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27 of 28 people found the following review helpful
4.0 out of 5 stars How to think in (mostly) ten dimensions
This book, from a mathematician, covers the period from the first proof that Calabi-Yau spaces actually might exist to their current central place as a preferred model for String Theory's extra dimensions. Shing-Tung Yau is the Fields Medallist godfather of the eponymous manifolds and Steve Nadis had the unenviable task of writing it all down so that the rest of us could...
Published on 17 Nov 2010 by Amazon Customer

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6 of 8 people found the following review helpful
1.0 out of 5 stars Post docs only
I also gave up - less than half way through. I am near the end of a maths degree which has included quantum theory and relativity and other than the historical side of how the theory developed I found the substance of the book incomprehensible. Don't think this is going to be like Greene, Penrose or Hawking. It's not.
Published on 28 Jan 2011 by CMN


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27 of 28 people found the following review helpful
4.0 out of 5 stars How to think in (mostly) ten dimensions, 17 Nov 2010
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Amazon Customer (Wells, UK) - See all my reviews
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This book, from a mathematician, covers the period from the first proof that Calabi-Yau spaces actually might exist to their current central place as a preferred model for String Theory's extra dimensions. Shing-Tung Yau is the Fields Medallist godfather of the eponymous manifolds and Steve Nadis had the unenviable task of writing it all down so that the rest of us could have a prayer of understanding it. He also did the interviews and fleshed out the physics side. The best way to review this book is just to explain what it says chapter by chapter.

Chapter 1: The universe is a big place, maybe infinite. Even if its overall curvature suffices to close it, observations suggest that its volume may be more than a million times the spherical volume of radius 13.7 billion light year we actually see. The unification programme of theoretical physics doesn't really work, however, if it's confined simply to three large spatial dimensions plus time. It turns out that replacing the point-like objects of particle physics with tiny one-dimensional objects called strings, moving in a 10 dimensional spacetime may permit the unification of the electromagnetic, weak and strong forces plus gravity. Well, today it almost works.

We see only four space-time dimensions. Where are the other six? The suggestion is that they are compactified: rolled up to be very small. But that's not all, to make the equations of string theory valid, the compactified six dimensional surface must conform to a very special geometry. That is the subject of the rest of the book.

Chapter 2: Yau was born in mainland China in 1949. His father was a university professor but the pay was poor and he had a wife and eight children to support. When Yau was 14 his father died leaving the family destitute: Yau's destiny seemed to be to leave school and become a duck farmer to pay his way but in a flash of inspiration he decided instead to become a paid maths tutor, teaching as he was learning. Yau's astounding talent led him from this humble background to the University of California at Berkeley by the time he was 20. As well as autobiographical details, this chapter also outlines the idea of a metric on curved spaces, introducing Einstein's theory of gravity.

Chapter 3: Yau's early work at Berkeley was in the area of geometric analysis, used in the proof of the Poincare conjecture (1904). This conjecture states that a compact three dimensional space is topologically equivalent to a sphere if every possible loop which can be drawn in that space can be shrunk to a point without tearing. The conjecture was proved in 2002 by the controversial Russian mathematician Grisha Perelman. Work in this area set the scene for Yau's celebrated proof of the Calabi conjecture: that what subsequently became known as `Calabi-Yau' (CY) spaces actually exist.

Chapter 4: The Calabi conjecture is simple to state if not to understand: it asks whether a complex Riemann surface (conformal, orientable) which is compact (finite in extent) and Kähler (the metric is Euclidean to second order) with vanishing first Chern class has a Ricci-flat metric. All these concepts are explained in this chapter. One of the more interesting features of a space satisfying Calabi's conjecture (if it existed) was that it would satisfy Einstein's vacuum field equations automatically.

Chapter 5. Yau initially didn't believe the Calabi conjecture and at a conference held at Stanford in 1973 went so far as to give a seminar "disproving" it. Calabi contacted Yau a few months later asking for details and Yau set to furious work, the argument slipping out of his hands the harder he tried to make it rigorous. Yau concluded that in fact the conjecture must be correct and spent the next three years working on the problem. In 1976 he got married and on his honeymoon the last piece of the puzzle dropped into place. The conjecture was proved correct.

Chapter 6. What Yau had proved was a piece of mathematics but he was sure there must be applications in theoretical physics. However, nothing happened until 1984. Parallel developments in string theory (ST) had determined that ten dimensions were needed to allow sufficiently diverse string vibrations to occur to capture the four fundamental forces and to induce `anomaly cancellation'. The search was on for a six dimensional compactified space to complement four dimensional space-time. The chapter describes how physicists came to CY spaces via supersymmetry and holonomy.

CY manifolds within ST are very small (a quadrillion times smaller than an electron) and are riddled with multidimensional holes (up to perhaps 500). The way strings wrap around the CY surface, threading through holes, is intended to reproduce observed particles and their masses. This has proven a fraught task as it requires a very special CY manifold to even get close. Yau has estimated there might be 10,000 different manifolds but no-one really knows.

The chapter closes with a discussion of M-theory, Edward Witten's framework for uniting the five different string theories developed in the 1990s. M-theory is defined in 11 dimensions and includes `branes' of anything from 0-9 dimensions. Apparently the universe could have 10 and 11 dimensions simultaneously but the mathematics (via CY spaces) works better in 10.

Chapter 7 discusses a challenge to the applicability of CY spaces due to the quantum field theory requirement for conformal and scale invariance. The CY metric doesn't (without tweaking) allow for this. This research led to a concept called mirror symmetry which associates CY manifolds with distinct topologies with the same Conformal Field Theory (CFT). This proved important for calculation.

Chapter 8 talks about the success of ST in deriving the Bekenstein-Hawking formula for (supersymmetric) black hole entropy. The very large number of required black hole microstates are constituted by wrapping branes around sub-surfaces of a CY manifold to build the black hole. The chapter ends by extending these ideas to the celebrated AdS/CFT correspondence.

Chapter 9 notes that ST has yet to reproduce the Standard Model (SM) and recounts some of the attempts being made. Yau's favourite is E8 x E8 heterotic ST and the technique is to break the many symmetries of E8 down to the 12 required by the SM [SU(3) with 8D symmetry, 8 gluons; SU(2) with 3D symmetry, W+, W-, Z; U(1) with 1D symmetry, photon]. We are not there yet.

Chapter 10 talks about mechanisms to keep the compactified dimensions small when energetically they would prefer to be large. The CY manifolds are stabilised by quantised fluxes. Suppose there are 10 values (0-9) for a flux loop and 500 holes in a CY manifold then there are 10 ** 500 different stable states. This extraordinary crude estimate has been widely publicised as "The Landscape Problem" for those who were hoping that there would be exactly one CY model for the universe. Yau is unimpressed, never having believed in such uniqueness in the first place. Chapter 11 continues the theme of `explosive decompactification' and recommends not being around if and when it happens.

Chapter 12 surveys the search for hidden dimensions. They may be visible `out there' for telescopes to pick up. Alternatively there's the LHC. Chapter 13 is an essay on truth and beauty in mathematics.

The final chapter raises a deep question. CY manifolds are solutions to Einstein's gravitational field equations in a vacuum. But Einstein's theory is classical - smooth all the way down (except for rare singularities). However, the QM view of space-time at the Planck scale is anything but smooth: the term `quantum foam' has been coined. What kind of geometry - quantum geometry - could model this?

Yau's view is that at present no-one has much of a clue although he describes some ideas exploring CY topology changes via singularity introduction - the flop transition -which could shed some light on what quantum geometry could look like.

In summary this is not a book for the faint-hearted. It gives a mountain-top view of the research area which is Calabi-Yau theory and its application to String Theory. One never forgets however how much inaccessible mathematics and physics lies behind Steve Nadis's persuasive and fluent writing.
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4 of 4 people found the following review helpful
4.0 out of 5 stars I wish I was clever, 29 Jun 2011
By 
John Ferngrove (Hants UK) - See all my reviews
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As someone who did an astrophysics degree in the late seventies, I've tried over the years to stay in touch with developments in cosmology by reading the odd popular science book now and again. Inevitably, since the string theory revolution of the eighties, that has involved digesting a few of the more or less well known lay-expositions of string theory, and its associated ideas. With each such book I have been left with a dissatisfied feeling that I would like to have been given just a bit more of the relevant conceptual mathematical framework. Well, with this book I got my wish, and its fair to say that I've come away understanding about 10% of what I've read. It is apparent that my rusty undergraduate physics level maths doesn't even get you to the front door of where string theory picks up from.

Yet, despite all this, I found the book to be compelling, and I found a way of reading it that allowed me to take much of what passed on faith, and just enjoy the handful of concepts and images that I did manage to abstract from the flow. Somewhat like watching for patterns in clouds or fire. There is a surface level story of extraordinarily gifted people, mathematicians and physicists, all attending conferences, then beavering away at impossibly difficult proofs and calculations. The history of who proved or calculated what, when, thus enabling whoever else, to prove or calculate whatever came next is, to someone like me, a plausible human interest story with a certain level of excitement. But with regard to the maths, then, if I am honest, it was really just a question of hanging on by the coattails. I may vaguely be able to remember the significance of a complex Riemann surface, and I may be able to follow the simple 3D-or-less analogies that help me to conceptualise compact Kähler metrics with vanishing first Chern class. But I have no choice but to accept on faith the significance of whether an object combining these properties has a Ricci-flat metric, the Calabi conjecture, along with all the mathematical paraphernalia that the author draws upon to prove it. Likewise the ultimate utility to physics and string theory of the class of objects, the Calabi-Yau manifolds, that his proof bought to light. I cannot visualise these things, and indeed no one can. At this level there are just the equations, and the rules of their transformation into other equations, but there are no equations in this book, and even if there were, I would not be able to understand them. There is just too much background required. Still the grist of the book sparked a rich flow of insights, however trivial, to engage and, at times, delight my humble brain. Of particular interest was the description of how the controversial missing entropy might be `encoded' into black holes, through configurations of higher dimensional geometries.

So, that's what I think awaits the prospective reader. I can't conceive of having any kind of traction on this book without some level of maths background, but I don't know just how much maths you would need in order to be able to claim that you had understood everything you read. I suspect as much as would be needed to start delving into the actual equations. But, on the other hand, surprisingly, one doesn't need a complete grasp of everything in order to get a quite rewarding experience out of the book.

Finally, there is the question of what, having lifted the bonnet an inch on the mathematical nuts and bolts of string theory, do I think of string theory as a basis for progress in cosmology? Well, I think I'm slightly appalled actually. When I was studying cosmology, a theory of everything seemed just around the corner, and notions of aesthetic mathematical beauty still proved a useful guide that ultimately led to the building of the Standard Model we have today. It was thought for a long while that the final theory, when it came, would be brief and breathtakingly elegant. I'm sure that to mathematicians Calabi-Yau manifolds and their geometric ilk are beautiful things. But, that such a huge amount of mathematical apparatus should be required to describe the geometry of reality seems rather unpalatable and intuitively unlikely. The hallowed relation between truth and beauty seems to have left by the back door. Still, who am I to judge such esoteric matters?

P.S. Congratulations to reviewer Nigel Seel for his superb outline summary of the book's main argument.
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3 of 3 people found the following review helpful
5.0 out of 5 stars The development of the concept of hidden spaces in string theory - A historical perspective, 7 Oct 2011
By 
Rama Rao "Rama" (Annandale, VA, USA) - See all my reviews
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This is a fascinating story about the development of the mathematical concept of extra spatial dimensions known as Calabi-Yau spaces and its application in the string theory. The author speaks candidly, and describes his excitement at emerging new ideas in physics and mathematics, and how it progressed in string theory, and in the process changed his perspectives. Over the last 35 years this idea has shaped our thought on the nature of physical reality and involved an entire generation of theoretical physicists in research. This is partly autobiographical and hence makes it very interesting to read as he explains his odyssey. We get to read the contributions of leading physicists in this adventure; the growth of string theory as major force in theoretical physics. This is an outstanding book to read, but requires undergraduate level physics and strong interest in geometry.

A summary of this book is as follows: In string theory, the myriad of fundamental particle types is replaced by a single fundamental building block, a string. As the string moves through time it traces out a tube or a sheet (the two-dimensional string worldsheet), and different vibrational modes of the string represent the different particle types. The particles known in nature are bosons (integer spin) or fermions (half integer spin). By introducing supersymmetry to string theory both bosons and fermions could be accounted for, and with ten-dimensions, the mathematical requirements of string theory are completely satisfied. In addition, the anomalies and inconsistencies that plagued string theory are vanished. Until superstring theory came into existence, any predictions and calculations yielded nonsensical results, and were incompatible with quantum physics. The ten-dimensions consist of two sets four-dimensional spacetime we live in, and six-spatial dimensions in a hidden state in an invisible state because they are compactified to minute size. In this geometry, every point has a six-dimensional Calabi-Yau manifold in a compactified form, thus bringing physicists to the doorsteps of Calabi-Yau geometry.

Some physicists had originally hoped that there was only one Calabi-Yau manifold that would uniquely describe the hidden dimensions of string theory, but there are a large number of such manifolds each having a distinct topology. Within each topological class there are an infinitely large number of such Calabi-Yau manifolds. The Calabi-Yau space is further complicated by the fact that it has twisting multidimensional holes (about 500) running through the space. Another problem is; what makes the six-dimensions of space stable in a compactified form? It would be like constraining an inner tube with a steel belted radial tire. Just as the tire will hold back the tube as you pump air into it. All the moduli of the Calabi-Yau, both shape moduli and size moduli needs to be consistently stabilized. Otherwise the there is nothing to keep six hidden dimensions from unwinding and becomes infinitely large. It turns out that the D-branes of string physics can curb the tiny manifold's inclination to expand.

Some physicists have considered other types of spaces besides Calabi-Yau manifolds; they include non-Kahler compactification, and some non-geometric compactification postulates. In the beginning of the book , the author states : If Einstein's relativity is proof that geometry is gravity, string theorists hope to carry that notion a good deal further by proving that geometry, perhaps in the guise of Calabi-Yau manifolds is not only gravity but physics itself." In the latter part of the book the author takes a conciliatory mode by stating "Despite my affection for Calabi-Yau manifolds - a fondness that has not been diminished over the past thirty-some years - I am trying to maintain an open mind on the subject," ............."If it turns out that non-Kahler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I'm OK with that."

There are many success stories of mathematical reasoning; one such is the prediction of positrons by Paul Dirac. The biggest shortcomings of the Calabi-Yau space and the superstring theory and brane world is even though there is beauty and elegance in the setup but it still needs to make predictions which can be confirmed by the experiments. The results of LHC experiments so far have not resulted in satisfactory conclusions.

1. The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos
2. The Fabric of the Cosmos: Space, Time and the Texture of Reality (Penguin Press Science)
3. The Little Book of String Theory (Science Essentials)
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2 of 2 people found the following review helpful
5.0 out of 5 stars What an Excellent Read!, 12 Aug 2011
By 
David E. Perkins (East Herts, UK) - See all my reviews
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Having read a couple of String Theory pro and con books by physicists, I thought this might provide me with a less partisan perspective, though I was a bit worried I might have to give up on the maths, as I did with the (admittedly excellent) 'The Road To Reality'.

But no need to worry! Rather than reams of equations and 'homework' problems this was much more of a narrative account of this mathematician's exploration of the (his) geometrical base for the theory with virtually no maths. Excellent analogies were widely used, which I found unusually helpful too. Humour, historical background, and relationships with other researchers were also liberally used, which made it a very entertaining and informative book. Although narrative, it did cover quite a lot of the more abstruse aspects, but in an approachable way.

As a collaborative effort of two writers it was seamless too, and not disjointed at all.

So it seemed a pretty unbiased account by a mathematician who had only a limited investment (though a considerable and genuine one as he one a principal developer of the maths involved) in whether the maths fitted a particular physical reality. Quite happy that the maths field had received tremendous support from string theory work, yet not heavily committed to it except as a mathematical structure. If it didn't work out there, well it might somewhere else, and in any case the maths was the main thing - no research grant dependency here!

Provided an interesting perspective on the perennial question of 'why does mathematics describe the physical world so well?' Doesn't actually answer it of course, but does give a bit of different food for thought.

I found it quite a 'foil' to the previous books I had read, and would recommend it to anyone who has a passing sympathy with the mathematical enterprise and string theory. A longish book but VERY easy to get through.
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2 of 2 people found the following review helpful
5.0 out of 5 stars THE PERFECT SHAPE, 9 July 2011
By 
C. S. Ebrey - See all my reviews
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Although this book emanates from a mathematical context, it has been co-authored by Shing-Tung Yau and Steve Nadis and impeccably offers two complementary perspectives.

In doing so it provides a fascinating and unusual account not normally described in many general physics books extolling the viewpoint of the mathematician over that of the physicist, highlighting the fundamental differences in approach each takes to similar physical problems and how mathematics manages to overcome many of the obstacles faced by the physicist in explaining Quantum reality. This is incredibly interesting to discover and not something I have seen elsewhere.

However, the main premise of the book leaves you utterly breathless as both Steve Nadis and Shing-Tung Yau take us on a journey never before presented in a way that opens up the mysteries of Calabi-Yau manifolds and ten and eleven dimensional spacetime. Shing-Tung Yau describes the heartache as well as the ecstasy of discovering the intricacies of multi dimensional spaces.

There is little in the way of mathematics other than fairly straight-forward schoolboy algebra and precious little of that. However, the physical concepts he describes, textually, whilst written in plain English, are truly astounding and well worth taking the ride to discover.

This is a truly worthwhile read. Please do not be put off by others attempts to place this book in an elite league that can only be understood by physics professors. Nothing could be further from the truth. The only perquisite to enjoyment of this book is an open mind and a healthy desire to learn. Please enjoy this book for what it is, an outstanding view of quantum reality.
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7 of 9 people found the following review helpful
5.0 out of 5 stars Fabulous, 2 Dec 2010
I love this book. It is rare that somebody dares to try to popularize such an esoteric topic. I enjoyed it greatly. It gives a taste of the real mathematics and not just anecdotes about the people involved (there is some of that too, which is nice, in moderation). This book deserves to be a bestseller but I doubt the world is ready for that yet...
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6 of 8 people found the following review helpful
1.0 out of 5 stars Post docs only, 28 Jan 2011
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I also gave up - less than half way through. I am near the end of a maths degree which has included quantum theory and relativity and other than the historical side of how the theory developed I found the substance of the book incomprehensible. Don't think this is going to be like Greene, Penrose or Hawking. It's not.
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4 of 6 people found the following review helpful
5.0 out of 5 stars AN AMAZING, INTELLECTUAL TOUR DE FORCE!, 11 Dec 2010
THE SHAPE OF INNER SPACE is guaranteed to take readers places they've never been before, nor thought about before. That was certainly the case for me. Before I read this book, I had never heard of Calabi-Yau manifolds, and it had never occurred to me that someone could write an entire book on the subject--let alone a book as fascinating as this one. The authors did an exceptionally skillful job presenting complicated ideas from math and physics. I can't claim to have understood every single word, but I found the discussion totally inspiring. And the book left me with a new slant on the world--and, indeed, the universe--that I hope will stay with me for a long time. I recommend this book highly to anyone seeking a deeper grasp of nature.
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4 of 7 people found the following review helpful
1.0 out of 5 stars Not for the Lay Reader, 12 Jan 2011
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As one who received a scientific training, and has attempted to keep abreast of developments in fundamental science, even though my career did not take that path, I had hoped that this book might fill some gaps in my knowledge. Unfortunately I have been disappointed to the extent that very unusually, I gave up on it before reaching the end. I am sure that the authors would have been able to produce a reasonably accessible account of the career of the distinguished mathematician Yau, or to have produced an intelligible explanation of Calabi-Yau manifolds and their relevance to string theory. Instead they have attempted to do both, and to treat other mathematical problems as well. The result is certainly much too taxing for this lay reader. Much less significant, though irritating, is the rather low quality of production with a number of typographical errors, and the continual references to `math' which I presume to be an Americanism for the fraternity of mathematicians.
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1 of 4 people found the following review helpful
5.0 out of 5 stars AN EXHILIRATING AND CHALLENGING BOOK!, 25 Dec 2010
By 
Martin H. Court (Chicago, Illinois) - See all my reviews
Simply put, this is a sensational book. The authors expertly guide readers through some really difficult terrain concerning "extra" dimensions, string theory, geometry, and topology. Most of the subject matter was new to me, and I am amazed that, with some effort, I was able to understand this unfamiliar material quite well. I regard that as a tribute to the authors' considerable expository skills. I'm really glad a friend recommended THE SHAPE OF INNER SPACE to me. And I'm now returning the favor by recommending it to anyone interested in our universe and the possibility of there being higher dimensions that might control everything, behind the scenes as it were. If you stick with this book, as I did, you will be amply rewarded.
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