Sunday, September 5, 2010

Early thoughts

I have a rather clear memory of what might have been the first time I realised that I was capable of thinking. We were having a lunch with my family in the annex of our house, where we used to roast meat on Sundays. The conversation came to the subject of the existence of God, and whether it was possible to prove it or not. I said that I considered that such a proof was not even desirable since since according to Saint Paul's Apocalypse God would come back on Earth at the end of time if we managed by earthly considerations to prove that He's here, then that would be the end of time. My parents' and sister's reaction was that of surprise that such ideas could circulate in my head. This made me aware that a thought had been created within me that had not been introduced there from the outside.

Some years (?) later I came across Saint Anselme of Canterbury's "ontological argument" for the existence of God. Roughly, it resorted to the reductio ad absurdum and said that given that the Perfect Being must have all positive qualities, if He didn't exist, He would lack one of those qualities, namely the quality of existing. I imagined a refutation of it, which went as follows. If the existence of God could be proven, then we humans would have no choice but to believe in Him -- in which case He would also lack a quality, the generosity of letting His creatures decide freely if they want to believe in Him or not.

Beyond the naive scholasticism of those early thoughts, something in their spirit is still very much akin to what turns me on nowadays.

Had I pushed my reflections a bit further, I would have realised that I was dealing with an absurd absurdity. By the same type of reasoning it has to be true that God does exist but it must also be impossible for us to prove it. How can this be? And then of course it's Godel's incompleteness theorem that comes to mind. (Godel who btw wrote down an ontological proof of his own using modal logic, which he kept secret until he thought his last days had arrived...)

PS: Those memories came back to me today while I was playing with the following thought:
"The meaning of life is to search for the meaning of life."
I like it because of its circularity and because it's contradictory in the sense that it implies that were we to actually find the meaning of life it would be gone. In fact it's also contradictory in the sense that it undermines its own content: it gives you the meaning so why search for it? The entire substance of the sentence is sucked in by the word "search", which then operates just like a minus sign in an equation. One could for example symbolise it abstractly as

\[ \infty - \infty = 0.\]

I fancy writing it in repetitive style as well:
" The meaning of life
is to search for the meaning of life
is to search for the meaning of life
is to search for the meaning of life."
Just like that.

PPS: the etymology of the word "search" is agreeable since, along with chercher in French and cercare in Italian, it comes from the Latin word circus, the circle. :) Should we push the perversity yet another step further and replace it by the word "research"?

Monday, July 5, 2010

Laughing and research

What is it that makes us laugh in a joke, after all?
It seems to me that a joke always requires from the audience a certain amount of mental effort to see the meaning of the joke. There is a gap in the plot and it has to be filled for the joke to be funny.
For instance:

A guy comes into a bar, completely drunk, and asks for a drink. The barman tells him that he never serves drunk people. The guy goes away but comes back five minutes later. The barman refuses again. The guy goes away but comes back ten minutes later, and so on.
Finally the drunk guy asks the barman: "Is there a single bar in this city where you don't work?"

To laugh at this joke you need to understand that the guy thinks he is visiting different bars, but he is so drunk he always ends up in the same bar.

The misanthropically inclined mind will think that then the laughter is only a demonstration of pride to have been able to crack the mystery.

But another interpretation would be that cracking the mystery is a pleasure in itself, and the real cause of the laughter.

We researchers are devoting our time to crack mysteries, although of different proportions. We are laughing the great laughter.

Sunday, March 21, 2010

The geometry of 3-manifolds

If you want to get an idea of the famous Poincaré conjecture for which Grigoriy Perelman was recently awarded a Millenium prize, I highly recommend watching this very entertaining lecture by Curtis McMullen at Harvard in 2006.

I had never realised that tori of genus greater than one are hyperbolic! It makes sense, if you think that a sphere (genus 0) has positive curvature, a torus (genus 1) is flat, that tori of genus 2 and more have negative curvature. These higher tori can be constructed by gluing together the edges of polygons.

I wonder whether anti-de Sitter space can also be viewed as a high genus space. Are the many black holes in the universe implying that it has a high "genus"? But the notion of genus isn't clear in higher dimensions...

So I started (very naively) to think about black holes on a latex surface, like the ones shown in the lecture. Consider a sphere and put a stone at the north pole. The sphere will bend, just like spacetime bends around the sun. Now imagine that you had a way to increase the mass of the stone, it would create a well at the north pole which would eventually touch the south pole from the inside. What happens then? The well could continue to become deeper and deeper but now it would create a spike out of the south pole. This is certainly misleading. More likely, once the stone approaches the south pole from the inside, its mass sucks it in, creating a well (as well).

Ultimately, what you'll end up with is a torus. The Planckian mass is at the centre of this torus, where there is really no spacetime: that's the black hole!

I had an unexpected confirmation of this picture in my kitchen. There was a frying pan full of a layer of greasy water and I noticed that it tended to leave discs where there was no water. By slowly pouring more water into the pan, the discs would shrink until there reach zero size, at which point a circular wave was emitted. I had just witnessed a topology change;) I could also reverse the process by taking some water out of the pan and stirring up the water. When the water had stabilised enough, discs were suddenly appearing and expanding quickly. That's my version of frying pan black holes: try it at home!

PS: the key is the circular wave.

Monday, February 15, 2010

"Holographic Superconductos in M-theory"

Last Monday we had Jerome Gauntlett telling us about exciting applications of the AdS/CFT correspondence to Condensed Matter systems.

Some quantum critical points (second order phase transitions) are at strong coupling, and hence intractable with standard CM technics, but the AdS/CFT comes to the rescue. This happens for example for "heavy fermions" (quasi-particles condensing to give Cooper pairs) or for high $T_c$ cuprates that have a superconducting phase. Holographic descriptions of such systems have been put forth by Gubser and by Hartnoll, Herzog, and Horowitz. The strongly coupled CFT in 3d has a holographic AdS dual, which has a $U(1)$ gauge symmetry that can be spontaneously broken by a charged field $\chi$. A finite temperature the description is in terms of a black hole in AdS, and it acquires charged hair at low temperature that breaks the $U(1)$.

Almost all the work on this topic has been taking a "bottom up" approach, but Jerome took the "top down" approach: constructing holographic superconductors from M-theory using consistent Kaluza-Klein truncation with Sasaki-Einstein spaces. Here consistent truncation means that you can keep only a finite set of the light KK fields (you cut the infinite KK towers) but then any solution of the low dimensional theory involving those remaining fields has to uplift to an exact solution of the original higher dimensional theory. So consider a stack of M2-branes at the tip of a Calabi-Yau cone on a Sasaki-Einstein seven-manifold for which the near-horizon limit is $AdS_4 \times SE_7$ with metric \[ d s^2 = \frac{1}{4} d s^2 (AdS_4) + d s^2 (SE_7) \] with \[ d s^2 (SE_7) = d s^2 (KE_6) + \eta \otimes \eta~,\] where $\eta$is the contact form. There is an associated Killing vector, which (although it does not always have to close) generates the required $U(1)$ symmetry. The four-form field strength is $G_4 = \frac{3}{8} \text{Vol}(AdS_4)$. This geometry is dual to $\mathcal{N} =3$ SCFTs in $d=3$ (or $\mathcal{N} = 8$ if $SE_7$ is replaced by a seven-sphere).
Now, had the membranes been replaced by anti-membranes, $G_4$ would have the opposite sign but more importantly the dual SCFTs would have $\mathcal{N}=0$. The corresponding geometry is called skew-whiffed $AdS_4 \times SE_7$. It is this geometry that can reproduce and generalize the HHH models. The consistent truncation found by Jerome has an extra neutral scalar field $h$ which corresponds to a deformation of the skew-whiffed CFT by an operator $\mathcal{O}_h$. At low temperature this system exhibits a superconducting phase.
Jerome concluded by suggesting that if superconductivity had not already been discovered, such a study would have pointed it out to us. M-theory might thus lead to a similarly radical discovery of qualitatively new phenomena...

Sunday, January 24, 2010

David Blaine

I watched this TeD video where the magician David Blaine explains how he broke the Guiness record of apnea by staying a little more than 17 minutes under water.

To me, this guy is more than a magician, and more than an athlete, he's a pure artist. His determination to push the human body beyond its clinical limits is of course very spectacular and impressive, but it also says something about what it means to live. How much can one deprave the body from resources (food, oxygen) and still survive? There is also a sense of askesis very similar to what one can find in several religious traditions (fast, meditation, etc.)...

However, what struck me particularly was the fact that after a failed attempt he finally broke the record (live on TV) when he was convinced that he had no chance to succeed since his heartbeat was unusually high during the immersion. I think there might be some universal truth in this: great things are often achieved just at the moment when a long chain of intense efforts oriented to a precise goal ultimately gives way to the disappearance of all hope.