Brett McInnes' Official NUS [National University of Singapore] Webpage

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Sorry, I don’t have any notes of my own on bifurcation theory, but I found the following web page extremely helpful:

http://www.atm.ox.ac.uk/user/read/chaos/lect3.pdf

How to run a business and get rich
Very nice graphing software....when in doubt, graph it!
Integrals Explained ODEs Explained Fourier explained Linear Algebra in Action!
 Need Some Exercise?

Need some more exercise ? [MA1506 ]
A Very Nice Webpage With Lots of Helpful Calculators and Stuff

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ACADEMIC THINGS
Preprints on Arxiv
SINGAPORE'S FIRST TWO HIGH ENERGY THEORY TOPCITE PAPERS---ONE NOW WITH 150+ CITATIONS!
SPIRES
The Full List:
1.    Brinkman's theorem in general relativity and non-Riemannian field theories. Gen. Relativity Gravitation 12 (1980), 767.
2.    [with Klotz, A. H.] On the meaning of the metric hypothesis in the nonsymmetric unified field theory. Acta Phys. Polon. B 11 (1980), 345.
3.    An identification of the metric tensor of the nonsymmetric unified field theory. J. Phys. A 13 (1980), no. 12, 3657.
4.    A note on Browne's "source function" in five-dimensional relativity. Gen. Relativity Gravitation 13 (1981), no. 5, 495.
5.    [with Klotz, A. H.] The nonsymmetron. Acta Phys. Polon. B 13 (1982), no. 3, 181.
6.    A note on the geometry of connections in gauge theories of the electroweak interaction. J. Phys. A 15 (1982), no. 11, 3623.
7.    A New Approach to Symmetry Breaking, J. Phys. G, 8 (1982) 609.
8.    Geometric Symmetry Breaking and the Weinberg-Salam Model, J. Phys. G, 8 (1982) 621.
9.    Remarks on the geometric interpretation of superconductive flux quantisation. J. Phys. A 17 (1984), no. 16, 3101.
10.  On the nonexistence of 't Hooft-Polyakov magnetic monopoles in  nonsemisimple electroweak gauge theories. J. Phys. A 17 (1984), no. 17, 3287.
11.  On the affine approach to Riemann-Cartan space-time geometry. Classical Quantum Gravity 1 (1984), no. 2, 115.
12.  On the geometrical interpretation of "nonsymmetric" space-time field structures.    Classical Quantum Gravity 1 (1984), no. 2, 105.
13.  On the significance of the compatibility condition in gauge theories of  the Poincare group. Classical Quantum Gravity 1 (1984), no. 1, 1.
14.  Spontaneous Compactification and Singularities in Inner Space, Physics Letters B, 150 (1985), 113.
15.  Global Aspects of Spontaneous Compactification, Classical and Quantum Gravity, 2 (1985), 661.
16.   Spontaneous compactification and Ricci-flat manifolds with torsion. J. Math. Phys. 27 (1986), no. 8, 2029.
17.   On the Splitting Problem in Higher-Dimensional Field Theories, CQG 4 (1987), 329.
18.   Spontaneous splitting and internal isometries of superstring vacua. J. Math. Phys. 28 (1987), no. 11, 2564.
19.   Superstrings and holonomy groups of Kaehler manifolds. Classical Quantum Gravity 5 (1988), no. 4, 561.
20.   Group-theoretic aspects of the Hosotani mechanism. J. Phys. A 22 (1989), no. 13, 2309.
21.   The geometry of gauge symmetry breaking in the superstring context. J. Math. Phys. 30 (1989), no. 2, 498.
22.   Hosotani Breaking of E_6 to a Subgroup of Rank Five, Journal of Mathematical Physics 31 (1990), 2094.
23.   Methods of Holonomy Theory for Ricci-Flat Riemannian Manifolds, JMP 32 (1991), 888.
24.   Gauge Theory in Witten's Approach to the Generation Problem, Communications in Mathematical Physics, 138 (1991), 107-136.
25.   The Quotient Construction for a Class of Compact Einstein Manifolds, Communications in Mathematical Physics, 154 (1993), 307.
26.   Holonomy Groups of Compact Riemannian Manifolds: A Classification in Dimensions up to Ten, Journal of Mathematical Physics 34 (1993), 4273.
27.   Examples of Einstein Manifolds with All Possible Holonomy Groups in Dimensions Less than Seven, Journal of  Mathematical Physics 34 (1993), 4287.
28.   Complex Symplectic Geometry and Compact Locally HyperKaehlerian Manifolds, Journal of Mathematical Physics 34 (1993), 4857.
29.   Gauging discrete symmetries. J. Math. Phys. 36 (1995), no. 10, 5414.
30.   Holonomy Groups and Holonomy Representations, Journal of  Mathematical Physics 36 (1995), 4450.
31.   Multiple spin structures in higher-dimensional physics. Classical Quantum Gravity 13 (1996), 3175.
32.   Calabi-Yau Compactifications and the Global Structure of the Standard Group,  Journal of  Mathematical Physics 37 (1996), 493.
33.  On Space Forms of Grassmann Manifolds, Manuscripta Mathematica 93 (1997), 205.
34.  Obtaining Holonomy From Curvature, Journal of Physics A 30 (1997), 661.
35.  Alice Universes, Classical and Quantum Gravity 14 (1997), 2527.
36.  Disconnected forms of the standard group. J. Math. Phys. 38 (1997), no. 8, 4354-4362.
37.  Existence of parallel spinors on nonsimply connected Riemannian  manifolds. J. Math. Phys. 39 (1998), no. 4, 2362-2366.
38.  Methods of Alice Physics, Journal of Physics A 31 (1998), 3607.
39.  Metric symmetries and spin asymmetries of Ricci-flat Riemannian manifolds. J. Math. Phys. 40 (1999), no. 3, 1255-1267.
40. The Semispin Groups in String Theory,  J.Math.Phys. 40 (1999) 4699, see http://arxiv.org/abs/hep-th/9906059
41. Spin Holonomy of Einstein Manifolds, Communications in Mathematical Physics 203 (1999) 349.
42. Gauge Spinors and String Duality, Nucl.Phys. B577 (2000) 439, see http://arxiv.org/abs/hep-th/9910100
43. AdS/CFT for Non-Boundary Manifolds, JHEP 0005 (2000) 025, see http://arxiv.org/abs/hep-th/0003291
44. The Topology of the AdS/CFT/Randall-Sundrum Complementarity, Nucl.Phys. B602 (2001) 132, see  http://arxiv.org/abs/hep-th/0009087
45. Topologically Induced Instability in String Theory, JHEP 0103 (2001) 031, see http://arxiv.org/abs/hep-th/0101136
46. A Positive Cosmological Constant in String Theory Through AdS/CFT Wormholes, Nucl.Phys. B609 (2001) 325, see  http://arxiv.org/abs/hep-th/0105151
47. Exploring the Similarities of the dS/CFT and AdS/CFT Correspondences,  Nucl.Phys. B627 (2002) 311, see      http://arxiv.org/abs/hep-th/0110062
48. The dS/CFT Correspondence and the Big Smash,  JHEP 0208 (2002) 029, see http://arxiv.org/abs/hep-th/0112066
49. Stringy Instability of Topologically Non-Trivial AdS Black Holes and of Asymptotically deSitter S-brane Spacetimes,  Nucl. Phys. B660 (2003) 373,  see http://arxiv.org/abs/hep-th/0205103
50. What if w < -1?, page 265 of  the proceedings of the XVIIIth Colloquium of the Institut d'Astrophysique de Paris, July 2002, "On the Nature of Dark Energy",
        edited by P. Brax, J. Martin, J.-P. Uzan, see http://arxiv.org/abs/astro-ph/0210321
51. The Covariant Entropy Bound, Brane Cosmology, and The Null Energy Condition, JHEP 0212(2002)053, see http://arxiv.org/abs/hep-th/0212014
52. The Strong Energy Condition and the S-Brane Singularity Problem, JHEP 0306 (2003) 043,  see http://arXiv.org/abs/hep-th/0305107
53. de Sitter and Schwarzschild-de Sitter According to Schwarzschild and de Sitter,  JHEP 0309 (2003) 009,  see  http://arxiv.org/abs/hep-th/0308022
54. Orbifold Physics and de Sitter Spacetime, Nucl Phys B 692  (2004),  270-300, see  http://arxiv.org/abs/hep-th/0311055
55. APS Instability and the Topology of the Brane-World, Physics Letters B 593 (2004) 10, http://arxiv.org/abs/hep-th/0401035
56. Quintessential Maldacena-Maoz Cosmologies, J High Energy Physics 0404(2004)036, http://arxiv.org/abs/hep-th/0403104
57. Answering a Basic Objection to Bang/Crunch Holography, JHEP 0410(2004)018,  http://arxiv.org/abs/hep-th/0407189
58. Inflation, Large Branes, and the Shape of Space, Nucl.Phys. B709 (2005) 213-240,  http://arxiv.org/abs/hep-th/0410115
59. The Phantom Divide in String Gas Cosmology,  Nucl.Phys. B718 (2005) 55-82,  http://arxiv.org/abs/hep-th/0502209
60. The Most Probable Size of the Universe, Nucl.Phys. B730 (2005) 50-81,  http://arxiv.org/abs/hep-th/0509035.
61. Pre-Inflationary Spacetime in String Cosmology,  Nuclear Physics B748 (2006) 309-332, http://arxiv.org/abs/hep-th/0511227
62. The Geometry of The Entropic Principle and the Shape of the Universe,  JHEP10(2006)029,  http://arxiv.org/abs/hep-th/0604150

63. Unitarity at Infinity and Topological Holography,  Nuclear Physics B754 (2006) 91-106,  http://arxiv.org/abs/hep-th/0606068
64. Inaccessible Singularities in Toral Cosmology,  Class. Quantum Grav. 24 (2007) 1605-1613 http://arxiv.org/abs/gr-qc/0611101
65. Arrow of  Time In String Theory, Nuclear Physics B782 (2007) 1-25,  http://arxiv.org/abs/hep-th/0611088
66. Initial Conditions for Bubble Universes, Physical Review D77 (2008) 123530-123545, http://www.arxiv.org/abs/0705.4141
67. The Arrow of Time in the Landscape, R. Vaas (ed.): Beyond the Big Bang. Springer: Heidelberg 2009, http://arxiv.org/abs/0711.1656 

68. Black Hole Final State Conspiracies, Nucl. Phys. B807 (2009) 33-55, http://arXiv.org/abs/0806.3818

69. Horizon Complementarity and Casimir Violations of the Null Energy Condition, http://arxiv.org/abs/0811.4465

70. Instability of Near-Extremal Black Holes Dual to Strongly Coupled Field Theories on Flat Spacetime, http://arxiv.org/abs/0905.1180

 I was an invited speaker at the following conference:

INITIAL CONDITIONS IN COSMOLOGY

The conference was held in beautiful Wuerzburg, Germany. The talk was based on Paper 65, above.

I was invited to Cambridge

to speak at this: The Very Early Universe 25 Years On , celebrating the 25th anniversary of the famous Nuffield conference at which the foundations of the theory of Inflation were laid.....and it was a great conference! See the PROGRAMME page for the slides of the talks, including mine. I was one of only two speakers from Asia.

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CURRENT RESEARCH INTERESTS:

I'm currently mainly interested in Cosmology and Black Hole Theory.

The most sensational scientific discovery in the last 40 years [at least!] is the finding that the expansion of our Universe is not slowing down, as you might expect, but rather *speeding up*---accelerating!  One of the reasons this is so exciting is that it means that we do not understand  what most of the Universe is made of!  When I say "most", I really mean it --- see this pie chart:  The coloured sections represent the fraction of the contents of the Universe corresponding to three classes of stuff. The yellow stuff is ordinary matter. The red stuff is "dark matter", which is rather like ordinary matter but we can't see it and are not sure exactly what it is. The blue stuff is DARK ENERGY, which is some kind of EXTREMELY strange stuff utterly unlike ordinary matter. For example, its internal pressure is gigantic compared to its density --- and negative!  This stuff nevertheless permeates the entire Universe, including the room in which you, gentle reader, are now sitting; it constitutes the ultimate invasion of privacy. [You don't feel it, for a variety of interesting reasons: one is that its density is incredibly small, about 5  x 10^{-30} times that of water! Another is that it is extremely uniform, so it varies, if at all, only very slightly over distance scales which we can experience.]  The yellow stuff includes everything you see around you. You see how thoroughly insignificant all that really is. 

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The rest of this section is about things I have worked on and am still interested in.

• HOW COLD CAN A BLACK HOLE GET?

And who cares, you ask. Well, there is a special kind of black hole that can [apparently] be made to reach absolute zero! These are black holes in “anti-de Sitter spacetime”; all you have to do is to throw charged particles into them and they get colder and colder. Now this spacetime is nothing like our own spacetime, but it is nowadays a subject of intense interest in String Theory. String theory works exceptionally well in AdS spacetimes. It turns out that, in string theory, there is an equivalence between a gravitational theory in AdS and a *non-gravitational* theory living in a space of one dimension lower, which is attached as a sort of boundary, as in the picture below:

This means that one can use facts about *gravity in five dimensions* to learn things about other theories in four dimensions, where we live. So you can think of us living on the boundary in the picture, but using the five-dimensional interior to understand physics in our world. What sort of physics? Well, one branch of physics which is not yet fully understood is the theory of the strong interaction, the force that holds quarks together inside protons, neutrons, etc. It turns out that when you squash protons and neutrons together at fantastically high pressures, they “melt” into a state in which the quarks are no longer bound together. This may be what happens in the core of a neutron star, which is what is left over after many supernovae [see the other picture]. These objects are so dense that even a part of the core the size of a 10-cent coin would be much more massive than the entire suburb where you live; the cores of these things have the highest pressures in the Universe…. It is not really understood what happens to quarks under such pressures. But AdS spacetimes give us a way to study this. You just have to put a black hole into the five-dimensional space and charge it up. But that means, as we said, making it very cold. So that’s why we care about very cold black holes in five dimensions --- they may tell us about what goes in in the cores of neutron stars in *our* universe! This is a fascinating example of mathematics being developed for one purpose finding a use in a totally different, completely unexpected direction.

My work shows that in fact if you try to make this particular kind of black hole too cold, it just disintegrates. So there is a limit to how cold it can be. On the neutron star side of the equivalence, this means that something strange must happen to quark matter at low temperatures but with very high values of something called the “chemical potential.” This fits in with earlier guesses about what happens under these extreme conditions. See paper 70 above.

• IS THE UNIVERSE FINITE?

This is a very old question and it is one that I can't seem to get away from. My recent work on black holes made me aware of some general principles [``horizon complementarity"] which have been discussed in relation to black holes but which may also be relevant to cosmology. The problem with ``general principles", of course, is that it is hard to extract quantitative information from them. The work reported in paper 69 does exactly that, and the numbers are looking extremely interesting. I have two ways of bounding a certain parameter; the two are  completely different. One of them based on the detailed geometry of a certain remarkable manifold [the ``Weeks manifold"], and the other is based on a detailed study of the stability of certain ``branes". Amazingly, the two procedures agree, at the level of a few percent!

The basic idea is that, in a Universe which accelerates forever [as ours might well do], we are cut off from all possible communication with most of the Universe. If someone too far away sends you a message, it will *never* arrive, no matter how long you wait! “Horizon complementarity” is the idea that those parts of the Universe which are cut off in that way are not needed to describe the physics in our part of the world. You can get some idea of what this means by looking at this famous Escher picture: the space it portrays is infinite [you have to imagine each bat or angel as being exactly the same size as all the others…] but clearly once you understand how things fit together at the centre of the picture, then you understand the whole thing --- there is nothing new as you move outward. The same goes for cows, etc etc etc.

 In view of that, you might as well declare that the outer parts of the diagram are just identical copies of the part at the centre: physicists say that you have “compactified”. But if you do that kind of thing with physical fields, strange quantum-mechanical effects arise --- something called the “Casimir effect”. This effectively means that you have *negative energy densities* in your spacetime, and that can lead to instabilities if you have too much of it. But the amount you have depends on the way you compactify. So you have a sort of balancing act to do, and things work out properly only if you choose your parameters in just the right way. That’s what 69 is about.

• THE BLACK HOLE INFORMATION PROBLEM

When things fall into black holes, the information they contain is apparently lost. That may not seem to be a problem, but in fact it leads to one of the deep mysteries in physics, one that has defeated some of the best minds: the "Black Hole Information Loss Paradox."  I don't know how to solve it either, but I have an idea that might make a small contribution towards solving it. See paper 68, above. The basic idea is due to Horowitz and Maldacena, who proposed that the "lost" information is actually saved by means of a special version of "quantum teleportation".

It's a brilliant idea, and in my opinion the most likely solution of the puzzle. But it is generally thought that there are serious problems with it. Paper 68 points out that the problems may not be as bad as people think, if you are willing to accept the possibility of pretty weird behaviour inside certain kinds of black holes [in Anti-de Sitter spacetime]. My theory of the arrow of time predicts exactly that kind of behaviour......and the paper appeared in Nuclear Physics B. By the way, Nuclear Physics B is an outstanding journal and it is extremely difficult to get papers accepted by them.

• THE ARROW OF TIME IN THE LANDSCAPE

While I was in Wuerzburg in September 2007, I was invited to contribute a chapter to a book, to be published by Springer Verlag  in 2009, titled "Beyond the Big Bang". Writing that chapter gave me an opportunity to think through a lot of things which I had previously been unclear about, so I am very grateful for the invitation. See paper 67 above. The upshot is the following picture. Our Universe is believed by many to be a baby universe, which split off from some larger, older universe. There could be a lot of babies around, all with different properties. This is like the different micro-environments in a landscape here on earth, as in this beautiful landscape by Pieter Bruegel [the elder]:

String theory does allow for a very large variety of possible universes, brought into existence by the birth of baby universes. This ensemble of many universes is called the String Landscape. Now while the Landscape allows for a large number of possibilities, that doesn't mean that "anything goes". The extremely rich chemistry of carbon allows for a fantastic number of carbon-based objects [like us] to be constructed, but nobody would use this as evidence that organic chemistry is a waste of time because it allows *anything* to be built out of carbon compounds. We probably need a "large" Landscape to account for the famous Cosmological Constant, but for some purposes the Landscape is actually very very small. One of those purposes is the problem of accounting for the Arrow of Time, the fact that time seems to pass, resulting in a future which differs from the past. It was shown a long time ago by Roger Penrose that our Universe is ``special" --- that is, its initial conditions were somehow selected fantastically accurately --- so accurately that there is no hope whatever of finding a universe like ours in the Landscape by mere chance. I argue that the origin of this specialness is that the ``mother universe" of our Universe was even more special, and her mother was

more special still.....and so on....right back to the Mother of All Universes, which I call Eve [Seen here courtesy of Albrecht Duerer and Lukas Cranach.]. So the problem is to understand the extreme specialness of Eve. Here "special" means "incredibly smooth". Using recent developments in string theory, I argue that Eve was "born" when time itself emerged from some purely stringy state which itself was timeless. A subtle argument [discussed below] shows that Eve had to be born in a state of extreme spatial smoothness. This is supposed to explain the existence of an Arrow of Time in Eve, and thus in at least some of her descendants. The overall picture of the Landscape is then like an ancestral tree, with babies branching off Eve, and inheriting an Arrow. Some of the descendants, by the way, will die [if their vacuum energy becomes negative] but Eve herself may well be immortal, as promised by you-know-who, pictured above.

• BAD BABIES

 Babies are bad.......  ....see paper 66 above.
Well, not that kind  of baby actually, but rather baby *universes*. It's possible for a tiny region of our Universe to split off  and go its own way. Thus even if you start with one universe, you might end up with lots of them. Some people like this idea, because it may give you a way of explaining the value of the cosmological constant: each time a baby splits off, the cosmological constant decreases, so eventually you might arrive at the absurdly small value we observe today. The question, however, is whether a baby universe can ever look like *our* universe --- if it can't, then the possible existence of babies is of no concern to us. Now [see below] *our* Universe began in an extremely special way --- with a very nice smooth geometry. That is extremely hard to organise, especially for a baby universe. For baby universes start out small, which may mean that whatever goes into making them may get squashed when the baby is born. Squashed things, however, are rarely nice and smooth. The simplest way to avoid this problem is to start out with a big universe that is *already* very smooth, and try to arrange things so that the smoothness is "inherited" by the baby. Paper 66 is concerned with this very tricky question of inheritance. It turns out that string theory *may* allow for the existence of a very exotic form of matter which does permit inheritance [by forbidding the baby to be too small]. So now we have passed the buck back to the mother universe: the baby can be smooth if the mother was smooth. But how can the mother be smooth in the first place? That was explained in paper 65.......see below. [By the way, saying that the baby is "small" when it is born is slightly misleading....actually the baby is indeed very small when seen from the *outside*, and it is true that to get inside it from the outside you would have to be squashed. But the baby itself is actually *large* when viewed from the inside --- in fact, it is *infinitely* large inside! Strange but true.] [Another note: some people object to the name "baby universe" for the things I describe as such. They want to confine this term to apply to a different kind of universe which [maybe] can split off and then remain completely inaccessible to the mother universe. I must say that these people have  very peculiar notions regarding the raising of children. One lives in hope, at least, that children can be influenced by their parents long after they are born. In Asia, in fact, children remain firmly under the influence of their parents for an indefinite period. Similarly, the point I want to make is that "baby" universes are permanently exposed to influences from their mother.]

• WHY DOES TIME PASS?  

Another one of those questions that seemingly have no scientific answers, yet some cosmologists do have ideas about it! [See also the very useful AOT FAQ .] And so do I, and I will write about that when I can find the......you know.  Well, it seems that that  time has arrived. Everyone knows that  it is easier to make a mess than to clear it up. But if that is the case, then why do we ever see any situation that ISN'T a mess? It's easier to break an egg that to put it back together....but then, why do we ever see unbroken eggs? This may seem to be a strange question, but, if you think about it, you will see that it is at the core of one of our most basic experiences...the experience of time passing. We know that time passes precisely because things break and almost never unbreak. And it's easy to see why: there are more ways to have a broken egg than to have an unbroken one! Messed-up situations simply outnumber non-messy situations. In physics terms, this is why we have the Second Law of Thermodynamics: entropy [nearly] always increases in a closed system, simply because "special" arrangements are likely to evolve into "generic" ones; the reverse, while possible, is highly unlikely:

So it's actually easy to understand why time *passes*. The catch is that, by *exactly the same argument*, it's extremely tough to understand WHY the world was less messed-up in the past. By the way, in case you are worrying that "messed up" is in the eye of the beholder, if you go to this webpage Molecules and play with the applet there, you will get an idea of what I really mean by this. Alternatively, look at this [but turn off the "music"!]

Anyway, the point is that we recognise the past by the fact that it was less messed up. The question is why was the past so nice and non-messy. This is an extremely tough problem. Eventually you can trace it right back to the very beginning of time: the geometry of space at that time was fantastically nice and smooth. It didn't have to be: it could have been a weirdly distorted, horrible three-dimensional lump. But of all the infinitely many possible initial geometries, our Universe chose something smoother and more regular than anything that has ever existed since then. Why? This is the Question of Questions: we will understand why time passes only if we understand HOW THE UNIVERSE BEGAN!

One possibility is that if, when the Universe is born,  space is already infinitely big [this *is* possible!], then everything that can happen must happen somewhere, so just by chance there was a smooth patch somewhere, and that small bit of space grew into the observable Universe. Personally I think that this kind of argumentation is sheer nonsense. You could equally well argue that all the so-called "laws" of nature are an illusion: in an infinitely large Universe, everything that can happen will happen, so all of the lawful behaviour we see around us, and that has been going on for billions of years, is JUST A COINCIDENCE, something that is bound to happen in an infinitely large world. This sort of argument is so lame that I adduce it as evidence that the Universe *cannot* be infinite. Jokes aside, though, can one do better? Can one put together an argument that says in effect that the Universe HAD to be born in an ultra-smooth state? I tried to do that in Paper 65 above. Basically the argument is very simple, though the details aren't. It turns out that the initial value problem for General Relativity has an unusual structure,

[see http://lombok.demon.co.uk/mathToolkit/home.svc for the nifty gadget that converts tex to gif]   and in order for it to be internally consistent, the scalar curvature of the initial three-dimensional space has to be related to the total energy density at that time. This is well-known, but what is not well-known to most physicists is that imposing conditions on the scalar curvature can, if the topology of the space is right, actually constrain the geometry in an amazingly drastic way. I argue that these amazingly drastic restrictions on the initial geometry are ultimately responsible for the Arrow of Time. The theory implies, by the way, that time does not reverse inside a black hole, as some have argued it might. The key point is that the Arrow of Time is ultimately a global, topological effect.

• DOES STRING THEORY REALLY MAKE COSMOLOGICAL SINGULARITIES GO AWAY? Well, I think it probably does, actually. String theory is supposed to be the fundamental theory governing the Universe. Classical General Relativity predicts that the Universe must have started as a geometric singularity, and it is generally believed or hoped that string theory makes spacetime smooth enough to avoid this unpleasant conclusion. I think that that is probably  what does happen. But it is still worth thinking about other possibilities, just to see whether there is really no alternative. So  in paper 64 above, I suggested a new way of dealing with this situation: instead of making the initial singularity go away, I suggest that string theory just makes it irrelevant! It is still there, but it cannot be probed by anything that has definite properties -- a definite charge, orientation, etc. Central to this claim is a simple calculation I do for Spatially Toral de Sitter spacetime --- this is a spacetime containing nothing but dark energy, where the spatial sections are copies of a three-dimensional torus. The calculation concerns the strange story of a pair of  twins 

who are born with the Universe --- that is, they enter the Universe simultaneously at the initial singularity [apologies to the mother of the twins]. One of them is stationary, the other moves. Then we examine them much later, and find to our dismay that their ages are different. This is a famous observation that everyone learns when they are taught Special Relativity. What is spectacularly different here is that the twins not only have different ages: the one who stays home is infinitely older than the traveller! The travelling twin has discovered the ultimate anti-aging product. In order for this to work, however, she has to circumnavigate the Universe infinitely many times in what [to her] is a finite amount of time. That may sound impossible, but in fact it is almost inevitable: remember that, as we move back in time, the torus is shrinking, and in fact it shrinks so fast that it can be circumnavigated infinitely many times in a finite amount of proper time! [If you try to trace back without doing all those circumnavigations, you will find that the Universe is infinitely old. Actually, however, that is very difficult to do: the torus gets so small that the slightest motion zips you around it infinitely many times.] So the singularity is there, but it can only be reached in a finite amount of time if you are willing to travel right around the Universe a literally infinite number of times. The question is whether an actual physical object can really do that. I argue that it can't, for reasons that will become clearer when you read the next paragraph. So the singularity does exist, but it is physically irrelevant. I also try to argue that this conclusion remains valid even if the early Universe is dominated by [tachyonic] closed strings. See paper 64 for the details.

• CAN YOU PERFORM A DISCRETE OPERATION AN INFINITE NUMBER OF TIMES IN A FINITE AMOUNT OF TIME?? Consider the following situation: you take a lightbulb and turn it on for 1/2 an hour. Then you turn it off for 1/4 hour. Then back on again for 1/8 hour. And so on. Question: after exactly one hour, is the light on or off? The point is that after one hour you will have turned the light on and off a literally infinite number of times. The question has no answer: it must simply be impossible to perform such an experiment. Of course we can say that the bulb will break, that our hand would have to move faster than the speed of light towards the end of the hour, etc, and these things are true; but the interesting point is that we can deduce the impossibility of performing the experiment without knowing anything about the physics of lightbulbs! The argument is logical rather than physical. Anyway, the conclusion is that it is completely impossible to perform a binary operation an infinite number of times in a finite amount of time.  So what? Well,  suppose that our universe has non-orientable spatial sections: for example, suppose that the space in which we live is the product of a circle with a flat Klein bottle.  This is of course related to the well-known Mobius strip.
• Now because we know that the Universe is expanding, it follows that as you go back in time the bottle will get smaller and smaller. As you probably know [and if you don't, look at that web page], moving something around a closed path on a Klein bottle can reverse its orientation --- it can turn left-handed things into right-handed things and vice versa. This is a discrete operation. Now as you go back in time, it becomes easier and easier to circumnavigate the bottle. Question: if the Universe really shrinks to zero size a finite amount of time in the past --- as classical General Relativity implies --- then will you find yourself going around that loop on the Klein bottle an infinite number of times or not? This is a tricky question in General Relativity. Paper 63 above gives the answer: if the very earliest Universe [even before Inflation, see below] contains stuff like strings, then the answer is actually yes: you will indeed flip orientation an infinite number of times in a finite amount of time [as measured by you!]. As we argued above with the lightbulb, this simply does not make sense. One possible  conclusion is that the Universe could not have been of size zero when it was born; another is that the Universe is singular, but the singularity is "physically irrelevant" as in paper 64 discussed above......of course you could argue that all this just means that our world has to be orientable, but the argument also works with other discrete symmetries like charge conjugation, which is actually what is discussed in paper 63. The basic idea explored there is that the physics of the world in the remote future actually forbids nasty things from happening in the most distant past. This is connected with the idea called holography.

• WHY DOES THE UNIVERSE HAVE TO BE A TORUS?......Some of us believe that the Universe came into existence from "nothing" [see below]. But, strange as it may seem,  you can't create just anything from "nothing"! Profs Ooguri, Vafa, and Verlinde have proposed an interesting variation on "creation from nothing", which may shed some light on this. In the traditional approach due to Hartle and Hawking, one begins with half of a [four-dimensional] sphere, ie a positively curved space. OVV start with a space of negative curvature instead --- this turns out to be much more natural in string theory. But as OVV themselves note, it is really hard to get the universe to create itself in this way. Paper 62 shows how to solve this problem --- but the solution only works if the universe was born with the topology of a torus, as below. This might allow us to derive this shape, instead of assuming it.

•  CREATING THE UNIVERSE OUT OF NOTHING, NOT EVEN NOTHING!!....The most basic question of all: where did the Universe come from? Amazingly enough, people do have scientific ideas about this! One idea that was popular over 20 years ago, and which has suddenly enjoyed a new surge of popularity very recently, is that the Universe came into existence out of nothing at all. By "nothing" I don't mean the vacuum, empty space etc: I mean literally nothing! Amazingly, Quantum Mechanics allows us to formulate this idea mathematically, and so you can work out the probability of  various different kinds of Universe coming into existence out of nothing. Until now, for technical reasons it has always been assumed that when the Universe comes into existence out of "nothing", it has the shape of a three-dimensional sphere. For reasons explained below, I believe that 3-dimensional space is more likely to be a torus or something similar. I want to set up this             

mathematical machinery  to study what properties the Universe is likely to have [its size and spacetime geometry] under the assumption that it can be born as a torus. After all, the world is flat, as you can plainly see. [This is a joke.....] A flat three-dimensional torus is what you get if you take ordinary three-dimensional space and  identify opposite faces of a cube [in the most straightforward way]. The space is flat in the sense that the local geometry is the same as ordinary geometry; however, the topology is different. You might  think that changing the topology without changing the geometry should have no physical effects, but you would be very wrong. In particular, changing from a sphere to a torus tends to make the spacetime both singular and unstable in string theory. This may sound bad, but actually it's good, because it means that string theory constrains the spacetime much more strongly in the toral case, and so we can hope to get some predictions out. See paper 61 above. Readers who think that I have gone out of my mind should notice that, if so, I am in good company: there are famous professors at Harvard  who also believe that it is worth investigating the idea that the Universe came into being out of "nothing". Prof Tye and his co-workers  at Cornell University have done  a particularly nice job of  explaining this idea.

• THE MOST PROBABLE SIZE OF THE UNIVERSE.....when the Universe erupts out of absolutely NOTHING, that is, nothing, NOT EVEN NOTHING, how big is it likely to be?  Small? How small? This is the topic of Paper #60, a major piece of work which has appeared in Nuclear Physics B. This is closely connected with the question of initial conditions for INFLATION.

.  

Well, no, not that kind of Inflation..... it is generally accepted that, very early in the history of the Universe, the expansion was incredibly fast for a short time....we say that the Universe INFLATED. This simple idea allows us to explain some otherwise very puzzling facts....for example, the fact that the Universe looks the same in opposite directions. The idea is that although the Universe is huge, the part of it we can see was originally tiny, so it is not so surprising that it looks pretty much the same in all directions. However, there are technical complications here: if you believe that the Universe simply came into existence out of NOTHING, then it is a lot easier to believe that it was born small than large. In fact, my calculations show that it should have been born at a size which is about 100 times smaller than its estimated size at the time Inflation began. But assuming that it was born nice and smooth --- and this assumption too has to be thought about --- how do we stop it from becoming lumpy during the time when it expands by that factor of 100? Here the great cosmologist Andrei Linde comes to the rescue: he points out that if the Universe is a torus and if the spacetime geometry is right, then signals can propagate all the way around the torus many times during that period, nicely maintaining the uniformity until Inflation is ready to begin. Of course the catch is that the Universe does indeed have to be born at just the right size and with just the right spacetime geometry, for otherwise signals will not be able to circulate around the torus. In Paper 60 I showed that, taking into account certain quantum-mechanical subtleties discussed by Prof Tye and his group [see above] and by means of a  geometric trick, I could actually calculate the most probable initial size and the most probable initial geometry......and they come out just exactly right! [Of course in Quantum Mechanics, probabilities are all you ever get, but in this case the "just right" answers really are a lot more probable than any alternative.] The whole calculation is probably too simplified to be completely realistic, but things come out so beautifully that it seems that this may be a real contender for a proper theory of the very earliest Universe --- even before Inflation!

• THE PHANTOM DIVIDE IN STRING GAS COSMOLOGY.   It's generally agreed that our Universe has more than the four dimensions [3 space+ 1 time] that we seem to live in: ten, or possibly eleven dimensions are the favoured numbers in string theory. Why do we seem to live in only four? Robert Brandenberger and Cumrun Vafa had a brilliant idea about this: they suggested that our Universe is shaped like a torus [see below] now in string theory you will of course have lots of strings [and other things, "branes", which are like higher-dimensional strings]  in the very early universe and you would expect some of them to wrap around the cycles of the torus and try to stop it from expanding. So in string theory there is no problem with explaining why dimensions remain small --- the problem is to understand how some of them get big! Brandenberger and Vafa noticed that because strings are 2-dimensional from a spacetime point of view, they are likely to intersect in 4 dimensions.....2+2 = 4. Thus in 3 space dimensions, all of the strings winding in one direction can destroy all of the strings winding in the opposite direction. In higher dimensions this would not work, because the strings can avoid each other. Thus we have an explanation as to why there are 3 big dimensions! There are a lot of problems with this idea but it is still the subject of a lot of  good research. In paper 59 above I discussed this cosmology and some of  its possible instabilities, and how this is relevant to some recent claims about the details of how the Universe is expanding. I do think the BV idea is basically good, though the details may be a bit more complicated than people thought at first........in particular I certainly agree with BV that cosmologies in which 3 dimensional space is a torus [more generally, a flat compact 3-manifold] are the most interesting ones from the point of view of string theory.

• ACCELERATING UNIVERSES WITH A BIG CRUNCH.  People often assume that if the Universe is expanding at an accelerating rate, then it must go on expanding forever. That may well be true, but it does not follow logically.  The acceleration could come to an end one day, and be succeeded by a period of deceleration. If one formulates this idea in the most natural way, what it means is that the underlying cosmological constant of our universe is *negative*, not positive. This would be good, because in fact negative cosmological constant spacetimes [the simplest is called *Anti-de Sitter spacetime*] are much better understood in terms of string theory than their positive-cosmological constant relatives, the de Sitter spacetimes. The latter are still generally thought to be the basic models for cosmic acceleration --- see many of the entries below --- but I think it is still worth considering this alternative. So do Juan Maldacena [IAS Princeton] and his co-worker Liat Maoz, who in January 2004 put out a preprint which outlined some cosmological models based on AdS. It is actually possible to modify the Maldacena-Maoz cosmologies so that they *accelerate*, though only temporarily. So there is a possibility that our universe could be like this. If so, then the ultimate fate of the Universe is to be squashed in a "Big Crunch", a Big Bang in reverse.  Click on the Cap'n to see a movie of this.

          Everything would collapse, including space itself, as the saying goes. That is, your (great)^n grandchildren would see a *contracting* universe, getting smaller all the time, and shrinking to "zero" size in a finite time, as shown in the diagram above.  That may seem rather sad, but there are worse possibilities [as for example a Big Smash/Rip, see below]. However, a theorem due to Witten and Yau [both of  them Fields Medal  winners, by the way] forbids this kind of behaviour under apparently physically reasonable conditions. So how did M&M manage it? Explaining this is a rather subtle exercise in global  differential geometry. Explaining how it works in the more realistic case in which the Universe accelerates is even more tricky. This is the subject of paper 56 above.

• CRUNCH = BANG? The kind of cosmology I discussed in paper 56 has a rather strange property, which was the subject of paper 57 above. In quantum gravity it is often useful to *replace* the time direction of spacetime with another *spatial* direction, so that spacetime becomes a four-dimensional space. When you do that here, you find that the space has a boundary which apparently falls into two disconnected pieces. In paper 57 I argue that, if you change the geometry in a subtle way, then this disconnected boundary [which causes serious difficulties for the physics of these spaces] can be seen to be just an illusion, the result of a bad  choice of coordinates. One way of thinking about this is to say that I am insisting that you must bend around the two ends of the space and glue them together. But then when you go back to spacetime itself [ie turn that spatial direction back into a time direction] this seems to indicate that the Big Bang and the Big Crunch are one and the same: they are the same thing seen from different sides.

• THE SHAPE OF SPACE.  As explained above, paper 57 above argues that you have to glue the two ends of the "purely spatial" version of spacetime. If you take the two ends of a cylinder and glue them together you get a doughnut-shaped object called a torus. The object shown, with a picture of the cosmic microwave background printed on it [from the WMAP satellite, see below], is, as you can clearly see, a doughnut. [This is a lame joke for all you topologists out there.] Thus printing the WMAP picture on it is actually singularly appropriate from my point of view.  As you can see, Homer Simpson actually foreshadowed this idea: 

Stephen Hawking:  "Homer, your theory of a donut-shaped universe is intriguing".

 Unfortunately, I see that Homer has the wrong value for the cosmic density parameter [Capital Omega] --- it should actually be 1, or perhaps slightly less than 1,  not greater than 1. Doh!
 However, it gets even weirder: because the matter content of this cosmology is rather strange [it has to be, to get a non-trivial topology] it turns out that the gluing has to be done in a non-trivial way. In  fact  what you really get is a sort of  four-dimensional analogue of the famous Klein Bottle. Two Klein bottles are shown above: one is a bottle, the other a hat......if you want to gain some intuition about how Klein bottles work, and at the same time drive yourself completely insane,  try playing "tic-tack-toe"    [= "noughts and crosses"] on a Klein bottle. Anyway, the point is that a Klein bottle is defined in much the same way as a torus, but you do a reflection of one end before joining them. Now I want to do this in four dimensions, starting with flat three-dimensional spaces. There are in fact 10 topologically distinct flat compact three-manifolds. One of them is the torus, and it's obvious that you can do the Klein bottle construction for it, but the other nine are more tricky [they are themselves obtained by performing various kinds of twists one dimension lower down]. It turns out that the construction won't work for most of them........ which is good, because it means that all of this may help us to find out which of the possibilities has actually been realised in our Universe! In fact, you end up with just three candidates: the torus and the ones named by the famous geometer [etc] John Conway the "dicosm" and the "didicosm". So in this roundabout way we have narrowed the topology of space down to just these three candidates. The didicosm is a particularly fascinating candidate for the shape of 3-dimensional space, and one can only hope that Nature has had the good taste to select it. All this is explained in paper #58, above.
Underlying all this is the extremely deep [and hard!]  theorem, due to Gromov and Lawson,  which says that,  no matter how you twist and turn it, you can never turn a flat space into one with positive scalar curvature: also, the only way for the scalar curvature to be zero is for the space to be *exactly* flat. In other words, flat spaces, despite their apparent  simplicity, actually have an extremely deep geometry, and it is very nice indeed that they seem to be the underlying structure both for *space* and for *spacetime*.

• THE TOPOLOGY OF DE SITTER SPACETIME. The basic spacetime corresponding to an accelerating Universe is deSitter spacetime, which is what you get if you solve the Einstein field equations [under the usual assumptions in cosmology, to wit, local isotropy and homogeneity] for a universe which contains nothing but the "dark energy".  There is in fact a fascinating ambiguity in the definition of de Sitter space, connected with its *topology*.  This has been known for a long time---in fact, de Sitter himself was well aware of it. de Sitter thought that the spatial sections of his spacetime were *not* spheres, as people think today, but rather copies of the *Real Projective Space*  RP^3,  which is what you get from a 3-dimensional sphere after topologically identifying opposite points. It seems that he got this idea from Schwarzschild [*the* Schwarzschild, of later black hole fame],  who,  writing about the possible geometries of 3-dimensional space over 100 years ago [in 1900], considered RP^3 and even flat tori, but he rejected the 3-sphere on the grounds that we should not consider something that complicated  [sic!] unless it were really necessary! [He did not like the sphere because two coplanar geodesics intersect twice instead of once, as in RP^3. Thus, the sphere forces conditions at the north pole to be the same as the those at the south pole, which does not make sense physically.] Schwarzschild lived in happier times, before the advent of the modern superstition which holds that simply connected spaces are in some mysterious way better than all other spaces......anyway I think that Schwarzschild and de Sitter were right. Certainly the structure of a black hole in RP^3 de Sitter space makes a lot more sense than the picture you get if you try to put a black hole into ordinary de Sitter space. To see what such a black hole might be like, look at this delightful picture, drawn for me by Wanmei:

This is called a *Penrose Diagram*. Here time is up and down. The wavy lines represent the singularities inside a black hole [at top] and a white hole [at bottom]. The stars at right are copies of the *TWO*-dimensional real projective space. [The ones on the left are in another universe, which, sadly, we can never visit, because the singularity will tear you to pieces if you try to go there!]  If you try to travel through the RP^2 on the right, with your back to the black hole,  you will immediately re-emerge [still on the right] but now *facing* the black hole....and you will be upside-down! [You can see this if you remember the definition of RP^2 and if you have a good visual imagination.] Of course, to you, it will seem that the world has suddenly turned upside-down! This is portrayed in  the following diagram,  in which the images represent the cosmic microwave background radiation [as seen by NASA's WMAP satellite, see below] and in which Felix [see http://www.everwonder.com/david/felixthecat/about.html ] is a cosmological explorer in a topologically non-trivial de Sitter spacetime. [If your screen is narrow, you may not see all three images on the same line---imagine it! ]   If you look closely you will see that the image on the right is upside-down, as above. In reality this picture is not quite right, because poor Felix would never be able to return home---the acceleration of the Universe implies that the Universe is getting  larger faster than he can travel! See paper 53 above for details, including references to the fascinating history of  "de  Sitter spacetime". This paper was accepted, with laudable punctuality, by JHEP,  one of the two top theoretical/mathematical physics journals.

• BRANE WORLDS and DESTROYING THE UNIVERSE!  As I said, I think that de Sitter and Schwarzschild were right --- the spatial sections of  de Sitter spacetime are copies of  RP^3,  not the sphere S^3. Another way of approaching this is to think of de Sitter space as a BRANE [short for membrane] in a higher-dimensional spacetime. See the picture below. A lot of people nowadays are working on this idea --- that our universe floats as a sort of   [four-dimensional!]  "surface" in a five-dimensional spacetime. There are lots of advantages to doing so --- for example, Neil Turok of Cambridge and others have pictured the Big Bang as a collision of two "parallel universes" in five-dimensional space!  Now from a theoretical point of view, it is interesting to do this because it imposes some discipline on people like me who want to play with the topology of  de Sitter space. For if you embed four-dimensional  de Sitter space in five-dimensional  ANTI-de Sitter space --- which is the negatively curved version of de Sitter space --- then you cannot modify de Sitter topology without doing something drastic to the larger space. It turns out that string theory implies that, if you try to modify the geometry of the five-dimensional space in which our four-dimensional universe sits, and if you make changes which are too drastic [eg, if you make spikes like the one at the tip of a cone] then you can generate a violent disturbance of the geometry  which can spread through the fifth dimension, and which can DESTROY OUR UNIVERSE [I just love that idea!]. So maybe it is not such a good idea to fool around too much with the topology of our world --- in other words, maybe we can use string theory to *deduce* what the topology of the universe is by requiring that it must not be destroyed by a tidal wave of geometric distortion propagating through the fifth dimension. The relevant mathematics here is an old theorem due to Elie Cartan, which says essentially that a finite group acting on the spatial sections of anti-de Sitter has to have a fixed point, even if the geometry is deformed a little. See paper 54 above, which has finally been completed after much confusion on my part. It's a complicated one!  Nevertheless it has been published  in Nuclear Physics B, the other top theoretical physics journal.

• BRANE WORLDS AND THE DODECAHEDRAL UNIVERSE.   Last year, the justly famed WMAP satellite [see below for a picture] reported results about the anisotropies in the cosmic background radiation. Much of that, though extremely important, was expected. But some of it wasn't; in particular, some of the data related to the largest angular scales [say, in two directions in the sky more than 60 degrees apart] was very peculiar....because of what didn't seem to be there! Roughly speaking, the very largest "sound waves" seemed to be absent. [I should stress that at this moment, it is very controversial whether these waves are really missing.]  In October 2003, a group of people including the well-known mathematician Jeffrey Weeks [one of the recipients of the famous "genius awards" given by the MacArthur Foundation] suggested a truly beautiful and exciting explanation of the low power of the quadrupole: they suggested that the really big waves could not be there because the Universe was too small to hold them! Better yet, they suggested a specific geometric model for such a "small universe", namely the  *Poincare homology sphere*.                                                                           

 This is what you get if you take a  dodecahedron, like the ones shown here, and then imagine that if you try to escape from it, you instantly find yourself back inside, but rotated through an angle of 36 degrees! [This is just a more complicated version of the experiences of Felix the Cat, above.See http://www.nature.com/nsu/031006/031006-8.html .  Unfortunately, subsequent analyses of the data have shown that this wonderful model probably isn't true. But *why* did Nature miss out on this  golden opportunity to use one of the most beautiful of all spaces? Recently [see paper 55 above] I tried to explain this using the DESTROYING THE UNIVERSE idea described above. You can easily modify de Sitter spacetime so that it has the Poincare space as spatial sections, and it is not too hard to extend this into the fifth dimension in the way I described above. The question is whether this triggers off a disturbance in the five-dimensional spacetime that might destroy our world. To determine this, you have to work out whether certain symmetries of anti-de Sitter spacetime are preserved or not. The group that defines the Poincare space --- it is called the *binary icosahedral group* --- is quite a lot more complicated than the ones I discussed in paper 54, so I had to develop some new method to handle it. The first thing to do is to simplify the problem. It's clear from the picture that the dodecahedron has a huge group of symmetries --- in fact it has 60 [rotational] symmetries. *But*  if you go to   http://www.toonz.com/personal/todesco/java/polyhedra/dodecahedron_tetrahedron.html  you will find a beautiful picture of a tetrahedron inside a dodecahedron, and if you play with the applet you can easily convince yourself that all of the symmetries of the tetrahedron are also symmetries of the dodecahedron. Using this, and using something called "quaternions", see http://mathworld.wolfram.com/Quaternion.html , I showed that certain symmetries of the tetrahedron break all of the relevant symmetries in anti-de Sitter spacetime, and therefore the same must be true of the dodecahedral symmetry group.  And combining all this stuff, that's why we can't live in the Poincare homology sphere!  That is, if you try to put that kind of  Universe into a suitably modified version of  anti-de Sitter spacetime, then string theory tells you that the situation is unstable and your carefully constructed Universe self-destructs --- and you wouldn't be sitting there reading this....intriguingly, however, the details of the argument suggest that we *might* live in a simpler kind of version of the sphere, called a *lens space*. In such a space, Felix might see something like this on his return from his voyage:                                                                                                                                                                                                                                                                                                                         

Stay tuned for the next episode.....meanwhile, a compressed version of the paper I wrote about this has been published in Physics Letters B [see paper 55 above].    

• BRANE WORLDS AND RELATIONS WITH THE AdS/CFT CORRESPONDENCE.   

  The Escher woodcut [above; thanks to http://sunsite.icm.edu.pl/cjackson/escher/index.html] shows the hyperbolic plane, a "Euclidean" version of Anti-deSitter space [which is a maximally symmetric spacetime of constant negative curvature]; it is clear that it is very natural to regard this space as the interior of a manifold-with-boundary, the boundary [which is infinitely far away from an internal point of view] having the topology of a circle. [Note: in physics, "Euclidean" means "with a positive [or perhaps negative] definite metric", as opposed to the Lorentzian signature of  ordinary spacetime. Thus "Euclidean" spacetimes are in fact usually curved!]  This works in higher dimensions too: an (n+1) dimensional hyperbolic space has a natural "boundary" which has the topology [and conformal geometry] of an n-dimensional sphere. Now String Theory, which is the focus of virtually all work on the problem of reconciling General Relativity with quantum mechanics, works particularly well on Anti-deSitter space. The AdS/CFT correspondence, put forward by Juan Maldacena,  gives a duality between a gravitational theory in the "bulk" [where the angels and demons live] and a kind of gauge theory on the spherical boundary at infinity. The illustration shows that even though the bulk has negative curvature, its boundary at infinity is *positively* curved. This is interesting in physics because, as I said,  string theory seems to work best on manifolds of *negative* curvature, but astronomical observations suggest that our world is *positively* curved. [This is deduced from the fact that the expansion of the Universe is *accelerating*, one of the great discoveries in the history of astronomy.] The illustration suggests how this apparent contradiction might be resolved. Namely, it is clear that it is natural to divide the picture into an infinite set of concentric circles [spheres in higher dimensions]. So maybe our four-dimensional, positively curved Universe sits inside [as a "membrane" or "brane-world"] a five-dimensional, negatively curved space like the one above.

• Positive Scalar Curvature and Applications in String Theory. Actually, the boundary in the above picture doesn't really inherit a Riemannian structure from the "bulk", only a conformal structure [because it is infinitely far away]. So it is perhaps not quite clear what is meant by "positive curvature". However, a famous theorem due to Yamabe, Schoen and others shows that each conformal structure is represented by a metric of constant *scalar* curvature, so we can talk about the sign of the scalar curvature of  the boundary. This leads to an interesting connection between the very highly developed and somewhat mysterious theory of  positive scalar curvature in Riemannian [and spin] geometry and string theory. For example, some compact manifolds simply *cannot* accept a metric of positive scalar curvature, no matter how they are deformed---so a boundary geometry like the one above is simply impossible, for *topological* reasons. [Actually, it's even more interesting, because the existence of a positive scalar curvature metric can depend on the *differentiable structure*.] So the requirements of string theory can actually impose conditions on the [differential] topology of spacetime, through the sublime mysteries of positive scalar curvature. On the other hand, some interesting manifolds, such as the underlying manifold of the K3 "surfaces", simply cannot be represented as the boundary of anything, so the interesting question arises: how can AdS/CFT possibly work for them?  .

• Riemannian Holonomy Groups and Physics Applications. Since the string theory revolution of 1985, physicists have been interested in Riemannian holonomy theory, which is the study of what happens when vectors are parallel transported around closed loops on curved spaces. Since the loops are *closed*, obviously holonomy theory is most interesting on manifolds which are *not* simply connected.  Surprisingly, many physicists and mathematicians are apparently not fully aware of this. Berger's famous classification theorem explicitly assumes that the manifold is simply connected, so it has to be extended. [Happily [?] the famous six-dimensional  Calabi-Yau manifolds used by string theorists to reduce the theory from 10 dimensions to 4 have the *very unusual* property that their holonomy groups do not detect whether they are simply connected. This statement is *false*, for example, for 8 dimensional Calabi-Yau manifolds! ] The problem of  understanding this extension and its physics applications is still not completely solved. One has a classification [due to me!---I mean in the non-simply-connected case of course], but in some cases there are no known examples. The point  is that string theory does care about the global structure of the holonomy group [whether, for example, it is a *connected* Lie group]. The classification of *spin* holonomy---which is what you get if you parallel transport *spinors* instead of vectors---is also a very deep and interesting problem.

• Alice Physics. As is well known, a surface or higher-dimensional space can be non-orientable; that is, you cannot distinguish right-handed from left-handed on such  a space. On an "Alice space" [don't blame me---I did *not* invent this terminology] it is similarly impossible to distinguish particles from anti-particles, a situation which does not hold at present but which might have in the early days of the Universe. In those days, a journey around the Universe might have converted you from matter to anti-matter! [Hence the Alice-in-Wonderland terminology---Alice got turned into antimatter---remember? Nor do I....] 

Getting the Universe to be "Alice-like" at one stage of its history but not later is a surprisingly delicate mathematical exercise, which is particularly subtle and interesting on the real projective space RP^3 [described above;  note that RP^3 *is* orientable in the usual sense.].  I will soon be returning to this subject, because I believe that the mysterious de-localised charge ["Cheshire Charge" --- see the picture of the eponymous cat above] that one gets in Alice physics is relevant to the question of the existence of charged black holes in RP^3 de Sitter space....

• Topology of  T-Duality in String Theory.  T-duality is one of the famous dualities of string theory, which show that the five apparently different string theories are fundamentally all equivalent to each other. For example, the E8 x E8 and "SO(32)" theories are T-dual. However, it can be non-trivial to get this particular duality to work properly on space-times of non-trivial topology, basically because E8 x E8 and Semispin(32) [the globally correct version of "SO(32)" ]  *don't*  have a common Lie subgroup with the Lie algebra of SO(16) x SO(16). The details of the global group theory are rather fascinating; one would have thought that there could not be much new to say about these classical groups---but there is. My idea was to turn this question into an obstruction problem: you can get around the Lie-theoretic problem in the same way that one studies the existence of a spin structure in Riemannian geometry. That is, you have to verify that a certain Z_2 cohomology class, analogous to the famous Stiefel-Whitney obstruction to spin structure, vanishes, which indeed it usually does. See papers 41 and 42 above.  I believe that I have found something here which is absolutely fundamental to string theory, and---amazingly, in view of what is at stake, and in view of the fact that many people are well aware that the "SO(32)" in string theory is really Semispin(32), which is very different--- there was  nothing else in the literature about this problem at that time. Subsequent work by David Morrison and others suggests that string theory automatically fixes this problem in an extremely deep and [in my opinion] not yet completely understood manner. I would like to insert a  picture here, because to me this work was as interesting as anything in cosmology,  but I really cannot begin to imagine one that would be relevant.....

• Cosmology in String Theory.

 The basic cosmological model corresponding to a positive cosmological constant [now apparently observed, as mentioned above, in the form of the cosmic acceleration] is deSitter space. But it is hard to obtain deSitter space from string theory. In particular, one needs to understand the "dS/CFT" correspondence better. One key to a deep understanding of AdS/CFT was Maldacena and Witten's  use of "Euclidean" techniques. I want to use Euclidean techniques to investigate dS/CFT---but the problem is that the usual Euclidean version of deSitter space is a sphere, not a space like the one illustrated at the top of this page! The trick here is to realise that "Riemannian" metrics don't really have to be *positive* definite---negative definite works equally well. This apparently trivial generalisation allows you to use the picture at the top of the page for a Euclidean version of deSitter space too---but you have to use an unorthodox foliation of that picture. [Instead of the obvious foliation by spheres, you foliate by hyperbolic spaces of one dimension less---think of what you get by visualising "phases of the moon" moving from left to right across the woodcut.] The nice thing about this foliation is that the *apparently* disconnected conformal boundary of deSitter space is revealed, in the Euclidean picture, as being connected. [The point is that the slices, being hyperbolic, are "infinitely large", so they extend right out to the boundary.] This is in agreement with the Witten-Yau theorem.  Thus the one-to-one bulk/boundary correspondence is maintained in the Euclidean formulation.  See paper 47 above.

• And you thought the Big Crunch was bad.....how about a Big Smash?

Evidence for a positive cosmological constant keeps piling up. To be more precise, the evidence keeps piling up for some kind of  "dark energy" with a *negative* pressure which is close in absolute value to its [positive] density. Of course, to say that the ratio of pressure to density is observationally close to -1 is to say that it might be slightly greater than -1.....or maybe slightly *less* than -1. In fact, there has been a lot of work on the former, which can be modelled by a "quintessence" field, but very little on the latter. Now in fact the "less than -1" cosmologies, called "Phantom Cosmologies" by R Caldwell http://www.dartmouth.edu/~caldwell/ [who was one of the quintessence pioneers] are very interesting indeed, for several reasons.

Caldwell's *particular*  phantom cosmologies have the bizarre property that they come to an end eventually, like the familiar "Big Crunch" cosmologies----but in the phantom world, the end comes not because the Universe collapses to zero size, but rather because the distance between any two given galaxies  becomes infinitely large in a finite proper time!  [I like to call this the "Big Smash". Caldwell's "Big Rip" would really destroy the universe, but I think that it is more likely that the universe would shatter into disconnected pieces---something more like this:

which is why I prefer "smash" to "rip". Of course, I don't know what happens to the pieces---but, since the pieces are themselves infinitely large, I would expect them to shatter too, and that the process would continue forever.]  More bizarre still is the fact that in these cosmologies, the density *increases* as the Universe expands, and in fact it diverges as the Universe expands "infinitely fast". That seems a little too extravagant for me, so I have constructed a family of  [extremely simple] Phantom cosmologies with no beginning or end---they are in fact asymptotically deSitter in a certain sense. Like deSitter space, they have a disconnected conformal boundary, but *unlike* deSitter space, their Euclidean versions *still* have a disconnected boundary, so they are ideal for exploring the question as to whether the one-to-one bulk/boundary correspondence can really be maintained. In fact, I claim that this disconnectedness of the Euclidean boundary is a signal that, contrary to appearances, these spacetimes are "effectively disconnected". Using recent ideas of Andrew Strominger and Vijay Balasubramanian et al, I argue that, in these spacetimes, time flows away from *both* boundaries. But there is an alternative: maybe it is OK to have a disconnected boundary if there is a quantum-mechanical "entanglement" between the conformal field theories on the two connected components. This would be analogous to Maldacena's rather amazing analysis of the quantum mechanics of asymptotically anti-deSitter black holes. If this version is correct, then the evolution of our Universe is controlled by strictly quantum-mechanical effects in theories which are defined on our *infinite* future and *infinite* past! This ties in with the recent efforts of Turok and Steinhardt et al to abolish the Big Bang and replace it with a Big Bounce. My paper on this is number 48 above. A less technical and more literate version, complete with insults directed at the Null Energy Condition, may be found at http://arxiv.org/abs/astro-ph/0210321. This is a version of the talk I gave at the Institut d'Astrophysique de Paris, which is a nice place, in July 2002.

• Topologically non-trivial black holes. Here is a [computer-generated, of course] picture of a black hole, though  not a topologically non-trivial one! [By the way, if you could really see this scene you would be dead: even so far away, the tidal "forces" generated by the hole would be enough to kill you, whether or not you tried to resist being pulled in.]

The event horizon of a black hole is the boundary of the region from which there is no escape. In "ordinary" spacetime, that surface is a 2-dimensional sphere. In five dimensions, which is particularly interesting in string theory, it would be a 3-dimensional sphere. But in a background with a negative cosmological constant---again, this is interesting in string theory, particularly in connection with the AdS/CFT correspondence---the topology doesn't have to be spherical. However, in the context of string theory, I have shown that there is a very severe restriction on the possible topology of the event horizon---only a tiny minority of topologies are possible for a *stable* black hole. This uses a kind of  string-theoretic instability discovered by Seiberg and Witten, involving the unstable production of "membranes". Now actually this kind of argument is usually regarded as dubious, because, according to Einstein's equations, the unstable production of membranes [or anything else] would be expected to modify the geometry of spacetime, and this modification might invalidate the condition for the Seiberg-Witten instability to be present----in other words, the instability would be expected to limit itself, as instabilities normally do. The amazing and beautiful thing here is that this *does not* happen---no matter what the membranes do to the geometry of spacetime, the Seiberg-Witten condition continues to hold! Only if the membranes change the *topology* of spacetime can this conclusion be avoided---but it is well known that such topology changes also lead to very unpleasant phenomena such as closed timelike worldlines.. A similar argument imposes restrictions on the structure of asymptotically deSitter versions [if any] of the spacetimes corresponding to the so-called "spacelike branes" introduced  by Gutperle and Strominger; again, the conclusion holds *even if back-reaction is taken into account*. The amazing thing here is not just that such a statement is true, but also that it is possible to prove it! [It is certainly not possible to predict in detail what will happen to the spacetime geometry when back-reaction becomes important.]  The proof involves an application of the concept of "enlargeable" manifolds in global differential geometry. It's very satisfying that such pure [and extremely deep] pure mathematics can be used for something so physical.

• Holography and the Null  Energy Condition.

 If you want to prove something about the structure of spacetime ---for example, if you want to prove the Penrose-Hawking theorems, which claim that singularities in spacetime are inevitable---then you need to assume something about the kinds of matter that can occur in the real world. For many years we were all trained to believe that if pressure can be negative at all, it cannot be very negative. The usual assumption was that, in units such that the speed of light is unity, the pressure cannot be less than (-1/3) times the density. [In these units, the ordinary pressures that you encounter are vastly smaller than this [in magnitude] so this is a very reasonable thing to claim---apart from the fact that "negative pressure" is pretty weird anyway!] But the discovery of cosmic acceleration proves that this is wrong; the dark energy has a pressure which is negative but almost equal in magnitude to its density! Question: why can't it be even lower? Answer: maybe it can!  However, I have recently argued as follows. HOLOGRAPHY is the extremely fashionable idea [ see picture above! ] that [roughly speaking] claims that everything happening inside a *volume* can be understood by studying its surface. That is, the stick figure can be completely understood by studying his shadow on the wall. Clear, eh? Anyway, I claim that holography  *predicts* w >= -1, where w is the pressure-to-density ratio. See paper 51 above. This paper has a good list of references for papers discussing violations of the General-Relativistic energy conditions, a field of research that is just catching on, especially in the astrophysics community, which tends to be more open-minded about such things than the mathematical physics community.

• S-Brane Singularities. One of the big problems in string theory is to get it to describe a spacetime which represents a "time-dependent" situation---such as the expansion of the Universe. One way of approaching this is the "Spacelike Branes" of Gutperle  and Strominger, mentioned above.  A brane is a higher-dimensional version of a string. Normally one thinks of a string as a "surface" in spacetime, because the string sweeps out a surface as time passes; similarly a brane would normally acquire one more dimension, corresponding to time. A Space-like brane or S-brane is even weirder: it is completely transverse to the time direction, so it only exists for an instant. The ultimate hope is that by not including time from the outset, the theory of S-branes will shed some light on the way "time passes" in string theory.  This is an extremely hot topic of research right now. Well, like any other brane, an S-brane modifies the structure of  the spacetime in which it lives its brief  life. Unfortunately the known S-brane spacetimes are singular, which might be ok, except that the singularities don't seem to be really related to the S-branes themselves. People hoped that the singularities would go away if we just included back-reaction, but recently it was shown that this does not happen as long as the Strong Energy Condition holds. Now in cosmology the Strong Energy Condition is dead, so you might hope to get rid of the problem there. Alas, my most recent work suggests that even breaking the SEC may not be enough to solve the problem. However, I do suggest 2 possible ways out: first, my old favourite, go ahead and break the *Null* Energy Condition, and/or do some pretty drastic re-interpreting of the Penrose diagram, so that conformal infinity becomes a manifold with boundary instead of a manifold. I then invoke a theorem of Gromov which essentially says that, on manifolds with boundary, there is no obstruction to positive scalar curvature. If all that seems too weird, then I invite the reader to contemplate the possibility that nasty, possibly naked singularities are unavoidable for a large family of S-brane spacetimes. That would be bad news. For more details, including some rather disgusting pictures of naked singularities, see paper 52 above.

LINKS

OTHER STUFF
THE JOURNAL OF HIGH ENERGY PHYSICS---GO HERE FOR THE LATEST IN THEORETICAL PHYSICS
What I do in a parallel Universe

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PHOTOGRAPHS:

Pictures of me:
Strangely penetrating gaze of  guess-who as a small child [19??]
A mountain of sand in Namibia [2001]
Me, at summit of that mountain of sand  [2001]
Quadbike in Namibian desert--a lot of fun until I crashed it  [2001]
Pictures of Wanmei:
Photo of wanmei in Venice [2002]
Wanmei on the Grossglocknerhochalpenstrasse [2002]
Wanmei's First Snowball [2002]
Wanmei In Innsbruck [2002]
Wanmei in Kitzbuhel [2002]
Wanmei in a Tea Plantation, Malaysia [2002]
Wanmei enjoying a fireplace in the Tropics [Malaysia!] [2002]
Another photo of Wanmei [2003]
Wanmei at 3000 metres in the Dolomiti [2003]
Wanmei in Corvara,  in the Dolomiti [2003]
Wanmei at Bled,  in Slovenia [2003]
The castle overlooking the lake at Bled [2003]
Wanmei at San Gimignano, a beautiful small town in Tuscany [2003]
Wanmei at one with the tourist zeitgeist in Pisa [2003]
Wanmei at Lake Bohinj, Slovenia [2003]
Wanmei in Siena [2003]
Wanmei in Florence [2003]
Wanmei with me [wearing a Hannibal Lecter hat] on the Ponte Vecchio, Florence [2003]
Wanmei at the Tre Cime di Lavaredo, the Dolomiti [2003]
Dolomitenblick [2003]
Wanmei in Piran, Slovenia [2003]
Another view of Piran [2003]
Wanmei at the Weissensee, in Austria [2003]
CROCODILES IN AUSTRIA?! [2003]
Wanmei in Rome, at St Peter's [2003]
Wanmei at the Colosseo [2003]
Oops! Forgot about the swatch [2003]
Wanmei in Indonesia [2004]
Wanmei in Jacuzzi [2004]
Vietnamese Wanmei [2004]
Wanmei in Trieste [2004]
In nearby Muggia [2004]
In Miramare [2004]
In Rovinj, Croatia [2004]

Also in Rovinj [2004]
The Pordoi Pass, near Arabba,  Dolomiti [2004]
Lake Como [2004]
Varenna on Lake Como [2004]
Did I mention that we liked Varenna? [2004]
Bellagio on Lake Como, with the Hotel Metropole [2004]
Dinner at the Metropole [2004]
The Giant's Tooth, Italian Alps [2004]
Courmayeur, Val d'Aosta [2004]
Lake Garda [2004]
Sirmione [2004]
Mali Losinj, Croatia [2004]
Venice Again! [2004]
Venice is So Cacciatore, as Granny Used to Say [2004]
She's Been There, Done That [sort of] [2004]
What News from the Rialto? [2004]
Me Rowing in Austria [2004]
Wanmei Rowing in Austria [2004]
The Passenger [2004]
The Weissensee at Stockenboi [2004]
Wanmei and Friend on the Nockalmstrasse [2004]
The Valley of the Soca, Slovenia [2004]
Prizewinner! [2004]
Wanmei finally ceases to fear dogs! Armidale, Australia [2004]

Evil Creatures
Inspecting a Koala [2004]
Wanmei learns to ride a horse [2004]
Valentine's day, Venice, 2005
Hard at work at the International Centre for Theoretical Physics, Italy 2005
The view from our apartment in the Via Zamboni, Trieste, Italy 2005
Another view 2005
Wanmei learns to skate, Weissensee, winter 2005
Getting a bit more confident 2005
View from our hotel,  Weissensee, March 2005
Cold butt good...the ice was 1 metre thick on the Weissensee 2005
ICEPRINCESS 2005
In Saudi Arabia...not really: the Isle of Rab, Croatia, 2005
On the car ferry, leaving Rab [2005]
At the Plitvice lakes [world heritage site] Croatia 2005
And you thought the one in Pisa was leaning....London 2005
No, I'm not squashing her arm! Paris 2005
Lake Bohinj, Slovenia 2005
Feeding the trout in Lake Bohinj 2005
Looking guilty at Lake Bohinj 2005
Vintgar Gorge, Slovenia 2005
The Mondsee, near Salzburg, Austria 2005
This has something to do with Julie Andrews. Or so I'm told. Salzburg 2005
View from my balcony
Scenic WC in Bukit Tinggi, Sumatra, May 2006
Wanmei about to attack the Suao-Hualien Highway, Eastern Taiwan, July 2006
At Taroko Gorge, Taiwan
View From Jioufen, Taiwan

Bangkok, December 2006
Demonic Wanmei
Wat Phra Keao
Maya bay, Krabi, Thailand Feb 2007
Snorkelling
Asteroid falling into sea
Sawtell, NSW, Australia, June 2007
On the beach...in winter.
Armidale, Australia, winter [June 2007]

Rothenburg ob der Tauber [September 2007]

Rothenburg again

And again

Hanoi, December 2007

Selling fruit in Vietnam

Viet Xmas

Explaining The Arrow of Time, Cambridge, December 2007

St Paul's Cathedral Devoured, London, December 2007

Wanmei at Angourie, NSW, Australia, June 2008

Disciplining my Nephew the old-fashioned way, June 2008

Bikes are now out of date, June 2008

Wanmei deep under Zacatecas, Mexico, July 2008

Wanmei far above Zacatecas, July 2008

GORDITAS! Zacatecas

Guanajuato, Mexico July 2008

Guanajuato again

The street of the dead dogs, Guanajuato

The street of the singing frogs

One for all you Malcolm Lowry fans [Monterrey, Mexico]

Plaza de Baratillo, Guanajuato

Banyan Tree Resort Feb 2009

View from Balcony at Banyan Tree

Jude Lüwen McInnes, Born 23/5/2009

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WEDDING PHOTOS :[2003]
I swear I wasn't that nervous [2003]
The Flower Girl
Beauty and the Beast....and if anyone else calls her a beast....
It's all uphill from here on
Reload That Matrix!
Well, it's still a great picture of the bride anyway.....
The Bodyguard
Wanmei!
Just keep your eye on the bride, ok?
Staying Cool by the Pool
Jasmine, who [with James]  prevented all disasters, and to whom we are eternally grateful
A loving embrace
Wanmei from a different angle!

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Brett McInnes
Department of Mathematics
National University of Singapore
Singapore 119260
Republic of Singapore                     

matmcinn@this.really.isn't.here.nus.edu.nor.this.either.sg                  

(65) 65162763 (office)
(65) 67359296 (fax)
(65) 6779-5452 (another fax)

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Last Modified: Some time in 2009