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Holographic Duality in Condensed Matter Physics
Holographic Duality in Condensed Matter Physics
by Jan Zaanen
Edition: Hardcover
Price: $75.99
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2 of 2 people found the following review helpful
5.0 out of 5 stars Excellence to the Nth Power, July 10, 2016
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In this incredibly well written book, where the authors display rampant enthusiasm as well as technical competence, readers can find an overview of the research conducted in the last decade in the application of the AdS/CFT correspondence to condensed matter physics. Naturally a background in condensed matter is required to appreciate the contents, but even those readers without such a background but who are strong in quantum field theory can obtain much from studying the book, especially since the authors spend two chapters detailing the need for looking at holographic duality in condensed matter physics.

On can naturally wonder if the AdS/CFT correspondence could be used to shed light on one of the oldest problems in condensed matter physics, namely that of predicting the existence of a periodic lattice instead of assuming its presence a priori. This is a difficult problem that has been studied mostly from the mathematical community than physics, and predicting the existence of a periodic lattice in the boundary from (gravitational) properties of the bulk seems speculative, but no more exotic an idea that perhaps the many that one find in this book. These ideas are fascinating and involve such concepts as strange metals, quantum (holographic) matter, the electron star, and holographic hydrodynamics.

Another feature of this book that enhances its didactic quality is that the authors delegate more specialized computations to “Boxes” which the reader can either study in detail or skip on a first reading. It is a monograph so there are no exercises, but readers will find ample challenges by checking the derivations and those in the quoted literature (which is vast). Readers who expect a general solution to the problem of high-temperature superconductivity will not find it in this book, but the discussions of this problem in the book are insightful and thought provoking.

For those readers who are very knowledgeable about the issues in condensed matter physics but are not experts in general relativity, conformal field theory, and string theory will still be able to gain insight into the reasons why the AdS/CFT correspondence works as it does. For such readers, a side-effect will be the acquisition of knowledge about gravitational physics and the reasons for the “top-down” approach using supergravity and the theory of Dirichlet branes. Conversely, those readers who are expert in the physics of black holes will get some surprises regarding the black-hole uniqueness theorems. Even readers who are specialists in fluid dynamics will find a highly interesting discussion on how the Navier-Stokes theory appears on the boundary. Solving the turbulence problem using holographic duality may be on the horizon in years to come. If so, this would be an awesome development and would no doubt encourage even more research in holographic duality. And last but not least, the discovery of experimental evidence of quantum entanglement on a macroscopic scale would have enormous ramifications not only for the fundamental physics of condensed matter but foundational issues in quantum physics and quantum information theory.

The existence of bound states and the performing non-perturbative calculations are two outstanding problems in quantum field theory. The AdS/CFT correspondence and gauge duality has been applied to the quantum field theory of strong interactions (quantum chromodynamics) via the quark-gluon plasma in heavy ion collisions. Perhaps even more light can be shed on these problems by a study of what happens in condensed matter. Clearly there is a large intellectual momentum in applications of holographic duality at the present time, and no doubt this will continue for decades ahead.

For all these reasons and more, the authors are clearly excited about the prospects of holographic duality, but all through the book and at the end they remind the reader to exercise caution and skepticism regarding just what has been done so far in this field. It will be interesting in the upcoming years to see whether or not holographic duality will remain and grow as both a computational tool as well as conceptual one, or be relegated to a “theory bubble” that contributed more to the field of gravitational physics than condensed matter.

Note: This book was read and studied between the dates of January 2016 and July 2016.

From Hire to Liar: The Role of Deception in the Workplace
From Hire to Liar: The Role of Deception in the Workplace
by David Shulman
Edition: Paperback
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2.0 out of 5 stars Anecdotal storytelling only, June 26, 2016
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Any reader in the workplace, whether employee, manager, or owner, can identify with some of the stories told by the interviewees in this book. There is no doubt from an anecdotal point of view that deception is practiced in the workplace, whether covertly or overtly. The reviewer could list many stories, similar to those in the book, which shed light on attitudes and strategies deployed by both employees and management when it comes to deceptive practices. The reading and telling of such stories can indeed serve as a catharsis, of a kind that the author classifies with terminology that inadvertently creates the impression that the content of the book is legitimate from the standpoint of research in sociology. But the content of the book is biased to the population of individuals who feel that deception is more efficacious than truth. It is extremely biased in fact.

One can easily conclude from reading the stories of the interviewees how unassertive they are when it comes to asking for guidance from management to complete their tasks. It would seem that deception is practiced most often from the weak-minded and the timid; from those who pay excessive loyalty to authority. There is extreme bias in this book to those individuals that have these character traits. No stories are told from the standpoint of those who are confident in their abilities and are not ashamed of seeking assistance from the immediate management when they judge it is needed. The author it seems is not willing to grant that honesty will lead to anything productive in the workplace. The cynicism displayed in this book counteracts any value it has in studying the role and efficacy of deception in the workplace.

In order for the content of the this book to be legitimate from a scientific point of view, the author would have to bring a lot more that anecdotal evidence into the discussion. He would have to compare the role of both deception and honesty in the workplace, by showing the efficacy of each in workplace interactions and strategies. Such a study would involve control groups, and monitoring the behavior of workers and management to the extent that might be too intrusive, and may indeed violate the need for firms to maintain their secrecy from a competitive standpoint. As it stands the book is a collection of the author’s opinions, and even though many readers, including the reviewer, can identify with the many stories told by the people interviewed for the book, any designation of the role, need, and permanence of deception in the workplace as established truth will have to await a future and more rigorous study.

Gravity's Ghost: Scientific Discovery in the Twenty-first Century
Gravity's Ghost: Scientific Discovery in the Twenty-first Century
by H. M. Collins
Edition: Hardcover
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2 of 2 people found the following review helpful
4.0 out of 5 stars Interesting in light of recent announcements, June 12, 2016
In the light of the recent announcement by the executive director of the LIGO project that “We have discovered gravitational waves” this book seems very appropriate if not somewhat ironical. Those scientists/readers who understand general relativity and the Einstein prediction of gravitational waves should view the recent announcement and the contents of this book with the usual skepticism that is required by the scientific enterprise. Such skepticism should be considered normal for both professional and non-professional scientists, the degree of which should be no lesser or greater for gravitational wave experiments than it is for other scientific experiments, no matter what their scale in terms of funding or estimated impact.

It can be the case that scientific projects that are very expensive are treated unfairly when it comes to the degree of skepticism and scrutiny exhibited towards them. Conversely, these types of projects, sometimes called “big science”, can be viewed as being more exemplary or plausible since, the argument goes, if they were not, funding agencies would exhibit so much generosity to them. In either case “big science” has gotten a lot of press, and this book, although highly interesting and thought provoking, could be viewed fairly as being one of a collection that focus on the “sociology of scientific discovery”.

The author though has thankfully not given opinions from the standpoint of an armchair sociologist, but rather as someone who was “embedded with the troops” of LIGO, and was able to collect opinions and anecdotes that assist the reader in forming their own opinions on whether the search for gravitational waves is sound science. There are many interesting sources of debate in the book, such as the discussion on the “5-sigma” issue in statistics and the use of probability and statistics in general. Also, the author gives what in the opinion of this reviewer is one of the best summaries of the proper ethics of scientific discovery in the 21st century.

It remains to be seen of course whether the announcement of the discovery of gravitational waves will stand scientific scrutiny, and if not, whether the LIGO project and others affiliated with it will survive from a financial viewpoint. With the extreme budget constraints at the present time, and the emphasis on practical applications of science, a negative assessment of the announcement will no doubt only assist in the demise of “big science”.

Note: This book was given to the reviewer as a gift from a relative, and so will not show up in this venue as a “verified purchase”. It was read and studied in its entirety between the dates of April 2016 and June 2016.

No Place to Hide: Edward Snowden, the NSA, and the U.S. Surveillance State
No Place to Hide: Edward Snowden, the NSA, and the U.S. Surveillance State
by Glenn Greenwald
Edition: Paperback
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1 of 1 people found the following review helpful
4.0 out of 5 stars Will reading/reviewing this book put you on the "terrorist watchlist"?, May 14, 2016
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The answer to the question posed in the title of this review is a definite yes, and follows vacuously if, as the author of this book asserts, that the National Security Agency (NSA) collects data at the scale claimed. This scale covers every person on the planet, and therefore the NSA views everyone as a potential terrorist. Information on everyone is collected and then stored in the NSA’s data repositories, which indeed must be massive in order to store this magnitude of data. Edward Snowden, a former contractor for the company Booz-Allen-Hamilton (BAH), has revealed that the NSA is engaging in this activity. Snowden was placed by BAH in the NSA to engage in security work some of which was classified as Top Secret. As this book and ensuing documentaries reveal, because of moral and ethical reasons Snowden made the decision to abrogate the trust the NSA placed in him and put himself at risk for an extended prison term because of the information he chose to place in the public domain.

The author of this book makes a good case for the morality behind Snowden’s decision, and therefore readers who formally viewed Snowden as a “criminal” or a “narcissist” may make a different assessment of his character after finishing the book. There is no doubt that Snowden did not view his decision as one of self-aggrandizement, and in fact exercised restraint in the sense he chose to not reveal information that could put people at severe risk. It might be a leap to call Snowden a “hero”, but he certainly possesses a level of intestinal fortitude that is unmatched by anyone in the government, whether the United States government or otherwise.

The details of some of the revealed information are included in this book as images, with some of them actually looking as though they originated in PowerPoint presentations. That this may be the case reflects the obsession that many, especially those connected with the Department of Defense (DoD), have in using PowerPoint to not only summarize ideas but also to codify the information in them as valid, authentic, or profound. Therefore it was difficult for the reviewer to accept the author’s (implicit) premise that the information in these images as reflecting anything of genuine importance to those readers who want to understand the extent of the NSA’s illegal surveillance. Typical government/DoD PowerPoint presentations, even though quite impressive from an artistic point of view, reflect a naiveté about scientific and technological matters in general. There is no reason to believe that this is not the case also for the NSA, in spite of the imputation of technical and mathematical competence given to it.

And this raises the further question as to the efficacy of the NSA in doing the analysis and data mining that would actually put individual privacy at risk. Many in the press have claimed that the NSA hires thousands of mathematicians and analysts and this is no doubt true. But what is not obvious from the press or from this book is the extent that this technical pseudo-army is able to extract damaging information about citizens or indeed any really useful information at all. The mere presence of thousands of analysts and mathematicians in a government agency may reflect the usual practice of patronage and other faulty and unethical governmental hiring practices rather than actual technical competence.
But one could also argue, and it seems to be the belief of the author, that the NSA is technically competent to use the gathered information to find individuals who it deems are “harmful” to the interests of the United States. Then since the NSA has no qualms in violating the constitutional rights of US citizens by collecting “metadata” then it would not be an unreasonable assumption that it would collect the actual content of phone calls, Email messages, and so on. It would also not be unreasonable to assume that the NSA would deliberately alter the content of messages and phone calls in order to embarrass certain individuals or groups as part of their security strategy or simply from just pure meanness. There is ample precedent in history for the meanness of governments.

Climate Change: A Very Short Introduction (Very Short Introductions)
Climate Change: A Very Short Introduction (Very Short Introductions)
by Mark Maslin
Edition: Paperback
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2 of 4 people found the following review helpful
3.0 out of 5 stars A fair introduction, April 24, 2016
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Since this book is so short it should not be viewed as one that discusses in detail the data and modeling that supports the doctrine of climate change (the use of the word “doctrine” here is meant to convey the idea that the field of climate change has become one that is approaching religious dogma, and like the latter permits no dissent from its claims). Readers cannot expect the author to give all the details and supporting data that will validate his assertion of anthropogenic climate change. What they can expect and receive is a brief outline of what has become one of most rancorous debates in the history of science, and definitely one where the level of vituperation has gone beyond all measurable bounds. One might with an element of humor argue that the amount of hot air produced by both sides of the debate is itself responsible for a surge in global temperatures.

Even though the author is clearly biased when discussing climate change, it should not be concluded that this book does not have some degree of merit for those readers who have no political or financial agenda but simply want the raw, naked truth about climate change. The book will at least give such readers some impressions about the thought patterns of the adherents of the doctrine of climate change, and why they have been proselytizing both politicians and the public about this doctrine. In this regard it is not uncommon to observe harsh debate among scientists about a particular public project, one example being the construction of the Superconducting Supercollider in the area of physics. But the doctrine of climate change has enlisted a massive army of supporters, some of them popular television personalities and comedians, who clearly do not have the scientific background in climate science or meteorology to assess the validity of climate change, but yet are adamant in smearing the reputations and character of those individuals who are challenging its doctrines.

The author is certainly correct when he asserts that science advances by utilizing detailed observation and experimentation, and that this involves an on-going necessity to acquire new data and perform new experiments. The predominant reliance on models though does not by itself satisfy this paradigm, since scientists use models to get an idea about how a physical system is going to behave when they don’t have the experimental data available. In the Manhattan project for example, models based on Monte Carlo simulation were invented to study neutron diffusion because the experimental data on neutron diffusion was lacking. Modeling of course, and not just Monte Carlo simulation, is now widely used in many areas of science, medicine, finance, and technology, and the validation of these models by data coming from real measurements is something that its practitioners widely recognize to be necessary if they are to part of the scientific process. In this regard, those scientists who do not support the doctrine of climate change (called “deniers” by the author) point out that in the last fifteen years the experimental data indicates a flatness in global temperature change, which has the effect of invalidating the predictions of the climate models.

An entire chapter of the book is devoted to modeling, and the author points to several different models being used to assess the likelihood of future climate change. The author points to the fact that the IPCC wants to assume that the output of each of these models is “equally valid”. A more reasonable approach, and one that the author implicitly endorses, is to use some of the tools from a field called uncertainty quantification to more meaningfully quantify the discrepancies in output from these models. And he is well aware of the difficulties in using modeling to make policy decisions, and in that respect he is being more reasonable than other individuals in the climate change debate who it seems do not want to face up to the uncertainties created by the use of models.

There are a few places in the book where the author is somewhat careless, as for example omitting error bars in graphs of experimental data, but as a whole the reviewer, who is just getting started in studying the data and conclusions of climate science, holds that this book is much better than others in framing the debate and in its elucidation of some of the key physics. Hopefully, and this is definitely against the seeming trend, both advocates and “deniers” of anthropogenic climate change will in the future be able to sit down at the same table and conduct themselves in the best tradition of scientific endeavor, namely that of having no respect for authority or political ideology, but only for facts.

The Will to Survive: A History of Hungary
The Will to Survive: A History of Hungary
by Sir Bryan Cartledge
Edition: Paperback
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1 of 1 people found the following review helpful
4.0 out of 5 stars A fascinating history of Magyarorszag., April 16, 2016
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The author opens this book by saying that “nations need myths”, a statement that may immediately raises questions in the reader’s mind as to the ability of the author to distinguish myth from real history. Such doubts, if they exist, do not really need to be addressed since books of history should be read and studied with an attitude of strong skepticism. The history of history-writing is frequently marked by sycophancy to political leaders or ideology, and therefore a highly critical frame of mind, indeed one that has a degree of skepticism that is even greater than what is exhibited in the scientific community, is required. That being said, those engaging in the study of books of history should not expect that historians can be free of bias, no matter how they discipline themselves in this regard.

The author of this fascinating and enjoyable book has biases to be sure, and from time to time imputes moods and other mental attitudes to historical figures that would be difficult if not impossible to verify. For example, his bias against socialist and communist doctrine, comes out towards the end of the book, but also manifesting itself in various places in the early chapters. But the supplied references and the manner of writing is very helpful for readers who have a sincere desire to understand the history of Hungary and the role it played in the history of Europe. And since the author played an active personal role in Hungary in modern times, this would justifiably lead the reader to believe in his analysis of the history of Hungary in the last decades of the twentieth century.

It is refreshing to read that the early Hungarians did not willingly drink of the rancid ale of Catholicism, but instead had to be dragged to it by the likes of Stephen I and Andrew I. As was typical of the time, the Catholics obtained converts by force and not by persuasion, and as the author explains the Catholic church was supported financially by tithing. The resistance to religion is a sign of the Hungarian penchant for independence, but unfortunately they surrendered this independence at various times throughout their history. A modern example was their choice to ally with the Nazi regime, and this alliance resulted in them being under the yoke of the Soviet Union for many decades.

Another helpful feature of the book are some of the discussions of economic development in Hungarian history. The author cannot of course give the details of this development without swelling the book to an unmanageable size. It is incredible to read that serfdom existed even up to the twentieth century in Hungary. Warfare and the desire to control surrounding areas and expand borders quite expectedly cause d Hungarians many hardships.

One need only witness the overabundance of the fruit and vegetable markets in Budapest today to reach the conclusion that the Hungarians have come a long way, and hopefully they can sustain the cosmopolitan attitudes that brought them to their current status of relative opulence. A country can sustain and enjoy its cultural and historical heritage without demanding other cultures to recognize it as such. It can have and preserve its own unique language without insisting on the purity (or superiority) of its vocabulary and grammatical constructions. It can be surrounded by nations that may not match up to its scientific, artistic, and literary accomplishments and still exercise humility when assessing these achievements. If Hungarians would have thought of themselves as having no special status relative to other countries and cultures, they might have avoided a great deal of heartache and pain throughout their history.

Lagrangian Intersection Floer Theory (Set) (Ams/Ip Studies in Advanced Mathematics)
Lagrangian Intersection Floer Theory (Set) (Ams/Ip Studies in Advanced Mathematics)
by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono
Edition: Hardcover

3.0 out of 5 stars Challenging, April 9, 2016
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Anyone who approaches this book needs to have a solid understanding of symplectic geometry, differential and algebraic topology, algebraic geometry, abstract algebra, global analysis, and complex manifolds. It will also help if prospective readers had some knowledge of topics from physics and mathematics such as mirror symmetry, string theory, the theory of operads, and non-Archimedean geometry. The book is not written with the goal of explaining the key concepts, but rather as a report detailing the contributions of the authors to the subject. There are an enormous number of references included at the end of the Volume 2, and a study of some of these will be needed if the contents of these volumes are to be appreciated beyond that of a mere formal understanding. The books are another example of the typical way in which modern mathematical monographs are written, namely that of expounding and not of explaining, and this makes it difficult for those who sincerely want to understand the subject matter, even though they have no intentions of conducting research in the relevant area. Therefore readers who want to be challenged intellectually will definitely find ample opportunity for such in these two volumes.

Readers will need an understanding of Floer homology, which is essentially an infinite-dimensional version of Morse-Novikov for the ‘symplectic action functional’. The goal is to interpret the contractible 1-periodic solutions of a Hamiltonian as the critical points of the symplectic action functional on the universal cover of the space L(M) of contractible loops in a symplectic manifold. The universal cover is the unique of covering space of L(M) whose group of deck transformations is the image of the Hurewicz homomorphism from the second homotopy group of M to the second homology group of M.

The symplectic action function acts on the covering space and takes values in the real numbers. The critical points of this action correspond to periodic solutions and the focus is then on the upwards gradient flow lines of the symplectic action functional with respect to the L^2 metric on LM induced by an almost complex structure on M. These are the solutions u(s,t) of the Floer equation that have domain 2-dimensional Euclidean space and have period 1 in t. These solutions approach the critical points as s approaches infinity in the distant past and future, and these solutions form the moduli space of solutions of finite energy, the latter of which is defined to be difference of the action functional on the respective critical points in the distant past and future. The dimension of this moduli space is the difference in the Maslov index (some readers may understand this to be the Conley-Zehnder index) of the respective critical points in the distant past and future.

Along these lines readers will need to have an appreciation of what it means to formulate and study moduli spaces. Generally speaking, a moduli problem is one of classifying certain geometric objects up to a notion of equivalence. Constructing a moduli space involves putting an an algebraic structure of M that should capture the way objects very in families. The ‘moduli problem’ is very familiar in the area of algebraic geometry where one has a family of objects over a base scheme, and with a notion of a pullback of families along morphisms. The pullback along the identity morphism gives the same family, and pullbacks of compositions of morphisms are merely the pullback of the first followed by the second. Two families over the base are equivalent if there is a map between the bases and the pullbacks of the map on these two families are equivalent. A ‘universal family’ is a family U where for every family X over the base there exists a unique morphism K from the base to M such that the pullback of K on U is equivalent to X as families over the base. The base is then called a ‘fine moduli space’. For every base B there exists a bijection between the set of families over B and the set of morphisms from B to M.

Along these same lines, and also motivated by what is done in algebraic geometry, the theme of compactification of the moduli space plays an important role in these two volumes. The compactification relies on an important concept called ‘stability’ and in algebraic geometry typically arises when studying what are called ‘n-pointed smooth rational curves’. These are projective smooth rational curves with n marked points and maps between these curves preserves marked points. One can also have families of n-pointed smooth rational curves where each fiber will have marked points. There exists a fine moduli space M(0, n) for classifying n-pointed smooth rational curves up to isomorphism, and there exists a universal family of n-point curves U(0, n) with respect to M(0, n). This means that for every family of projective smooth rational curves over a base B with n disjoint sections is induced by a pullback along a unique morphism from B to M(0, n).

But M(0, n) is not necessarily compact (for example M(0, 4)) and to compactify correctly one must include curves that are ‘reducible’. These are the ‘stable’ curves, and intuitively involve configurations where there is a “break” in the curve. Prospective readers of these volumes can find ample pictorial examples of stable curves in the literature on algebraic geometry. More formally an n-pointed curve is ‘stable’ if does not have a map that fixes each marked point: some of these marked points, or all, must move under this map. ‘Stabilization’ of a curve involves adding points so as to obtain a stable curve. Of particular relevance to the content of these volumes is that a stable curve of genus zero is a finite collection of 2-spheres that are glued together at certain points in these spheres and have marked points coming from these spheres.

The notion of stable curves in algebraic geometry carries over to symplectic geometry and the theory of J-holomorphic curves, and therefore in these volumes, but with an added complication that is referred to as “bubbling”. The presence of bubbling is important because it contributes to an obstruction in defining Floer cohomology. Generally speaking, J-holomorphic curves are maps from Riemann surfaces to smooth manifolds M of even dimension whose exterior derivatives satisfy a Cauchy-Riemann equation involving an almost complex structure J on M. Similar to what is done in algebraic geometry, one constructs moduli spaces of J-holomorphic curves for the case where M is a compact almost complex manifold, and then attempts to show these moduli spaces are compact.

Showing compactness entails studying sequences of J-holomorphic curves, and if the gradient of these curves is uniformly bounded in the L^p norm on compact subsets then a successful compactification of the moduli space means there exists a subsequence converging uniformly on compact subsets to a J-holomorphic curve. It is the case where the gradient is unbounded and the energy is uniformly bounded that results in the bubbling of a non-constant J-holomorphic curve. The use of the word “bubbling” here comes from the fact that one can find a subsequence which converges to a J-holomorphic curve on the 2-sphere. There are many interesting examples of bubbling in the research literature as prospective readers will find, but it must be said that some of these examples do not look sphere-like in appearance. A J-holomorphic curve can have several different bubbles, depending on the the scaling that is chosen, and if the genus of the curve is to be preserved during the limiting process, then this is referred to as stability, as in the case of algebraic geometry.

The phenomenon of bubbling puts a constraint on the construction of Floer cohomology which the authors handle via the use of A-infinity algebras. The discussion of these algebras (more precisely that of filtered A-infinity algebras) is the most interesting part of these volumes, and is a reflection of the fact that to study Lagrangian intersections one must develop a theory at the cochain level. Filtered A-infinity algebras have an infinite number of operations that are must satisfy certain identities, and because of the complexity of these operations readers who are familiar with operads will be more comfortable with them. The authors show how to associate to a Lagrangian submanifold a filtered A-infinity algebra over a particular type of Novikov ring, and this algebra is interpreted as a quantum deformation of the rational homotopy type of L. The physicist reader may object to the use of the term ‘quantum’ to describe what are essentially all nonlinear contributions coming from pseudo-holomorphic disks, and such an objection would be valid. There is nothing “quantum” about these deformations in the sense that they exhibit interference or entanglement, two properties that are manifest in quantum physics. One could perhaps view them as a “semi-classical” phenomena manifested as “disk instantons.”

The authors spend a considerable amount of time detailing when it is possible to define the cohomology of the filtered A-infinity algebra, which essentially involves defining it relative to a particular type of bounding cochain that satisfies an A-infinity version of the Maurer-Cartan equations. The obstructions to defining Floer cohomology for Lagrangian submanifolds involve deformations of this bounding cochain. The authors gives the details on how to define Floer obstruction classes and show a connection with the famous Gromov-Witten inveriants.

A special type of algebraic object that arises in Morse theory for multi-valued functions on loop spaces is the Novikov ring, and is used throughout these two volumes. A perusal of Novikov’s original paper on the Morse theory of closed 1-forms will assist in gaining an intuition on this ring, and the authors of this work spend a fair amount of time in discussing it, even though for the most part their discussion is purely formal. Essentially what Novikov did is to take a closed 1-form w on a manifold M and integrate it over paths in M. This results in a multivalued function S which becomes single-valued on some covering p of M with a free abelian monodromy group given by taking the pullback p* of w. The number of generators of this monodromy group are the number of rationally independent integrals of w over integral cycles in M. The surfaces of steepest descent for critical points of S on the cover define cell complexes C which are invariant with respect to the action of a generator t: covering space-> M of the integers Z. This implies that C is a free complex of Z[t, 1/t] modules generated by S on the covering space. The object Z[t, 1/t] is the ring of Laurent polynomials and serves as an elementary example of the Novikov ring. A Novikov complex is then the collection of finitely generated modules over the Novikov ring whose generators are in 1-1 correspondence with the critical points of the 1-form.

Another concept in these volumes that readers may find challenging to understand is that of a Kuranishi structure. To get some insight into what a Kuranishi structure is, the reviewer had to perform a fairly time-intensive consultation of the research literature. Loosely speaking, a Kuranishi structure is generalization of an ordinary manifold in that it allows the presence of singularities. In the context of symplectic topology, one is interested in the construction of algebraic structures by extracting homological information from the moduli spaces of holomorphic curves in general compact symplectic manifolds. This is referred to as ‘regularization’ and is to be contrasted with other regularization approaches such as where almost complex structures are perturbed but are unable to yield a smooth structure on the moduli space. Such a regularization procedure makes used of some kind of perturbative technique that will permit transversality. The construction of the algebraic structure also should be independent of the choices involved in obtaining transversality. The elaboration of these requirements leads to the performing of regularization via Kuranishi structures.

Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World
Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World
by Leslie Valiant
Edition: Hardcover
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5.0 out of 5 stars Thought provoking learning theory for evolutionary biology, March 27, 2016
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When one is confronted with a “your money or your life” proposition from a gun-wielding thug in a dark alley, there is no temptation to poll the data on dark alley robberies in order to calculate the probability that the thief will pull the trigger. Unless one is trained to deal specifically with threats and stressful situations such as this, one quickly hands over the wallet or the purse. No sophisticated time intensive algorithms are in play in this situation. The pattern matching of the gun image and the affective capabilities of the brain take over here, provoking with incredible speed an appropriate fear response. The feedback received from this situation is that of walking away unscathed.

The point to be made here is that if one is to view the brain as a computational entity that deploys various algorithms to deal with situations like this and survival in general one must come to grips with the computational complexity of these algorithms. One must acknowledge that survival entails in many instances that thought processes operate on time scales that can are very short, as well as time scales that can be very long, i.e. require much deliberation and a quantitative assessment of risks.

The author of this book is well aware of the issues with computational complexity and via the idea (which he invented) of ‘probably approximately correct’ or PAC learning for short, has given the evolutionary biologists an interesting and provocative view of evolutionary processes that addresses some of the gaps in the Darwinian paradigm.

The book is highly interesting and its perusal will not only help the reader understand the issues at stake in the Darwinian view of evolution but will also assist the uninitiated reader in the understanding of PAC learning itself. In this regard the author devotes a portion of the book to PAC learning and examples are given that illustrate it. A very plausible case for the role of PAC learning in evolutionary process is outlined and should be understandable to anyone with even a modest background in computer science and mathematical logic. Readers with more background in learning theory and artificial intelligence can still appreciate the book even though the rigorous formalism has been omitted in order to appeal to a wider audience.

Whatever the eventual impact this book has on evolutionary biology it raises issues that should be addressed when contemplating the Darwinian paradigm, and beyond that it addresses requirements that every algorithm developer confronts in everyday practice. These involve the running time of algorithms designed for practical use, the data needed for these algorithms that can frequently be corrupted or sparse, and the overhead generated by the algorithm, especially those deployed on information networks.

Morse Theory and Floer Homology (Universitext)
Morse Theory and Floer Homology (Universitext)
by Michèle Audin
Edition: Paperback
Price: $89.99
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5.0 out of 5 stars Written with meticulous detail, March 26, 2016
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One of the unique features of Morse theory that makes it somewhat rare in modern mathematics is that it has a very intuitive geometric content. Critical points, level surfaces, tangent spaces, transversality, and differentiable functions are all things that are easily visualized even if stated in the rigorous language of differential topology, a field that Morse theory is part of, and one that has considerable applications not only in mathematics but in physics, economics, and network science.

From the standpoint of the field of physics, Hamiltonian mechanics, and its dual, Lagrangian mechanics are also very understandable from a geometric point of view as soon as one gets use to thinking in terms of phase spaces and least action principles. Indeed, the time evolution of conservative Hamiltonian systems preserves the area in phase space, and phase space diagrams can reflect the periodicity (or lack thereof) of the time dependence of mechanical systems.

The authors of this book bring together Morse theory and Hamiltonian mechanics in the context of symplectic topology and nonlinear partial differential equations, summarizing in meticulous detail a subject that began with the late mathematician Andreas Floer. The goal of the book is to prove the Arnold conjecture, which essentially gives a bound on the number of fixed points of particular types of Hamiltonian diffeomorphisms on a symplectic manifold.

To achieve this goal requires studying the solutions of a particular type of partial differential equation, called the ‘Floer equation’, which defines the flow of the gradient of functional acting on the space of loops on the symplectic manifold. As the authors show, the conjecture is straightforward to prove for the case where the Hamiltonian is independent of time. The general proof of the Arnold conjecture requires many pages to complete, but the authors guide the reader every step of the way, and spend time in discussing the required symplectic geometry and global analysis needed for working in the context of infinite-dimensional Banach manifolds.

The authors give very understandable discussions of the notions of the Maslov index and the phenomenon of “bubbling” in symplectic topology. Readers familiar with the Yang-Mills equations in high energy physics may have ran into the bubbling phenomena when studying these equations. In this book it arises when proving the compactness of the space of solutions to the Floer equation. Bubbling is shown to arise because of the behavior of the gradient of symplectic balls with small radius, and is prohibited by making the ‘asphericity’ assumption, namely that the symplectic manifold not contain any spheres of nonzero symplectic area.

Almost complex structures are intimately connected with symplectic topology and in this book appear in the Floer equation via an endomorphism of the symplectic manifold that squares to -1. The famous J-holomorphic curves are solutions of the Floer equation when the Hamiltonian is zero, where the Floer equation reduces to the ordinary Cauchy-Riemann equation.

When studying the book it is readily apparent that the proof is harder than its Morse theory analog, and the reviewer found it was easy to get lost in the proofs of the lemmas and propositions at a first reading. This is especially true in Chapter 9, which is probably the most important one in the book since it shows the needed property of the differential of the Floer complexes, namely that is squares to zero. With patience though, readers will be able to get through this chapter and get a deep appreciation of the topic of Floer homology.

For those readers who are interested, the book can also serve as an excellent warm-up for a future study of Lagrangian intersection theory and Floer cohomology, a topic that has seen a lot of development in recent years, but is formidable in its level of abstraction. Some of the exercises in the back of the book are geared towards developing an understanding of Lagrangian submanifolds of symplectic manifolds.

Climate Change: The Facts
Climate Change: The Facts
by Mark Steyn
Edition: Paperback
Price: $15.84
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7 of 7 people found the following review helpful
4.0 out of 5 stars A good start for those genuinely interested in climate science, March 19, 2016
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For those interested in an alternative view on climate change and its asserted man-made causes this book will be a good start, even though at times it engages in the same kind of vituperation that is characteristic of those individuals who support the doctrine of climate change, and rally behind it with unequaled enthusiasm. Indeed, the exchange of harsh words on both sides of the “aisle” of climate change reminds one of the shouting matches that took place in the “debate” on AC and DC power that took place early in the twentieth century. The dialog in the AC/DC power “debate” was unpalatable from both a social and scientific point of view, and served no useful purpose in arriving at an eventual decision. It seems at times that the “debate” over climate change is competing for the chance to be one that is marked with an overabundance of ridicule and rage.

There are a lot of gems in this book from the standpoint of the reviewer who is just beginning to examine the scientific evidence both for and against man-made (anthropogenic) climate change. Each one of these however needs to be critically examined along with the supporting references that are quoted in the book. As such one should not expect a book of this length to contain the evidence that supports the claims that are made between its covers. Readers therefore who want to understand the arguments against (and for) anthropogenic climate change, and like the reviewer have a strong scientific background in areas not dealing with, both serving as a foundation for climate science, can expect to spend a considerable amount of time in obtaining this understanding.

There are no short-cuts in scientific understanding, and unfortunately there are those on both sides of the “debate” that do not even have even a rudimentary understanding of the fundamentals of physics or data analysis. How to deal with uncertainties in measurement data and how to deal with errors in simulation models are just two examples, and both of these are actively discussed in the book. An understanding of these fundamentals is absolutely crucial if one is to make claims on the role of humans in climate change or global warming.

It is fair to say that climate change is now political doctrine, and like most doctrines permits no deviation from its policies and edicts. It eschews (and fears) critical thinking, and puts emphasis on marketing and propaganda, rather than on experimentation and facts derived from careful analysis. For those with a sincere desire for the raw, naked truth behind climate change, and climate science in general, this book will hopefully be one of many others that will assist in fostering a healthy skepticism about the subject.
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