Busy Week

Well, a lot of things happened this week. First of all, you don't need to look for Grigory Perelman anymore. Yes, he was found, he won the Fields Medal and, as expected, he declined it. The guy is tough. Anyway, he had his reasons (at least he said that). Read the detailed news here: Maths genius declines top prize.

There is also a very important news. NASA researchers are saying that they proved that dark matter exists beyond doubt by indirect observations. They crossed data from three different sources (three telescopes) and measured the gravity in a huge collision. Dark matter was proposed as a kind of matter that only interacts by gravity (no electromagnetism, no weak force, no strong force...) to explain why galaxies have the observed structure although we cannot see the matter which should be there to explain that. There are alternative theories though, but the researchers say that the main explanation has to be dark matter according to the measurements. In short, what they saw was an event where electromagnetism separated the ordinary matter from dark matter, allowing guys to measure it. The complete story is here: Cosmic smash-up provides proof of dark matter.

This week, although I knew its existence for a long time, I tried to explore Wikibooks, a project related to Wikipedia where people share knowledge in the form of free books. There is an incredible variety and they are extremely interesting. I recommend it strongly!

Another interesting news is an article about the bombing plot here in the UK. Everybody heard about that. The police discovered a plot to detonate liquid explosives in planes going to the USA. It seems like an action movie and, if this article make sense, it really is: Mass murder in the skies: was the plot feasible?. I am not a chemist to know if the article is correct. If there is one reading this, please give your opinion.

And for those researchers interested in helping the human space colonization, the Mars Society is looking for volunteers for an experiment: Four-Month Mars Mission Simulation at the Flashline Mars Arctic Research Station: Hard Work, No Pay, Eternal Glory. I would be a volunteer, if I was not married...

To finish this random collection of paragraphs today, the guys of the 'Delta de Dirac' rock band contacted me. I heard their music and its the kind of rock that I really like. Their site (finally I know) is www.deltadedirac.com. You can download a demo of their music there. They told me that their guitar player studied electronic engineering, that's the reason of the name. And their logo is amazingly cool too! :)

Picture: Busy Life, by Edwin Gardner.

Wanted: The Russian who smashed Poincare

Jules Henri Poincaré was a French mathematician AND physicist AND philosopher who made a lot of important (and beautiful) contributions to all those branches of science. In some sense though, his most famous contribution was an (until now) unproved conjecture which has his name: the Poincare Conjecture. As you can read in the Wikipedia link, the modern version of the conjecture seems very simple:

Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere.

I'll explain it in a moment. First, let me say that it is a conjecture because Poincare asserted it without proving. He was led to think that it could be true by clues in his work, but was unable to prove it in that time. The second thing is that it is so important to mathematics (and probably to physics too) that it is one of the famous Millenium Prize Problems from the Clay Institute, which include among others, the P=NP problem, the solution of Navier-Stokes equation and the Riemann Hypothesis. It means that whoever solves one of them is entitled to receive the US$ 1,000,000.00 prize!

Poincare conjeture talks about topology. A 3-manifold is a mathematical set that looks like a 3-dimensional euclidean space if you look very close (what is called locally in technical language), like the Earth looks like a flat locally, but is really a sphere. The 3-sphere is just the familiar sphere in 3-dimensions, i.e., the whole sphere, including its interior. In mathematics, when you say that two things are homeomorphic, you are saying that, in practice, they are the same. Now, the notion of being simply connected. A set is simply connected if any closed curve (e.g., a circle) in the set can be continuously shrinked to a point without breaking. The classical example is the sphere against the torus (i.e., the doughnut). Inside a sphere, you can shrink any closed curve to a point without breaking, in a torus, if the curve goes around the hole, you cannot shrink it to a point, you are stopped by the hole. The term compact is a little technical and in some sense means that the set is not infinite. The non-boundary condition means that you cannot found the boundary of the set. The circle, seen as a one-dimensional object, has no boundaries. Indeed, it is a one-dimensional manifold which is compact, has no boundary but fails to be simply connected. Actually, you can say that it is a 1-dimensional torus.

As you can see, although it is simply stated, it has a lot of technical subtleties and stood without proof until the russian mathematician Grigory Perelman posted some papers in the arXiv saying that he had found a proof. The problem is that the papers where very compact and took some time to be dissecated. And now, that it seems that they were and the proof seems correct, Dr. Perelman simply disappeared! You can read the detailed story in the NY Times: Elusive Proof, Elusive Prover: A New Mathematical Mystery.

The detailed version of the Poincare Conjecture is in the Clay Mathematics site, as well as the papers explaining his proof. I'm linking them here just in case:

- Ricci Flow and the Poincare Conjecture, John W. Morgan, Gang Tian (math.DG/0607607)

- Notes on Perelman's papers, Bruce Kleiner, John Lott (math.DG/0605667)

- A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow, Huai-Dong Cao, Xi-Ping Zhu

If you find Dr. Perelman, please send an email to the CLay Institute! All mathematicians in the world want to find them. It is very probable, as it is what he likes most to do, that he is looking for mushrooms in some Russian forest.

Picture: The Guy.

MathJokes

I've just received them by email. :)

The Dirac Delta

I've been doing replica calculations since I arrived in UK. Appart from the replica trick itself, these calculations involve a lot of interesting and useful mathematical tricks and techniques. One of them, which I particularly like, is related to the Dirac delta function.

The Dirac delta is not a function in the correct sense of the word, it is a distribution and is a very odd object. It is defined rigorously as a limit of functions. There are different functions that gives the Dirac delta in some limit, but (in my opinion) the most elegant is the Gaussian distribution. It goes like this:


The main characteristics of the delta is that its integral over the entire real line is one and it has the filtering property:


The legend (as I couldn't find more accurate historical details) says that Dirac introduced the delta to study point charge distributions in electromagnetism. Soon, a lot of other applications were found for it. I stumbled with one in the replica papers. It is usual in replica calculations to do the following transformation:


Let me explain. In the above equation, s is a random vector with probability distribution P(s) and x a constant vector. What I'm doing is a change of variables under the integral. Now, the new one-dimensional (and this is important!) variable v is a random variable and its distribution will be


The average of the delta over s. If you look more closely, there is a very interesting thing ocurring here: if you consider the equations above without the probability distributions like


The integral of the delta is nothing more than the Jacobian of the transformation. The interesting is that it is a Jacobian of a transformation between variables with different dimensionality. I'll leave to the reader the pleasure to see that all the Jacobian properties and delta function properties match beautifully indeed. I also bet that this can be related with Gauss theorem (or the generalized Stokes Theorem) although I haven't done the detailed calculations.

Now, changing the topic a little, I found this nice site about Singular Value Decomposition, a kind of matrix decomposition which is important in numerical calculation, but not only there. I remember that once I entered the office of (again) Prof. Caticha and he was playing with SVD. He transformed a picture in a matrix, calculated the singular values and took the resulting matrix as the original ignorating the less important singular values. The result was an image almost equal to the original one: a kind of compression technique.

Picture: I also discovered that 'Delta de Dirac' is the name of a Mexican band of progressive rock and the picture is of one of its CDs. Unfortunately, I didn't find their website. If someone knows, please tell me.

Links & News

A friend sent me an interesting site today with lots of physics flash applications. They range from chaos maps to a little calculation of a Stern-Gerlach experiment.

I also found two very interesting news which I am listing below. The third one concerns the question Hawking posted on Yahoo Answers!: How can the human race survive the next hundred years? Full Hawking's own answer is here.

Okay, here are the news:

1. Scientist thinks invisibility possible in future
2. House and Garden: Architects design a living home
3. The great man's answer to the question of human survival: Er, I don't know

Picture: from the story House and Garden above.

Cubes 2

I know, this sequence of titles looks like that terrible movies Cube and Cube 2: Hypercube (I apologize to those who liked them...), but Prof. Caticha answered my email and, as I predicted, there is a more elegant formula for expressing the n-cubes algebraically. It is not to much different from the one I posted, but I will describe it anyway. Instead of initiating with the representation of the 1-cube as I showed in the last post, you can initiate it by representing the 1-cube as the Pauli matrix


Also, changing the notation of the unity matrices such that I_n is the (n+1)x(n+1) unity matrix just to make the final formula prettier and following exactly the same procedure as before, the expression for the n-cube becomes


Well, I have nothing specific to speak this week, but I was browsing the arXiv and found some odd papers. What I mean is that they have curious titles, but I really still haven't read them to say something about their contents. I'm listing them here if someone wants to check. :)

1. Football: a naive approximation to the effect of increasing goal size on the number of goals - J. Mira (physics/0607183)
2. A fixed point in Coptic Chronology: the solar eclipse of 10 March, 601 - John Ray, Gerry Gilmore (astro-ph/0607520)
   

I have found some interesting links this week:

1. California and Carnegie Planet Search
2. Wave Packets Animations
3. Find a Postdoc
   

and also some news:

1. Mysterious quasar casts doubt on black holes
2. Shadowy T-rays: Hunting Tumors and Exploring the Universe
3. Medical 'Miracles' Not Supported by Evidence
   

Picture: Hypercube, by Shem Booth-Spain