Carl Ludwig Siegel: number thoery, pig and nazional socialism

via Lang, S. (1994). Mordell's review, Siegel's letter to Mordell, diophantine geometry, and 20th century mathematics. Mitteilungen der Deutschen Mathematiker-Vereinigung, 2(4), 20-31.
This is the letter that the german mathematician Carl Ludwig Siegel wrote to Louis Mordell about the book Diophantine geometry by Serge Lang. Here I proposed you an extract in which we can read Siegel approach to mathematics and also his political position about nazism:
When I first saw [Lang's Diophantine geometry], about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!
The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.
Unfortunately there are many "fellow-travellers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"
I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up.

The poet and the pendulum

Looking upward, I surveyed the ceiling of my prison. It was some thirty or forty feet overhead, and constructed much as the side walls. In one of its panels a very singular figure riveted my whole attention. It was the painted figure of Time as he is commonly represented, save that, in lieu of a scythe, he held what, at a casual glance, I supposed to be the pictured image of a huge pendulum, such as we see on antique clocks. There was something, however, in the appearance of this machine which caused me to regard it more attentively. While I gazed directly upward at it, (for its position was immediately over my own,) I fancied that I saw it in motion. In an instant afterward the fancy was confirmed. Its sweep was brief, and of course slow. I watched it for some minutes, somewhat in fear, but more in wonder.
- from The pit and the pendulum by Edgar Allan Poe
After the abandonment of Tarja Turunen, the Nightwish engaged Anette Olzon as a female voice for five years, from 2007 to 2012. Again the separation it was not the best, but the fact is that Anette, despite the apparent sympathy, certainly did not enter the hearts of the fans, evidently still loving Turunen. His place was then taken by the dutch Floor Jansen, who turned out to be a worthy substitute for Turunen, but in the meantime Olzon sang one of Tuomas Holopainen's most interesting and inspired songs, evidently inspired by Edgar Allan Poe and to his mystical story The pit and the pendulum:

Gerbert's satanic signs

In the history of numbers, Gerbert of Aurillac, better known as Sylvester II, the 139th Pope of the Catholic Church, takes on a curious role.
He was an eclectic character: enthusiast about science and mathematics, it is handed down that he was the introducer of the Arabic numbers in Europe:
Gerbert was a figure of utmost importance as a religious, politician and scientist, who could not be ignored by his successors to the papal throne. He was considered the greatest intellectual exponent of the 10th century and one of the most important of the Middle Ages, a multifaceted and profound connoisseur of the arts of trivium and quadrivium. Thanks to his contact with the most advanced Islamic culture, Gerbert introduced in Europe the use of the clock, of a siren running on water vapor, and was the inventor of complicated musical and astronomical instruments. He used these inventions in Reims for teaching in the cathedral school. For example, Gerbert had built a complex system of celestial spheres designed to calculate the distances between the planets and, again in astronomy, asked in a letter of 984 to Lupito of Barcelona for the translation of an Arabic astronomy treaty, the Sententiae Astrolabii. Always in Reims he had a hydraulic organ built that excelled on all the previously known instruments, in which the air had to be pumped manually, and that in the sixteenth century was still visible in Ravenna. In the field of mathematics, the introduction of Arabic numerals in Europe has long been attributed to Gerbert, a merit of difficult attribution: surely the young aquitan knew them at the Hatto's school in Vich, but nothing authorizes us to think that he then made them know in the old continent. Certainly, Gerbert had the great merit of contributing to the studies on the astrolabe and of reintroducing the abacus in Europe, of which, according to an ancient chronicle, he would have learned the use by the Arabs.
The Arabic numbers were then considered demonic signs, so it should not be surprising that Pope Innocent X, in 1648, decided to resume the body with the aim of finding out if there was any trace of these sings on his predecessor. The exhumation was thus narrated by Cesare Rasponi:
When we dug under the portico, the body of Sylvester II was found intact, lying in a marble sepulcher at a depth of twelve palms. He was dressed in pontifical ornaments, his arms crossed over his chest, his head covered by the sacred tiara; the pastoral cross still hung from his neck and the ring finger of his right hand carried the papal ring. But in a moment that body dissolved in the air, which still remained impregnated with the sweet perfumes placed in the urn; nothing else remained but the silver cross and the pastoral ring.
The Arabic numbers derive from the Indian Brahmi symbols probably dating back to 300 BC and were spread mainly by the Arab mathematicians al-Khwārizmī and al-Kindi. Despite the meritorious work of introduction of Gerbert, it was only with Leonardo Fibonacci that, at the turn of the 1200s, the Arabic numbers were adopted in Europe in a systematic and widespread manner.

The Riemann Prize to Terence Tao

Excuse me for the delay, but I read the press release only today. So I proceed to publish:
Terence Tao, a world renowned mathematician based at the University of California in Los Angeles, USA, has been announced as the first recipient of the Riemann Prize in Mathematics, awarded by the Riemann International School of Mathematics (RISM).
Terence Chi-Shen Tao is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, geometric combinatorics, arithmetic combinatorics, analytic number theory, compressed sensing, and algebraic combinatorics. Tao was a recipient of the 2006 Fields Medal and the 2015 Breakthrough Prize in Mathematics. This prolific mathematician has been the author or co-author of 275 research papers, his most impressive results being those on three-dimensional Navier-Stokes existence and smoothness.

Maths in Europe: The ultimate question

If you are a reader of the Hitchhiker's guide to the galaxy, you probably know that 42 is the answer to the Ultimate Question of Life, the Universe, and Everything. The choice of the number by Douglas Adams was quite random, excluding the simple fact that the number liked the writer. Yet the 42 was the protagonist of a recent news related to one of the open problems of mathematics:
Is there a number that is not 4 or 5 modulo 9 and that cannot be expressed as a sum of three cubes?
To find an answer to this question, mathematicians used numerical methods. In particular, Andreas-Stephan Elsenhans and Jorg Jahnel using a particular vector space, searched solutions of the following diophantine equation:
Continue on Mathematics in Europe

A primordial flash

The primordial universe continues to reveal its mysteries to us: a research group led by Christina Williams from the Steward Observatory of the University of the Arizona, using the 66 antennas that constitute the ALMA radiotelescope in Chile, has revealed a weak emission in radio frequencies, probably dued to the dust generated by stellar formation inside a giant primordial galaxy, about 12.5 billion years away from us, just more than a billion year after the initial expansion of the spacetime (the Big Bang).
Williams, C. C., Labbe, I., Spilker, J., Stefanon, M., Leja, J., Whitaker, K., ... & Weiner, B. (2019). Discovery of a Dark, Massive, ALMA-only Galaxy at z∼ 5–6 in a Tiny 3 mm Survey. The Astrophysical Journal, 884(2), 154. doi:10.3847/1538-4357/ab44aa (arXiv)

We know we don't know

A few days ago on Nature Astronomy it was published a paper by a team of italian researchers with an unequivocal title: Planck evidence for closed Universe and a possible crisis for cosmology(6). We can consider it as one of the first scientific articles that seriously takes into consideration a situation that it is becoming increasingly pressing: a crisis in cosmology.
The standard cosmological model, based on cosmic inflation(2) and on empirical constants that evaluate unknown physical quantities as dark matter and dark energy, although very well verified, has not yet passed the last step: the detection of gravitational waves in cosmic microwave background (CMB). One of the fundamental points of this model, but also of many of the surviving competing models, is the accelerated expansion of spacetime at speed greater than that of light which explains the flatness of the early universe.
This flatness emerges in particular when studying the cosmic microwave background, the residual energy of the initial expansion of spacetime. This radiation has come down to us from the point where it was produced, a little less than 14 billion years ago, crossing the whole universe. This means that in the signal detected there must also be gravitational lens effects(1) due to the amount of matter, usual and dark, present in the universe. These effects have long been known and calculated(4) and can already be seen in the image produced by Planck(5).

Perpetual motion

Popular Science's cover by Norman Rockwell, October 1920 - via commons
Like the research on the philosopher's stone, the mysterious alchemical material which should allow the transmutation of the elements, particularly of base metals into precious gold, there is the search for a tool that can generate perpetual motion, or a gear capable to move indefinitely without any need of power supply from the outside.
As we will see this research has well over a thousand years and continues today among people who genuinely (and a little naively!) looking to get what would be a considerable technological leap and scammers themselves. The best way to deal with all of these is to remember what Richard Feynman said some students who invited him to a demonstration for an engine running unless perpetual but rather long:
You have to ask yourself, 'Where is the power supply?'(1)
The magic wheel

Bhaskara's wheel
The first tool would have to create the perpetual motion was the so called magic wheel, a wheel that turns on its axis the movement of which would have to be powered by a lot of magnets. This instrument made its first appearance in the eighth century in Bavaria: designed to rotate in perpetuity was defeated in the long run, by friction, so that the magic wheel was overcome by the inevitable thermodynamic end. Although the times don't match, someone say around that this magic wheel from Bavaria is based on an earlier project proposed by the Indian mathematician and astronomer Bhaskara II, wholived in 12th century.
His most important work is the Siddhanta-Shiromani, the Crown of treatises, a poem where, among others results, he comes to approximate the derivative for the sine function: \[\frac{\text{d}}{\text{d} y} \sin y = \cos y\] He also made a demonstration of the Pythagorean theorem, and his path is crossed, as it can only in the tortuous paths of mathematics, with Pierre de Fermat, the amateur mathematician known to throw challenges to more titles colleagues, as in the case the best known Fermat's last theorem or for the following Diophantine equation: \[61 x^2 + 1 = y^2\] The latter, proposed in 1657, was resolved in 18th century by Euler, unless we consider the solution discovery by Bhaskara II already 6 centuries before.
As astronomer most of his contributions are contained in the aforementioned Siddhanta-Shiromani, where, as we have seen, he has developed some concepts about trigonometry, a branch of mathematics important, if not necessary to make observations as accurate as possible.
Bhaskara II, astronomically speaking, was heir of Aryabhata (fourth century) and Brahmagupta (seventh century) who they developed, about a thousand years in advance on European astronomers, a heliocentric model. Drawing on these theoretical and observational basis, Bhaskara II made a series of observations on celestial bodies, first of all on moon and sun.
As an engineer, however, it is best known for Bhaskara's wheel, a wheel whose spokes were partially filled with mercury. According Bhaskara it would be just that mercury to ensure the perpetual motion of the wheel(2).

The light limit of the neutrino

Neutrinos are the most light particles in the universe, but we don't know your mass. In the current state of the research, the only thing that we can hope to do is find upper and lower limits. And in the previous weeks we have some interesting news about the upper limit.
In april Physics Review Letters published a paper in which a team of researcher have compared constraints from physically motivated neutrino mass models (i.e., ones respecting oscillation experiments) to those from models using standard cosmological approximations. They founded an upper limit about $0.26 \, eV$, almost 2 million times lighter than an electron.
Loureiro, A., Cuceu, A., Abdalla, F. B., Moraes, B., Whiteway, L., McLeod, M., ... & Rollins, R. P. (2019). Upper Bound of Neutrino Masses from Combined Cosmological Observations and Particle Physics Experiments. Physical review letters, 123(8), 081301. doi:10.1103/PhysRevLett.123.081301 (arXiv
In the meanwhile, just ten days ago, the KATRIN's team (KATRIN, Karlsruhe Trtitium Neutrino experiment) announced the new experimental upper limit: $1.1 \, eV$.
Aker, M., Altenmüller, K., Arenz, M., Babutzka, M., Barrett, J., Bauer, S., ... & Besserer, U. (2019). An improved upper limit on the neutrino mass from a direct kinematic method by KATRIN. arXiv:1909.06048.
The research of neutrino mass becomes more and more interesting: if the study of theoretical models combined with astronomical data gives us an idea of the range to look for, experiments will say the last word.
The hunt to the neutrino is still open!

Maths in Europe: Lunar Arithmetic

One of the most popular expressions in Italy for giving strength to numbers is mathematics is not an opinion. The expression is exclusively Italian and mathematicians don't agree with this opinion, since they have fun inventing a large number of different mathematics. For example, a curious mathematics is what today called lunar arithmetic. In this kind of arithmetic, the sum between two digits gives the largest digit, while the product between two digits gives the smallest one. A particular consequence of the multiplication rule is the definition of prime numbers: in base 10 a lunar prime number is a number divisible only by itself and by 9, because the neutral element of lunar multiplication is 9.
Continue on Mathematics in Europe

Updates from outer space: from Earth to K2-18 b

There are some interesting news about the research of exoplanets, but the first step starting from our planet, the Earth. Indeed, Evelyn Macdonald and Nicolas Cowan used the satellite Scicast to detect the transit spectrum of our planet. The idea is to deduce the atmosphere composition, obtaining the Earth's organic signature, and in this way data to confront with exoplanets transit spectrum.
Macdonald, E. J., & Cowan, N. B. (2019). An empirical infrared transit spectrum of Earth: opacity windows and biosignatures. Monthly Notices of the Royal Astronomical Society, 489(1), 196-204. doi:10.1093/mnras/stz2047
About two weeks after the pubblication of the previous paper, an international team of astronomers have discovered water vapor in the atmosphere of K2-18 b, an exoplanet that orbit in the habitable zone of the red dwarf K2-18. While it was initially considered a mini-Neptune on its 2015 discovery, the improved data on K2-18b has classified it as a super-Earth, although its size and density make it unlikely to be composed entirely of rocky iron and silicates.
Tsiaras A., Waldmann I. P., Tinetti G., Tennyson J., Yurchenko S. N. (2019). Water vapour in the atmosphere of the habitable-zone eight-Earth-mass planet K2-18 b. Nature Astronomy. doi:10.1038/s41550-019-0878-9

Breakthrough Prize 2020: Physics and Mathematics

2020 Breakthrough Prize in Fundamental Physics to the Event Horizon Telescope Collaboration, for the first image of a supermassive black hole, taken by means of an Earth-sized alliance of telescopes.
Using eight sensitive radio telescopes strategically positioned around the world in Antarctica, Chile, Mexico, Hawaii, Arizona and Spain, a global collaboration of scientists at 60 institutions operating in 20 countries and regions captured an image of a black hole for the first time. By synchronizing each telescope using a network of atomic clocks, the team created a virtual telescope as large as the Earth, with a resolving power never before achieved from the surface of our planet. One of their first targets was the supermassive black hole at the center of the Messier 87 galaxy – its mass equivalent to 6.5 billion suns. After painstakingly analyzing the data with novel algorithms and techniques, the team produced an image of this galactic monster, silhouetted against hot gas swirling around the black hole, that matched expectations from Einstein's theory of gravity: a bright ring marking the point where light orbits the black hole, surrounding a dark region where light cannot escape the black hole's gravitational pull.
2020 Breakthrough Prize in Mathematics to Alex Eskin, for revolutionary discoveries in the dynamics and geometry of moduli spaces of Abelian differentials, including the proof of the "magic wand theorem" with Maryam Mirzakhani.
Eskin teamed with famed Iranian mathematician and Fields Medalist, Maryam Mirzakhni, to prove a theorem about dynamics on moduli spaces. Their tour de force, published in 2013 after five years of labor, is a result with many consequences. One addresses the longstanding problem: If a beam of light from a point source bounces around a mirrored room, will it eventually reach the entire room – or will some parts remain forever dark? After translating the problem to a highly abstract multi-dimensional setting, the two mathematicians were able to show that for polygonal rooms with angles which are fractions of whole numbers, only a finite number of points would remain unlit. Mirzakhani passed away in 2017, at age 40, after fighting breast cancer for several years.

via Breakthrough Prize

Spacetime analogies

The elastic sheet model is one of the most used model for telling general relativity. It allows to show as smaller balls "orbit" around the larger ball in the center of the sheet in a way similar to the planets, at least until the balls lose energy due to friction and finally fall into the "gravitational wall".
This way to see the general relativity is directly connected with the embedded diagrams and Flamm paraboloids, the mathematical way to see the spacetime deformations. But this analogy has some problems not only because is inaccurate like all analogies, but also becuase it could be confusing about distorced space and spacetime especially among students. So we can ask: why is the sheet deformed? Because of the weight? This fact implies the use of a circluar argument: usinf gravity to explain gravity! But if the ball isn't in spacetime, where is it?(1)

Come back on the road

Due to some family problems, 2017 was a difficult year and all my online activities, particularly blogs, had a considerable decrease in content. As for Doc Madhattan the effects were seen especially in the last months of 2017 and in 2018, when the last post came out in august.
In june of this year, almost a year later, I resumed posting also on this blog with the death of Murray Gell-Mann. However, I still had no idea if and how to regularly resume publications. Then the case wanted that, thanks to the first photograph of a quantum entaglement, I am selected among the contents of the week of Science Seeker, and this gives me the incentive to resume curating the contents also on this blog!
The program, which I hope to maintain, is to publish a couple of articles a month: in this way I should be able to keep up the pace, placing it between work and articles for my other italian blogs. As for the contents, I would like first of all to recover some of the notes that have remained in draft in all these months of silence, so try to write new contents from scratch. Probably I will also decrease the amount of mathematics present here on Doc Madhattan: the idea is also to take up my column on Mathematics in Europe, so probably the mathematical posts here will be an extract with the link to the complete article.
ON the other hand, physics will have all the space needed here on Doc Madhattan.
I hope that taking this road again may be even longer and more enjoyable than the one that was interrupted a year ago.

The great question about the Hubble constant

The Hubble-Lemaitre law is the mathematical formula bout the expanding universe. One of the collateral results of Einstein's theory of relativity was an expanding and non-static universe, a result that, in a first time, Einstein himself had disavowed. Yet various observations made in the second half of the 20s of the twentieth century instead confirmed the hypothesis of cosmic expansion(1, 2). \[z = H_0 \frac{D}{c}\] where $c$ is the speed of light, $H_0$ is the Hubble constant, while $z$ and $D$ are the light's redshift and the distance of the galaxy from the observer. The redshift, in particular, is due to the Doppler effect applied to electromagnetic waves. For example, when you hear the siren of an ambulance, it will seem to you stronger or weaker if approaching or moving away from your position. An electromagnetic wave, like light, instead will be closer to blue or red depending on whether it is closer to or away from the observer.
So, it has a certain importance to measure the redshift of the galaxies around us: evidently a null or little redshift was a clue to a static universe, otherwise we live in a dynamic universe, as you can see from the image present in the historical Hubble article(2):

Imaging quantum entanglement

The main object of a paper published a couple of days ago on Science is to find an answer to the following question:
what kind of imaging process could reveal a Bell inequality?
The experimental set-up used a $\beta$-Barium Borate crystal pumped by a (quasi-cotinuous) laser. The pairs of entagled photons generated are subsequently separated on a beam splitter and propagate into two distinct optical systems like LIGO interferometer.
The results is the production of some images that shots the Bell inequality violation, like the following image:
Moreover, our demonstration shows that one can detect the signature of a Bell-type behavior within a single image acquired by an imaging setup. By demonstrating that quantum imaging can generate high-dimensional images illustrating the presence of Bell-type entanglement, we benchmark quantum imaging techniques against the most fundamental test of quantum mechanics.
Moreau, P. A., Toninelli, E., Gregory, T., Aspden, R. S., Morris, P. A., & Padgett, M. J. (2019). Imaging Bell-type nonlocal behavior. Science Advances, 5(7), eaaw2563. doi:10.1126/sciadv.aaw2563

Murray Gell-Mann: Proposing quarks

In 1963, when I assigned the name "quark" to the fundamental constituents of the nucleon, I had the sound first, without the spelling, which could have been "kwork". Then, in one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word "quark" in the phrase "Three quarks for Muster Mark". Since "quark" (meaning, for one thing, the cry of the gull) was clearly intended to rhyme with "Mark", as well as "bark" and other such words, I had to find an excuse to pronounce it as "kwork". But the book represents the dream of a publican named Humphrey Chimpden Earwicker. Words in the text are typically drawn from several sources at once, like the "portmanteau" words in Through the Looking-Glass. From time to time, phrases occur in the book that are partially determined by calls for drinks at the bar. I argued, therefore, that perhaps one of the multiple sources of the cry "Three quarks for Muster Mark" might be "Three quarts for Mister Mark", in which case the pronunciation "kwork" would not be totally unjustified. In any case, the number three fitted perfectly the way quarks occur in nature.
- Murray Gell-Mann, The Quark and the Jaguar, 1995 - via