A flat torus in three dimensional space

Vincent Borrelli, Said Jabrane, Francis Lazarus and Boris Thibert realize for the first time the image of a 3d flat torus. But, what is a flat torus? In a mathematical point of view is a torus without gaussian curvature everywhere. For example: in a 3d space, you can bend a sheet of paper in a cylinder, but you cannot embed it in a torus without stretching the paper itself. But, if you live in a 2d space, a flat torus is a surface like the pacman labirint: you can enter in the right side of the paper and exit behind you from the left side:

We can define a flat torus starting from a couple of real numbers $u$, $v$ such that $u, v \in ]0, 2 \pi]$⁽¹⁾: \[(u, v) \rightarrow \frac{1}{\sqrt{2}} \left ( \cos (u+v), \sin (u+v), \cos (u-v), \sin (u-v) \right )\] The following is the best visualization of a flat torus before Borelli-Jabrane-Lazarus-Thibert:

The mathematical problem around the flat torus was challenged by Nash and Kuiper in 1950s:

Nash and Kuiper proved the existence of a representation that does not perturb the lenghts in the square flat torus. For a long time, this existence remained a challenge for the imagination of mathematicians. But proving and showing should sometimes be clearly distinguished in Mathematics. This is well explained by the thief allegory: Let us assume that a group of people is gathered around a jewel in a closed room. Let us further suppose that the light is turned off for a moment and that the jewel has disappeared when the light is again turned on. We then have the proof that a robber is hiding among the attendance but he can not be exhibited. Although the proofs of Nash and Kuiper are much more than an «existential» trap, their proofs do not provide a sufficiently explicit procedure that would allow for visualization or simply for a mental picture of a square flat torus.⁽²⁾

Between 70s and 80s the Abel Prize Gromov extracted a method from the work of Nash and Kuiper, proposing the so called convex integration, a very useful tool. Indeed it

(...) does not only yield the existence of a solution, it also provides us with an effective construction.⁽²⁾

Starting from this method, Borelli, Jabrane, Lazarus and Thibert realize an algorithm in order to picture the flat torus.

Mathematicians were puzzled by the works of Nash and Kuiper. These works could indeed prove the existence of objects whose regularity was problematic, if not paradoxical. They had to be smooth and rough at the same time... In effect, the mathematical analysis of the images reveals a surface belonging to two antagonist worlds; the smooth surfaces and the fractals,infinitely broken. When zooming in, we invariably observe ripples at smaller and smaller scales. Each ripple - called a corrugation – appears smooth when viewed alone, but the accumulation of those creates an object with a rough and fractal aspect.⁽²⁾

And in conclusion

Demonstrating that convex integration can be implemented open new perspectives in applied mathematics, notably for solving differential systems originating from Physics and Biology.
More specifically, our images reveal a class of objects whose structure lies inbetween smooth surfaces and fractals. Such objects could play a central rôle for shape analysis. They could also resolve some unexplained paradoxes.⁽²⁾

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Read also Wikipedia | phys.org

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(1) The Flat Torus in the Three-Sphere
(2) A flat torus in three dimensional space (pdf) | Hevea Project

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Borrelli, V., Jabrane, S., Lazarus, F., & Thibert, B. (2012). Flat tori in three-dimensional space and convex integration Proceedings of the National Academy of Sciences DOI: 10.1073/pnas.1118478109

GILDA or the future of the financial resources for italian research

GILDA is an italian beam line based on Grenoble, France. It has some financial problems, and the two responsibles (and the only researchers that work on the experiment in this moment) write a letter to the president of CNR, L.Nicolais. I share it for my readers (via Peppe Liberti)

To the President of CNR, Prof. L. Nicolais

Dear President
Since 1994 GILDA has provided to the user community an access to a high quality third generation synchrotron radiation source. The project has been a success considering the high number of publications, exceeding 500 units with a rate of about 40 publications per year in the last decade. In these years we have actively contributed to the diffusion of the Synchrotron Radiation techniques in the Italian community and formed an entire generation of scientists that now can be found not only in universities and research institutes but also in the staff of several synchrotron sources. In all the reviews of the beamline activity, carried out by external referees, it has been underlined the quality of the science produced at GILDA and its fundamental role in the overall ESRF activity.
In spite of that, the financial and human resources dedicated to this activity have plunged in the last years, leaving at present GILDA in severe difficulties. At present GILDA is left with only two units of personnel at Grenoble and ageing instrumentation so putting at a serious risk the efficient operation and quality of support that have characterized our instrument during its several years of life. The present shortage of local staff prevents GILDA from coming back to operation in the forthcoming 2012-I run with a severe damage to the vast community (material science, biology, chemistry, physics, earth and environmental science , archaeometry) waiting to carry out experiments there.
Time has come to invert the trend and to provide to GILDA all the necessary resources for continuing, in an effective way, its action for at the benefit of the italian and international scientific communities. This, also in the light of the present situation of ESRF where all the older beamlines have undergone (or are about to carry out) major refurbishment programs creating in this way an impressively competitive environment. For this reason, we ask to CNR to define a long term strategy for the activity of GILDA. This will include the necessary financial and human resources to carry out a comprehensive refurbishment program of the beamline and to ensure, in the years to come, the consistent staff and funds for an effective operation.
Without a prompt and massive intervention it will be inevitable, in the forthcoming months, to definitely close down the GILDA project that will have heavy consequences on the activity of the italian and international scientific community at ESRF.

Sincerely Yours
The GILDA responsibles
Francesco d'Acapito
Settimio Mobilio

Gravity vs height

(powered by scilab)

Source code:

G = 6.67428 * 10^(-11);
R = 12745594/2;
M = 5.9742 * 10^(24);
function g = myg (x)
g = G*M/(R+x)^2
endfunction
xdata=linspace(0,2*R,5000);
plot(xdata,myg)
eps = 10^(-1);
d0 = sqrt(G*M/eps) - R

How much would you pay to the universe?

There are some who question the relevance of space activities in a developing nation. To us, there is no ambiguity of purpose. We do not have the fantasy of competing with the economically advanced nations in the exploration of the moon or the planets or manned spaceflight. But we are convinced that if we are to play a meaningful role nationally, and in the community of nations, we must be second to none in the application of advanced technologies to the real problems of man and society.

Vikram Sarabhai, from the Indian Space Research Organization, to Wired

Mathematical egg

Equation: \[1.2 + \left ( \sqrt{\left (1 - \sqrt{x^2+y^2} \right )^2} + 1 - x^2 - y^2 \right ) \cdot \left ( \sin \left ( 10 \left ( \frac{3x + y}{5} + 7 \right ) \right ) + \frac{1}{4} \right )\] Instruction for Google search box:

1.2+(sqrt(1-(sqrt(x^2+y^2))^2) + 1 - x^2-y^2) * (sin (10 * (x*3+y/5+7))+1/4)

Result:

(via Google Italia)

Conversations with physicists

Using Google+ Hangout, CMS experiment realize a discussion session between two CMS physicists, Joe Incandela (spokesperson of CMS) and Sue Ann Koay and CMS followers on Google Plus:

Ian Stewart and the Black-Scholes equation

The Black-Scholes equation is an economical tool used in financial contracts. Following Ian Stewart (via Alexandre Borovik), this equation caused the economic crash and crisis. We can immediatly say that we must not use mathematics to describe the financial system, but the problem isn't in the equation, is in his use:

The formula was fine if you used it sensibly and abandoned it when market conditions weren't appropriate. The trouble was its potential for abuse. It allowed derivatives to become commodities that could be traded in their own right. The financial sector called it the Midas Formula and saw it as a recipe for making everything turn to gold. But the markets forgot how the story of King Midas ended.

The equation was derived by Black and Scholes in 1973, in the paper The Pricing of Options and Corporate Liabilities. In the same year Robert Merton in the paper Theory of Rational Option Pricing develop the mathematics under the equation and the options, starting from the model of Fischer Black and Myron Scholes. For their work Scholes and Morton won the Noble Prize in economics in 1997 (Black died in 1995).
Now, one of the background ideas in Black and Scholes original model is the brownian motion, a mathematical model used to describe the random motion of a particle in a fluid. So it could be right use a brownian model in the study of the financial networks, but is also a great simplification of the problem:

Large fluctuations in the stock market are far more common than Brownian motion predicts.

The case of the Jiggly Wires

Video by Gavin Wince shared by the auhor in the comments of this complete and ultimate post about OPERA's neutrinos by Matt Strassler.
On of the most interesting observation in the post was resumed in the following plot:

We could conclude that the problem was originated in 2008.
In every case I'm agree with this Strassler's questions:

A big question that the OPERA leadership that resigned today has to answer: why didn’t they do this cross-check before they made their result public? Did no one think of it til recently? And if not, why not? Was it harder than it sounds? Or did they just miss an obvious opportunity?