraic curves of genus g parametrized by some base space B, does there exist a natural map from B to a “moduli space” of curves that gives the essential information about F?

As is known, and as brought out in this book the answer to this question is in general no. If Mg is defined to be the moduli space of smooth curves of genus g then a family of curves with base B is a morphism from F to B of algebraic varieties whose fibers are smooth complete curves of genus g. Any map phi from B to Mg needs to be algebraic and in general F will not be the pull-back of any universal family over Mg. For g = 0, one-dimensional projective space P(1) there exists a trivial map to a point, but there exists complicated families with fibers isomorphic to P(1) because of a “large” automorphism group which can enable the construction of complex objects from simple ones. It is the presence of this automorphism group that makes it difficult to find a universal family of curves over Mg.

However, if the automorphism group is finite, then this can be dealt with by putting “marked” points on the curves. The number of marked points must be greater than or equal to 3 for the case of genus 0 and greater than or equal to 1 for the case of genus 1 curves. There are some straightforward examples of marking in the book, and the authors show just how one needs to change the moduli space Mg to M(g, n), where n is the number of marked points, in order to eventually lead to a theory where one can discuss intersections of curves and a formula for computing the number of points of intersection.

The first issue that must be dealt with is that families of curves over a base space B typically have singular fibers, and these fibers give valuable information about the geometry of the fiber. How are these singular fibers to be dealt with? The answer involves only worrying about the so-called ‘stable’ curves of genus g with n marked points. One thus obtains a ‘compactification’ of M(g ,n) which consists of stable curves, i.e only those curves that are complete and connected, have only nodal singularities, and only finitely many automorphisms. This procedure allows more control over the fibers over B.

Through helpful diagrams the authors show how to deal with the phenomenon where marked points can approach each other. In more advanced treatments of this subject, this procedure is called ‘normalization’ of the curve. In particular when the base B is one-dimensional a family of curves over B is a map from the fibers F to B, where the fibers are curves of the family. Marking n points on the curves gives essentially n sections of the map, i.e. n maps from the base to F. There may be a point in the base where these sections (i.e. the marked points) coincide, and this will result in a curve that is not stable. The ‘normalization’ procedure is to “blow up” this “bad” point on F, giving a new family of curves over B and at the bad point has an additional contribution called the ‘effective divisor’ and the resulting combination will be a stable curve with two marked points which is essentially the “stable” limit of the old curves as points in the base B approach the bad point.

In general then, if a marked point on a curve C approaches another, then C will “bubble” off a P(1) with these two points on it. For a family of curves with a smooth one-dimensional base B that are stable except at a bad point in the base, one can apply a sequence of blow-ups and blow-downs so that a new family is obtained which has stable fibers and where the fiber over the bad point is determined uniquely. This is called ‘stable reduction’.

The real goal behind all this marking and consequent stable reduction is to use the compactified moduli space to do intersection theory and arrive at a general formula for the number of points of intersection. This is done by looking at the line bundle of a (stable) curve C over the n marked points and intersecting the first Chern class of this line bundle. If pi: F -> B is family of stable pointed curves and phi: B -> compactification(M(g, n)) is the induced map then sections of the pull-back phi* of the tangent space at the marked points are vector fields on points s of the fibers of pi that are tangent to the fibers of pi. This procedure gives a section of the normal bundle to the points s in the fiber F, and the degree of this normal bundle is the self-intersection of the points s on F, and is equal to the integral over B of the first Chern class of phi* of the tangent space at the marked points.

The first Chern classes are the “psi’s” that one sees in the vast literature on quantum cohomology and its connection with intersection. The computation of the intersections of the psi’s with compactification(M(g, n)) is the subject of Gromov-Witten theory and the authors show how this is connected with the quantum cohomology and enumerative combinatorics. Readers with a physics background will find that the designation of this cohomology as being “quantum’ is only because of the historical origins of the subject in the area of quantum gravity. One should not in that regard view quantum cohomology as being a “quantization” of some underlying cohomology theory. It should rather be viewed as a deformation of the ordinary cup-product multiplication that is found in discussions on Chern classes of Grassmann varieties in algebraic geometry or in the Chow ring of P(r). The use of “generating functions” is also reminiscent of what is done in quantum field theory and quantum statistical mechanics, but since the resulting “quantum” product, which amazingly produces the right enumerative information, is commutative, the analogy to quantization is rather loose, given that quantization typically results in operations that are non-commutative.


FRYE Women's Lindsay Spur Boot
FRYE Women's Lindsay Spur Boot
Price: $262.77 - $437.50

1.0 out of 5 stars One of the spurs broke after just a few weeks, August 14, 2014
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Whatever the reason for purchasing this pair of boots, whether for yourself or as a gift to someone close to you, you should know that the spurs are NOT real metal and one of them broke in half not too long after being worn.

The spurs are apparently made of a cheap composite and will break easily. If you check online many owners of this boot have complained about the spurs breaking.

Try another style because for the price the spurs should be REAL metal.

Update (Sept 10, 2014): I contacted Frye and they at first agreed to replace the spur free of charge, and even pay for postage to send the boot to them. A few days later they reneged on this promise, and then only agreed to send me replacement spurs (which they did) with instructions to the effect that if I could not find a local cobbler, I could send the boot to them for repair. However I was to pay for the postage (both ways) and the cost of insurance.

The replacement spurs are the same material: a cheap composite that will easily break. One would expect more from Frye, who are supposed to stand for quality.
Comment Comments (2) | Permalink | Most recent comment: Sep 10, 2014 5:28 AM PDT


Our Mathematical Universe: My Quest for the Ultimate Nature of Reality
Our Mathematical Universe: My Quest for the Ultimate Nature of Reality
by Max Tegmark
Edition: Hardcover
Price: $22.68
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2 of 3 people found the following review helpful
4.0 out of 5 stars Confessions of a neo-Pythagorean, July 26, 2014
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The subject matter of this book is so interesting and important that some readers may wonder why the author decided to devote so many of its pages to personal anecdotes and sidebar story telling. But the pages that are devoted to physics and mathematics are well worth studying, even for readers who are not experts in these subjects but want to attain more insight into the controversy behind the topic of the multiverse, and why some physicists are embracing this very radical worldview in spite of no direct observational support. Neo-Pythagoreans (such as the reviewer) will be enraptured by the author's proposal, and will find him to be one of their conceptual kindred.

The author is definitely in the academic mind set, which comes out especially when he talks about the inability of the (now famous) Hugh Everett, the originator of the many-worlds interpretation of quantum physics, to find employment as a physicist. But there is life outside of the academy, and there is more to physics and mathematics as a profession than merely publishing papers and attending meetings and conferences. And one might argue that after reading this book that its proposals are best done by individuals who are not associated with academia, so as to avoid the conflicts and admonitions by colleagues that the author evidently faced as he frequently alludes to in this book.

Those readers intimately familiar with the philosophy of mathematics may find the author's view of mathematical truth as somewhat restricted, for the reason that although the author asserts that the ultimate nature of reality is mathematical, his conception of the existence of mathematical structures is more in line with the Platonist or formalist schools of mathematical truth. In his (interesting) discussion of the measurement problem as being one of the "crises" in modern physics he does show familiarity with the finitist philosophy of mathematics, but a more in-depth justification of his thesis would entail that he come to grips with the intuitionist school of mathematics, who have very demanding requirements for what it means for a mathematical structure to exist.

Indeed, in intuitionism proving that a mathematical structure exists must be done constructively, in that it is not enough to show that denying this structure will result in a contradiction, and that one must actually give an example of one of these structures or list procedures for its construction. Intuitionist frown upon for example the occurrence of multiplicities of "existence proofs" where not even one example is given of the object that is proposed to exists. One particular example of this is in the field of functional analysis, wherein myriads of theorems are stated that show the existence of fixed points under certain transformations, but where no explicit examples are given of a fixed point.

Intuitionism does have ramifications for the author's thesis, for asserting that one can find all mathematical structures present in the plethora of universes that are part of the multiverse would entail that these structures are not just "predicted" by certain theorems but also that they can be explicitly constructed. Requiring this would prune the number of universes that actually will inhabit the multiverse.

In a neo-Pythagorean conception of reality, which this book clearly represents, everything is mathematical. It remains to be seen if those with practical mind sets, such as experimental physicists, engineers, and technicians, who are used to working with measurement errors and statistical uncertainties, will find any common ground with the author's thesis. For them there is no risk in believing his thesis to be false. The practical realities of doing science will still be with them, whatever the nature of reality.
Comment Comment (1) | Permalink | Most recent comment: Jul 26, 2014 12:42 PM PDT


Our Final Invention: Artificial Intelligence and the End of the Human Era
Our Final Invention: Artificial Intelligence and the End of the Human Era
by James Barrat
Edition: Hardcover
Price: $16.25
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5 of 12 people found the following review helpful
1.0 out of 5 stars Won't do, June 8, 2014
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Anyone who has indoor plants has no doubt run into the problem of proper lighting, with the need sometimes to use artificial lighting. There are several ways in which this might be done, depending on the imagination of the plant lover: 1. One method is to put the plant under a lamp, which is then turned on and off by the plant lover. 2. For those who do not want to remember to turn the lamp on and off, there are devices on the market whose timing can be set by the user to turn a lamp on and off. The actual time that the lamp is on can be set in these devices, according to recommendations given by a plant expert of botanist.

3. Suppose now that this device was modified so as to contain information about the lighting needs of the plant, an Aphelandra squarrosa for example, and that the device was able to turn on and off and vary its lighting intensity based on the judgements of a plant expert. Suppose also that the device is able to compare the efficacy of its "light curve" on the health of the Aphelandra with others grown under light controlled by a device of the same kind. The actual comparison is done under the instigation of the plant lover, and the device can then change its light curve based on the results of the comparison.

4. Suppose that the device is further modified so that it can make the comparison itself, namely it judges whether the difference in light curves on the health of the plants is significant and then alters its own light curve appropriately. Its judgements are taken independent of the plant lover or plant expert, and are based on historical or experimental data it has access to.

5. As a further modification to the device, suppose it can now formulate a set of hypotheses that explain the effects of this type of artificial light generation on Aphelandra squarrosa. The device generates these hypotheses and formulates theories based on the instigation of the plant lover. For example, the plant lover may want to know how the health of the Aphelandra would be affected by changing the lighting conditions, without having to do the testing herself. The device can also formulate light requirements for plants other than Aphelandra squarrosa.

6. Suppose a further modification gives a device that can use the information on light curves of plants to understand the effects of light on other physical entities. The device can find common elements of behavior in the response of plants to light and the response of these other entities to light and formulate a set of hypotheses based on these elements. The device attempts to formulate these hypotheses based on the instigation of an interested human party. A typical plant lover would probably not want this kind of information, but a scientist or botanist might. The device would probably be too impractical to a typical plant lover and its additional ability therefore useless for general home use.

7. The device is further modified so that it is curious about the effects of light on entities, whether these entities are plants or something else. It tries to formulate theories on its own, independent of any external interested party. Such a device might be able to formulate procedures, based on genetic engineering, for altering the biochemistry of Aphelandra squarrosa, so as to make it more resilient as a houseplant, possibly needing less light or a radically different light curve.

8. The device is modified so as to be able to self-manage itself, such as its power requirements. In addition, it can send a set of instructions to a manufacturing facility that will manufacture copies of itself, or it might recommend its own design be altered and then manufactured, with recommendations being based on designs it generated.

It might be fair to say that these eight types of devices are very different, qualitatively speaking. The first type of device is incapable of solving problems but is more of a simple switch. The second type of device represents a machine that can find answers to domain-specific problems but does not compare these answers to any standards. Machines of this type do not attempt to check their answers or correct them. The third device represents machines that find answers to domain-specific problems and check their answers to these problems according to standards that are given to the machine from an external source or standard. The fourth device represents a machine that is able to check its answer to domain-specific problems and make judgments as to the quality of these answers, and do so independently of any external standards.

The fifth type of device represents machines that are able to judge the quality of their answers to domain-specific problems and then propose theories or explanations that subsume these problems, whereas the sixth type of device is able to solve problems having their origin in more than one domain, but their attempt takes place only under the instigation of an external inquirer. The seventh type of device expresses curiosity and creativity, can solve problems independently without any external instigations, and can develop theories of explanations around these problems. Finally, the eighth type of device represents machines that can self-manage and self-replicate,and have all the abilities of machines of the seventh type.

In analogy with human reasoning one might argue that as one goes from the first type to the last the intelligence increases. But if one insisted upon a quantitative measure of just how much "smarter" the last type of device is than the first, then this would be difficult, since no such measure has yet been devised in the field of artificial/machine intelligence.

And the lack of such a measure is the predominant reason why the thesis of this book is problematic and needs to be rejected. There are many places in the book where the the author speaks of "super intelligent" machines as being a thousand or a trillions of times more intelligent than humans, but no where in the book is there any discussion of how this is to be determined. The author does refer to machines taking IQ tests, and the reader is evidently supposed to surmise that it is the use of these tests that will enable one to determine the time when a machine "could match and then surpass human intelligence." No where in the book though is an example given of a machine, either existing or projected into the future, that has taken one or more IQ tests and therefore shown to be "intelligent" to the degree to which these types of tests measure intelligence (if indeed they do). This is also an indication of the great need for the field of artificial intelligence for a rigorous "theory of intelligence" that would allow researchers and engineers to assess more quantitatively the difference between what is called AGI (artificial general intelligence), and domain-specific intelligence.

Again, qualitatively speaking, one could argue that there are many machines today that exhibit domain-specific intelligence, such as those able to play chess and backgammon, perform financial analysis and trading, regulate and troubleshoot communication networks, and find interesting patterns in genome data. These are just a few examples, and apparently the author wants to base his case for what he believes will be "super intelligent" machines on the proliferation of these types of machines in everyday life, as indeed they are. It is true that are lives are dependent on the output of these machines, such as credit scores, financial trading, medical diagnostics, etc. It is quite a stretch though to argue that this massive proliferation of domains-specific reasoning machines will result in machines that can reason over many domains (AGI) without substantial re-writing of their "brains". The author is clearly fearful that this will occur, but he has given no absolutely no hint on how this is do be done.

Instead, the author relies on the opinions of experts who work in the field of artificial intelligence, and also gives figures on the funding levels of research in AGI. If one checks the reality of this funding, there are certain instances where one can verify the figures, but to say as the author does that "billions" are being spent on bringing about human-level intelligence in machines. In addition, opinions of experts are valuable in assessing their comfort level on advances in artificial intelligence, but if one is to build a sound case for the "intelligence explosion" that the author claims will happen, one will definitely need to offer a more quantitative case. The Vinge/Kurzweil conception of the "law of accelerating returns" and the associated concept of a technological "singularity" is with each passing year looking to be more of a sophisticated marketing campaign rather than sound science, and reliance on these conceptions is not bringing about a theory of machine intelligence that is practical and sound.

There are also a few other difficulties in the claim that super-intelligent machines are destined to be our "final invention", mostly coming from basic physics and the manner in which scientific research and results are obtained. There are thermodynamic considerations and energy requirements that need to be addressed if such machines are to operate creatively in bringing about new scientific knowledge and practical products. A "super-intelligent" machine engaged in scientific research will need to conduct actual experiments, this being essential to science rather than just thoughtful musings, and this will require space, instrumentation, and a substantial amount of energy. These kinds of machines will also be subject to the ordinary laws of thermodynamics, and will have to deal with the heat they generate when such an "intelligent explosion" occurs.

One might ignore all of these considerations and take the author's case as one that is more of a warning, just like some scientists had sounded off during the development of nuclear weapons. But to argue that super-intelligent machines are the biggest threat to our existence is to ignore the fact that it is the dumbest entity in the world today that has that privilege, namely the ordinary biological virus.
Comment Comments (5) | Permalink | Most recent comment: Sep 3, 2014 8:39 AM PDT


Thinking the Impossible: French Philosophy Since 1960 (Oxford History of Philosophy)
Thinking the Impossible: French Philosophy Since 1960 (Oxford History of Philosophy)
by Gary Gutting
Edition: Hardcover
Price: $41.99
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1 of 1 people found the following review helpful
5.0 out of 5 stars Written with great clarity, April 27, 2014
The title of this book automatically sets up expectations from the reader who is seeking points of view that address alternative ways of thinking and corresponding methodologies. But the subtitle makes it clear that such alternatives will be cast in the context of French philosophy, with all the historical and cultural biases and baggage that it carries. Some of this bias is the result of bad press, but some of it also originates in the writings of the French philosophers themselves, with this writing being sometimes convoluted and vague, and tied sometimes in literary knots. The author of the book recognizes this, and devotes some pages towards the end of the book commenting on the lack of "persuasive elaboration" that he feels accompanies the texts, articles, and public discussions of the main French philosophers. In spite of this lack, the author gives the reader a very understandable overview of the relevant history and main contributions of philosophers such as Emmanuel Levinas, Michel Foucault, Gilles Deleuze, Alain Badiou, Jacques Derrida, and Jean-Luc Marion
.
Of particular interest, and somewhat surprising for readers such as the reviewer who are non-experts in French philosophy is the powerful influence that various German philosophers had on this philosophy, and this influence going beyond just that of Hegel and Marx. When one studies the works of Jean Paul Sartre it is readily apparent that Hegel had a large role to play in shaping Sartre's thoughts, along with Martin Heidegger, but the author brings out much more lasting influence on the French philosophers, with that influence not just coming from how they dealt with the perceived weaknesses of Sartrian philosophy. In particular, Sartre's failure to reconcile his philosophy with that of Marxism served as an impetus for those after him to discuss the proper context in which social struggle and personal ethics were to be placed.

Those readers in the scientific community, especially those physicists who are perturbed by the 'deconstruction' movement that is associated with the philosopher Jacques Derrida, will find both reasons to become more angry and reasons to reflect on why Derrida and those who came before and after him were so adamant in their criticisms of "analytic" philosophy and capitalist society. The scientific reader may find there is some intersection in their thoughts and those of the French philosophers, especially in the area of ethics. That being said, the clarity of expression demanded by scientific discourse is not to be found in the mental machinations of Derrida et al, which in a certain sense is kind of upsetting, for there is no reason why clarity should not accompany a departure from traditional or entrenched thought patterns, that latter of which was supposed to be mission of the French philosophers.

This zealotry of the French philosophers is brought out with great clarity by the author, who does more than just associate their philosophy with the 1960's drive for the exotic, as has been frequently charged by the analytic community and conservative political thinkers. Some of the ideas of the French philosophers are quite interesting in this regard, particularly in how they dealt with Nietzsche's notion of 'repetition' and how it was viewed by some of them as being beyond a more conceptual categorizing as being qualitative or quantitative equivalence. In particular, the author discusses how the philosopher Gilles Deleuze dealt with repetition as being a kind of "radical novelty", which at first glance may seem to be quite useless if it were not for the fact that theories of human creativity could possibly make good use of it.

Lest the reader think that the French philosophers were part of a grand conspiracy to bring down Western capitalist society as some in the scientific community and conservative commentators have charged, there is ample proof in this book of their disagreements. These disagreements were no more rancorous than those that one can find between any two philosophers or academicians, and actually served to clarify many of the issues that were being bantered about. This clarification may not have been the intent of those participating in the argumentation, but it definitely has that effect on those who are trying to understand the French philosophy in the time period emphasized in this book.

The simple view that philosophers are merely interrogators of reality and seekers of truth will perhaps be changed by some who study this book. On the other hand, its study will be valuable to those who have a real thirst to understand the French philosophers and why their thoughts resulted in such a strong reaction from those outside of their circles. The debate over their ideas and influence seems has decreased in recent years, but could possibly be revived if the French philosophers would take the author's advice at the end of the book and use ideas and thoughtful elaborations that would allow anyone interested to "see through the pretensions of traditional truth".


Complex Topological K-Theory (Cambridge Studies in Advanced Mathematics)
Complex Topological K-Theory (Cambridge Studies in Advanced Mathematics)
by Efton Park
Edition: Hardcover
Price: $85.94
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1 of 1 people found the following review helpful
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
Price: $84.98
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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
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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).


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