What we can say about Google and 2024 Nobel Prizes

2024 Nobel Prizes spotlight breakthroughs in AI, neural networks, and protein folding advancements.

I apologize for the delayed publication of this post, but due to the problems with the security certificate that I was writing last week, I preferred to leave this article on hold, so I'm recovering it now.

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The Abel Prize 2025: Masaki Kashiwara

Masaki Kashiwara wins the 2025 Abel Prize for groundbreaking work in algebraic analysis, D-modules theory, and crystal bases in representation theory.

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Fields Medals 2022

I hope to write something about the Ising model in the next weeks, but in the meanwhile you can read something about E8 group. Below you can find the mathematicians that awarded the Field Medals 2022:

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Hugo Duminil-Copin

For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four.

June Huh

For bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture.

James Maynard

For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation.

Maryna Viazovska

For the proof that the E8 lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis.

The entropy and the halting probability problem

The third law of thermodynamics states:

It is impossible for any procedure to lead to the isotherm \(T = 0\) in a finite number of steps.

The theorem, discovered by Walther Nernst, is equal to say:

It is impossible for any process, no matter how idealized, to reduce the entropy of a system to its zero point value in a finite number of operations.

In classical thermodynamics we can define entropy, or the variation of entropy \(\Delta S\), with the following equation: \[\Delta S = \frac{\Delta Q}{T}\] where \(\Delta Q\) is the heat's variation and \(T\) is the temperature.

The Berry's phase and the black hole

In quantum mechanics a geometric phase, also called Berry phase, is a phase difference that a given physical system acquires during a cycle in which the system itself is under the action of an adiabatic process. This phase is linked to the geometric properties of the system itself (which is a simplification, but for our purposes there is no need to go into too much detail).
It was discovered independently by Shivaramakrishnan Pancharatnam in 1956⁽¹⁾, Hugh Christopher Longuet-Higgins⁽²⁾ in 1958 and subsequently generalized by Michael Berry⁽³⁾ in 1984. This phase, although geometric, has measurable physical effects, for example in an interference experiment. An example of a geometric phase is Foucault's pendulum.
The most famous version of this experiment, designed by Léon Foucault, dates back to 1851 when the French physicist, with the aim of showing the rotation of the Earth around its axis, suspended a ball of 28 kilograms of lead coated with brass over a surface of sand using a 67 meter cable hooked to the top of the dome of the Panthéon in Paris. The plane of the pendulum was observed to rotate clockwise at approximately 11.3 degrees per hour, completing a full circle in 31.8 hours. A more refined examination shows that after 24 hours there is a difference between the initial and final orientation of the trace left on Earth which is equal to

A chaotic balance

Our mathematical history begins in a discipline that, apparently, has very little to do with mathematics: biology. In 1975 on the journal Nature Robert May, an australian ecologist, publishes a review article with a rather indicative title: Simple mathematical models with very complicated dynamics⁽¹⁾. The heart of the paper is the following equation: \[x_{t+1} = a x_t (1 - x_t)\] The equation, or logistic map, this is its name, describes the rate of change of a population in function of the parameter \(t\) (the time), that varies in a discrete rather than continuous way, while \(a\) is a constant that identifies the growth rate of a population. Insteed \(x_t\) is the ratio between the existing population and the maximum possible population at time \(t\).
The model thus described is deterministic, i.e. the population at instant 0 determines the population at subsequent instants. The equation predicts the existence of a stationary state, i.e. a situation in which the population at time \(t + 1\) is equal to the population at time \(t\). This state is stable, that is, it is maintained for a sufficiently long time, but only for \(a\) lower than or equal to 3. However when the growth rate exceeds this value, the size of the population begins to oscillate between 0 and 1, apparently in a random way. But if we observe carefully, we notice small more or less periodic recurrences, which show how the behavior of the equation is actually chaotic.

Butterflies, hurricanes and... pools!

Chaos is nothing more than order seen from the opposite side.

This defintion by Fethry Duck in the italian story Il mobile caotico (The chaotic furniture) can be considered very centered on the heart of chaos. And the mathematical tool that we used to study it is the theory of chaos.

Flapping the wings

What best identifies chaos theory is the butterfly effect, which identifies in a simple and effective way the strong dependence of chaotic systems on initial conditions. The name was first used by Edward Lorentz, who published the first article on this effect in 1963⁽¹⁾.
The popular version of the butterfly effect goes something like this: The flapping of a butterfly's wings in Brazil causes a hurricane in New York and the use of the butterfly was probably suggested to Lorentz from Ray Bradbury's 1952 short story A sound of thunder in which an unwary time traveler, stepping out of the path set by the travel agency and thus stepping on a butterfly, even manages to change the result of the last US presidential elections, allowing a fascist to become the most powerful man on the planet!
From a scientific point of view, one of the most typically chaotic problems is that of weather forecasts, because of the large amount of variables that are present. The appearance of chaotic behaviors, however, would not be so scientifically interesting if it were not for one of their particular characteristics: the fundamental laws that govern, for example, time are deterministic and individually easily solved, but by combining together a large number of such equations, not only the resolution of the system is more complicated, so much so that it is necessary to use electronic calculators, but also the solution shows a chaotic behavior graphically well identified by the Lorentz attractors:

Maths in Europe: Seven cosmic messengers

Let us suppose we travel from Earth to the furthest observable point in the universe. We have seven satellites on our spacecraft, used to keep communications between us and the Earth. Let’s suppose that the speed of the satellites coincides with that of light, or in any case equal to a speed whose difference with c is negligible, while the speed of the spacecraft is $v = 2 / 3c$. The satellite, once it reaches Earth orbit, transmits the information we have loaded into its memory, then heads back to us to collect the new information. Meanwhile, within 24 hours of each other, we launch all the satellites.
The time each probe takes will be given by the formula \[t = \frac{y_1+y_0}{c}\] where $y_0$ is the distance traveled on the outward journey (or if you prefer the relative position of the spacecraft respect to the Earth at the time the first probe was launched), $y_1$ the distance of the return (or the position of the spacecraft when the first probe returns) and c is the speed of the probe.

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Maths in Europe: John Conway

Card Colm Mulcahy, an irish mathematician and John Conway's friend, on 11 april 2020 published on twitter the news of the death of Conway. His source was a close associate of his and confirmed by the family.
I written a little post in his honour on Maths in Europe, a ahort article about the free will theorem.
The theorem was proposed by Conway with Simon Kochen, inspired by the question about the interpretation of quantum mechanics. The statement is:

If the choice of directions in which to perform spin 1 experiments is not afunction of the information accessible to the experimenters, then the responsesof the particles are equally not functions of the information accessible to them.

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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.

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:

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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:

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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⁾.

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.

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The map of mathematics

The field of mathematics is massive, but a lot of people stop dealing with more complex maths after leaving university. Or, if you're not involved in an occupational field that needs mathematics, you might stop taking mathematics courses toward the end of high school. However, math is much bigger than most people realize. This creative map of mathematics made by Domain of Science puts the entirety of the field onto a single map.
The video traces the origins of math back to counting, which the video points out is not an exclusively human activity. It then quickly mentions how early civilizations tapped into counting and transformed it, most notably Arab scholars who created the first books on algebra. From there, mathematics and the sciences exploded during the Renaissance period.
After that point, the video starts to take off and diverge into a variety of mathematics we've come to know today. Maths diverged into two categories: pure maths and applied maths. Pure maths consists of the studies of numbers systems, structures (like algebra and group theory), spaces (goemetry and trigonometry) and more. Applied mathematics then taps into the application of pure maths. It includes the calculations that help determine how life functions, like mathematical chemistry or biomathematics. It also includes daily usage of math in economics, game theory, and statistics. By the end of the 11 minute video, viewers are left in awe (or simply overwhelmed) by all the possibilities that fall under the umbrella of "maths."
There were some mistakes made in the video that have since been corrected in the video's description. One of the biggest is that the video mistakenly mentions 1 as a prime number. However, prime numbers by definition are numbers bigger than 1 that can be divided evenly only by 1 or itself. The number must be greater than 1.

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Pi stories: the Cyclometricus and other tales

Fifth period

It was 1621 when the Cyclometricus by Willebrord Snellius, a pupil of Ludolph van Ceulen, was published. Snellius proved that the perimeter of the inscribed polygon converges to the circumference twice than the circumscribed polygon. As a good pupil of van Ceulen, Snellius managed to get 7 decimal places for the $\pi$ by using a 96-sided polygon. His best result, however, was 35 decimal places, which improved his master's result, 32.
The next improvement is dated 1630 by Christoph Grienberger, the last mathematician to evaluate $\pi$ using the polygon method, while the first successful method change came out thanks to the british mathematician and astronomer Abraham Sharp who determined 72 decimal places of $\pi$, of which 71 correct, using a series of arctangents. A few years later, John Machin improved Sharp's result with the following formula and that allowed him to achieve the remarkable result of 100 decimal places! \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\] Machin's approach proved successful, so much so that the slovenian baron Jurij Vega improved on two occasions the above formula obtaining a greater number of decimal digits of $\pi$, the first time in 1789 with a formula similar to Euler's one \[\frac{\pi}{4} = 5 \arctan \frac{1}{7} + 2 \arctan \frac{3}{79}\] then in 1794 with a Hutton-like formula \[\frac{\pi}{4} = 2 \arctan \frac{1}{3} + \arctan \frac{1}{7}\] The arctangent era continued with William Rutherford \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{70} + \arctan \frac{1}{99}\] and with Zacharias Dase \[\frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{5} + \arctan \frac{1}{8}\] Finally comes the british William Shanks that pushing the full potential of the Machin's formula managed to get 707 decimal places, of which only 527 were correct after Ferguson's controls in 1946. Here, however, we are going in the era of mechanical calculation, prologue to computer era.

The man who measured Mount Everest

Radhanath Sikdar was an Indian mathematician who, among many other things, calculated the height of Mount Everest in the Himalaya and showed it to be the tallest mountain above sea level.

Mathematical joke

A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians to derive humor. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathematical concept. Mathematician and author John Allen Paulos in his book Mathematics and Humor described several ways that mathematics, generally considered a dry, formal activity, overlaps with humor, a loose, irreverent activity: both are forms of "intellectual play"; both have "logic, pattern, rules, structure"; and both are "economical and explicit".
Some performers combine mathematics and jokes to entertain and/or teach math.
Humor of mathematicians may be classified into the esoteric and exoteric categories. Esoteric jokes rely on the intrinsic knowledge of mathematics and its terminology. Exoteric jokes are intelligible to the outsiders, and most of them compare mathematicians with representatives of other disciplines or with common folk.

Wigner's theorem

The Wigner’s theorem was formulated and demonstrated for the first time by Eugene Paul Wigner on Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum⁽¹⁾. It states that for each symmetry transformation in Hilbert’s space there exists a unitary or anti-unitary operator, uniquely determined less than a phase factor.
For symmetry transformation, we intend a space transformation that preserved the characteristics of a given physical system. Asymmetry transformation implies also a change of reference system.

Invariants

Invariants play a key role in physics, being the quantities that, in any reference system, are unchanged. With the advent of quantum physics, their importance increased, particularly in the formulation of a relativistic quantum field theory. One of the most important tools in the study of invariants is the Wigner’s theorem, an instrument of fundamental importance for all the development of quantum theory.
In particular, Wigner was interested in determining the properties of transformations that preserve the transition’s probability between two different quantum states. Given $\phi$ the wave function detected by the first observer, and $\bar {\phi}$ the wave function detected by the second observer, Wigner assumed that the equality \[|\langle \psi | \phi \rangle| = |\langle \bar \psi | \bar \phi \rangle|\] must be valid for all $\psi$ and $\phi$.
In the end, if we exclude time inversions, we find that the operator $\operatorname{O}_{R}$, such that $\bar{\phi} = \operatorname{O} _{R} \phi$, must be unitary and linear, but also anti-unitary and anti-linear. Consequence of this fact is that the two observers’ descriptions are equivalent. So the first observes $\phi$, the second $\bar{\phi}$, while the operator $\operatorname{H}$ for the first will be $\operatorname{O}_R \operatorname{H} \operatorname{O}_R^{-1}$ for the second.