# Mathematics: number theorist Don Zagier’s one-line lesson

Every year, the mathematics department of the Ecole normale supérieure de Lyon organizes a weekend for its students. The recipe is always the same: about forty students, a dozen researchers or teacher-researchers from Lyon and a single outside guest. The guest is a renowned mathematician, often forty years older than the students, with the mission of presenting his or her personal vision of mathematics. The idea is to bring together an accomplished researcher with future researchers who know almost nothing about mathematics.

The place is favorable to informal meetings: the magnificent castle of Goutelas, 80 kilometers from Lyon. For twenty years, the guests have represented the diversity of mathematics: geometry, combinatorics, number theory, analysis, etc. On arrival, the students are often impressed by the reputation of the guest, but they do not know that he himself is worried about his own responsibility. He has most of his career behind him and is facing young people who have not yet started it. But this mutual anxiety does not last long, the magic works, and we often see exchanges of great richness.

The most recent guest, from October 7 to 9, was Don Zagier, 71 years old, a number theorist and a bit eccentric. He has dual American and German citizenship and is fluent in a number of languages (including French). He currently teaches in Bonn and Trieste, after having been a professor at the Collège de France. He is fascinated by numbers and formulas of all kinds, which is not so common in mathematics, contrary to what the general public thinks. He made some fundamental discoveries, but anecdotally he published a one-line proof of Fermat’s two-square theorem: a prime number other than 2 is the sum of two integer squares if and only if the remainder of its division by 4 is 1. For example, the remainder of the division of 41 by 4 is 1, and indeed 41 is the sum of 16 and 25. A one-line proof? Maybe, but a line that is probably incomprehensible to many readers of Le Monde.

A one-line proof

Zagier set the scene for the students with an assumed bad faith, by placing number theory far above geometry and therefore, according to him, even higher than topology. Many of his predecessors had expressed radically different opinions. A central theme of his lectures concerned the so-called “modular forms”, unknown to students. He began with a quip that there are five operations in arithmetic: addition, subtraction, multiplication, division and… modular forms!
His lectures consisted in showing how number theory allows us to take a fresh look at the theory of knots. A knot is the topological object one imagines: a string tied in space. Zagier reminded us of what all professionals know: that a mathematical result is all the more interesting when it builds bridges between theories that were thought to be independent. The presentations were not always easy to follow for the students, but the important point is that they saw a mathematician at work, with his passion, his personal history, his anecdotes, his vision and his conjectures.

On Saturday evening, Zagier proposed an exercise to the students. The next morning, he announced that five students had found the solution, by different ways. He admitted that he himself had searched for three years before finding a solution. Let us rejoice: the succession is assured.

# ￼Behind the scenes of the Fields medals, James Bond atmosphere

https://www.lemonde.fr/sciences/article/2022/07/10/dans-les-coulisses-des-medailles-fields-ambiance-james-bond_6134197_1650684.html?contributions

On July 5, the International Mathematical Union (IMU) awarded its prizes, including the famous Fields Medals, to Hugo Duminil-Copin, June Huh, James Maynard and Maryna Viazovska. These medals are often referred to as the “Nobel of mathematics”, but this is not a good comparison. To start with a “detail”, a Nobel laureate receives 10,000,000 Swedish kronor (more than 900,000 euros), while the Fields Medal pays “only” 15,000 Canadian dollars (or 11,000 euros).
Most importantly, a Fields Medal winner must be under 40 years of age, which is far from the case for the Nobel Prize. Moreover, even if the medal recognizes remarkable work, it also expresses the jury’s conviction that the winner will have major successes in the future. It is a question of encouraging a very promising young mathematician to continue an exceptional career: it is thus also a bet on the future.

The work of the selection committee is therefore not easy, especially since there are other constraints. The official statutes specify that no more than four medals may be awarded, once every four years, and that they must reflect the diversity of mathematics: an almost impossible mission.

In the past, mathematicians often worked alone, but this is no longer the case today, and this is to be welcomed. Unfortunately, two or three collaborators cannot share a medal for joint work. In any case, this award ceremony is a traditional event in the mathematical community, which eagerly awaits the official revelation of the laureates and does not refrain from making predictions in the year preceding.
A month ago, a journalist from Le Monde asked me if I had any information about the winners. I told him that I did not have any, but that I was pleased, because it showed that the selection committee was working in a confidential way. I did send him a few names I had in mind. A week before the announcement, the reporter knew the names of the winners he had obtained, under embargo, but he could not reveal them. My predictions turned out to be correct…

A confidential job

I know absolutely nothing about the debates that took place and that led to this choice, but I can describe the functioning of this committee, since I was a member of it, eight years ago. Without a doubt, it was the most thorough selection of laureates that I have ever had the opportunity to participate in. The committee consisted of twelve members, born in eleven different countries. The chair is the president of the IMU, but the names of the other eleven members are kept secret until the official announcement of the winners.

When one is invited to participate in this committee, three years before, one receives very strict instructions, a bit like James Bond on a secret service mission! No information is passed on about the debates that took place for the previous medals. Then began the exchange of opinions by e-mail (on secure servers), a few video conferences and two face-to-face meetings (in Zurich and New York, in discrete locations). At the last meeting, we were left with a short list of twelve names, which had to be reduced to four. Each of us was asked to make an oral presentation of one of the mathematicians still in the running. Then, we had to discuss, seek consensus and vote, to arrive at a very satisfactory result.
This year’s winners know that they were selected after a long and rigorous process. They should be proud.

Translated with DeepL

# Mathematics and war, a whole history

https://www.lemonde.fr/sciences/article/2022/05/25/les-mathematiques-et-la-guerre-toute-une-histoire_6127648_1650684.html

Archimedes is often presented as the first great scientist to have used his scientific knowledge to build war machines. During the siege of Syracuse in 212 BC, it is said that he built giant parabolic mirrors to ignite enemy sails by concentrating the Sun’s rays. Although the anecdote is certainly not true, it illustrates one of the first uses of science in warfare.
However, Archimedes was also a “pure” mathematician to whom we owe treatises on geometry that have marked the history of science. When a Roman legionary came to disturb him while he was drawing a geometrical figure in the sand, he replied: “Don’t disturb my circles”, and the soldier killed him with a sword.
Much later, during the Second World War, the Manhattan Project in the United States brought together a considerable number of engineers, physicists and mathematicians in the greatest secrecy with the aim of building the first atomic bombs, which were far more powerful than Archimedes’ mirrors. On August 6 and 9, 1945, the bombs killed over 100,000 people in Hiroshima and Nagasaki. The whole world became fully aware of the determining role of the scientific community in the war.
At the end of the First World War, like the League of Nations, many scientific disciplines created international unions. The International Mathematical Union (IMU), for example, was founded in 1920 and organized a very prestigious international congress every four years to review the progress of mathematics: a sort of Olympic Games of mathematics.

No peaceful agreement

However, one should not believe in a peaceful agreement between all the mathematicians of the world, ignoring the wars and the political conflicts. For example, at the time of the foundation of the IMU, German mathematicians were not invited, and to mark the victory, the opening ceremony took place in Strasbourg, which has recently become French again. The congresses were cancelled during the Second World War and very disturbed during the Cold War.
In Cambridge (USA), in 1950, no Soviet delegates or delegates from communist Eastern Europe participated, although several had been invited. The Soviet Academy of Sciences had claimed that Soviet mathematicians had too much work to travel. The United States had initially refused the entry visa for the Frenchman Laurent Schwartz, a communist, who came to get his Fields Medal. In 1966, Alexandre Grothendieck refused to go to Moscow to get his medal. The congress that was supposed to take place in Warsaw in 1982 was postponed and was held the following year. The history of this international mathematical union is indeed quite chaotic.
Four years ago, Saint Petersburg won the competition against Paris for the organization of the congress in August 2022. Was it necessary to rebel against this choice? Some did at the time and proposed a boycott. When the war broke out in Ukraine, all mathematicians wondered whether the St. Petersburg congress would be confirmed, cancelled or postponed? Already, a number of invited speakers before the war had declined the invitation for political reasons.
The solution proposed by the IMU is surprising: the congress will take place, but in virtual mode, by videoconference, which is not easy in practice because of the time difference. But the speakers will be able to record their conference in advance if they wish. Will they take the opportunity to denounce the war?

Translated with DeepL

# The Abel Prize rewards Dennis Sullivan, charismatic mathematician

https://www.lemonde.fr/sciences/article/2022/03/23/le-prix-abel-recompense-dennis-sullivan-mathematicien-charismatique_6118837_1650684.html

The Abel Prize in mathematics was awarded on Wednesday, March 23, to the American Dennis Sullivan, 81, “for his revolutionary contributions to topology in its broadest sense, and particularly in its algebraic, geometric and dynamical aspects,” announced the Norwegian Academy of Sciences and Letters. While the Fields Medal is awarded to a mathematician under the age of 40, the Abel Prize is closer to the Nobel Prize (which does not exist in mathematics) and rewards a whole career.
Towards the end of the 17th century, Leibniz dreamed of manipulating forms, in the manner of the abstract symbols of algebra. He gave the name of analysis situs to this theory, which he could not develop and which was only firmly established at the end of the 19th century by Henri Poincaré. In this theory, which we now call topology, we consider that the surface of a sphere is equivalent to that of a cube, because we can deform one into the other, if we imagine them made of rubber. On the other hand, the sphere is not equivalent to an inner tube. One studies curves, surfaces and more generally much more complicated “varieties” in any dimensions. Among Sullivan’s major contributions is his theory of rational homotopy, which makes it possible to understand the topological structure of varieties by associating them with objects of an algebraic nature, which can in principle be computed, thus fulfilling Leibniz’s dream in a way.

Unsuspected bridges

Sullivan moved effortlessly from one chapter of mathematics to another and discovered unsuspected bridges that led him to entirely new points of view. For example, he established a “dictionary” between two theories that were thought to be independent (Kleinian groups and holomorphic dynamics). All he had to do was to translate a theorem from one theory to obtain the solution of an important problem in the other, which had resisted for nearly seventy years (the theorem of the non-wandering domain). He is neither a geometer, nor a topologist, nor an algebraist, nor an analyst: he is a bit of all of these at the same time. Very few mathematicians have such a strong sense of the deep unity of mathematics. For a few years, he has been trying to export his topological ideas to a major problem in fluid dynamics. The experts are not (yet) convinced, but this may lead to a resounding success.
Sullivan is also remarkable for his exceptional charisma. For many years he was a hub in the mathematical community. Always surrounded by a wide variety of researchers, especially very young ones, he has an incredible ability to listen, share, motivate and encourage. He is the opposite of the epinal image of the solitary mathematician. When he was a professor at the IHES, in Bures-sur-Yvette (Essonne), you had to see him at tea time putting in contact mathematicians of all backgrounds and all ages who did not know each other, in all simplicity. His seminar in New York was very well attended and had nothing to do with a traditional lecture: questions came from all sides and the lecturer had to be prepared to speak for many hours, until general exhaustion. He was one of the first to record these seminars on VHS video cassettes, starting in the early 1980s. They are now collectors’ items.
One day, I was sitting next to him during a conference where I couldn’t understand a word the speaker was saying. As I asked him if he did, he replied, “I don’t understand the words, but I listen to the mathematical music!”

# The “game of life”, fruitful imitation

https://www.lemonde.fr/sciences/article/2022/02/16/le-jeu-de-la-vie-imitation-feconde_6113863_1650684.html

The Arte.tv channel has put online ten videos of ten minutes each, proposing ten “trips to the land of maths”, directed by Denis van Waerebeke. Of course, they are not part of the live program, but they will remain available until the end of 2026, which is even better. Ten very successful trips, both aesthetically and conceptually. In ten minutes, “you encounter epic landscapes, dizzying ideas, and sometimes even useful things,” as explained in the preamble.

The videos are all worth watching, but here’s a teaser for one of them, titled “The Game of Life”. The question of whether a virus is alive is a bit artificial as long as there is no definition of the word “alive”. Without clear definitions, there can be no science. In 1940, John von Neumann proposed a very theoretical definition. First of all, a living being must be able to reproduce. But this is not enough, because we can easily imagine a robot with its own assembly plan that moves around looking for the parts necessary for its replication. It can then make a copy of itself. However, nobody would call such a robot alive.

A living being must do something else than reproduce itself: Neumann asks that it can simulate a Turing machine, in other words that it can do what our computers do. This is a very abstract definition of life! In 1944, the physicist Erwin Schrödinger, one of the founding fathers of quantum physics, published a book entitled What is life? Today, this book is obsolete because the structure and functioning of DNA was not known at that time. But it contained the essential idea that a living cell must contain some sort of reproducible code.

Create infinite shapes

Around 1968, the mathematician John Conway invented a very simple game that tried to imitate life. On a (infinite) board divided into squares, like a huge checkerboard, one places a few tokens that draw a certain shape. The rule of the game is as follows. For each token, you count the number of tokens that are adjacent to it. If this number is 2 or 3, we leave it in place. If not, it is removed from the board. On each empty square surrounded by exactly three tokens, you place a new token. The initial shape then becomes a new shape, and the operation is repeated… We thus see a succession of shapes. At the beginning, Conway worked with real chips on a go board, but very quickly computers allowed to simplify the work. By dint of testing, he discovered a number of configurations that seemed to oscillate and periodically return to their initial position.

In 1970, he offered a \$50 reward to the person who would discover a configuration whose size would tend to infinity during its development. He had to keep his promise because a “cannon” shape was found that regularly sends out “cannonballs”. Afterwards, there was a real craze for this game because it is very easy to program on a computer and anyone can play it. Download the free Golly application to have fun. In 1982, Conway demonstrated that it is indeed “life” as defined by Neumann. Today, the progress is incredible. Some configurations are made of a “membrane” that contains a filament of “DNA”. These virtual and abstract “living beings” can have a “sexual” reproduction in which the filaments mix. They can mutate and evolve, as in real life. You can find nine other mathematical journeys on Arte.tv.

# The unfortunate eclipse of cosmography

https://www.lemonde.fr/sciences/article/2021/12/21/l-eclipse-malheureuse-de-la-cosmographie_6106872_1650684.html

This Tuesday, December 21 will be the winter solstice in the Northern Hemisphere. At noon, the height of the Sun above the horizon will be the lowest of the whole year. In the following days, the Sun will rise only slightly higher in the sky and this explains the origin of the word: sol (for “Sun”) and sistere (for “stop”). This will also be the shortest day of the year. The following days will lengthen very slowly, which is much needed in the dark times we are living in. But this lengthening of the day is not symmetrical between sunrise and sunset: the Sun has already been setting later and later since December 18 and will continue to rise later and later until January 6.

In space, the Earth turns on itself in one day like a top, around an axis that passes through its poles, an axis that is not quite perpendicular to the plane of the Earth’s orbit around the Sun and that points to the North Star. Our winter solstice occurs when the direction of this axis is closest to the direction of the Sun.

Many people still think that winter is the time of year when we are farthest from the Sun, but this is of course absurd: when the Northern Hemisphere is in winter, the Southern Hemisphere is in summer. On the contrary, the Earth will be closest to the Sun on January 4th. If it is colder in France in December than in June, it is because the Sun’s light hits us more sharply. In our latitudes, we only receive during winter about 40% of the solar energy received during the summer.

Understanding your place in the Universe

All this has been well known for a long time, but unfortunately it seems that many of our fellow citizens are unaware of it. Is there a regression in the public’s scientific knowledge, even if it is only elementary? Since the beginning of the 19th century, high school mathematics programs have included a section called “cosmography”. It was about describing the main characteristics of the Solar System: the seasons, eclipses, the Sun and the planets, comets, etc. A famous mathematician, now deceased, explained to me that when he took the oral test for the agrégation in mathematics in 1948, the jury asked him to present the theory of the phases of the Moon, which would cause great difficulty to many candidates today.

The mathematics teachers did not appreciate this teaching, which they considered to be outside their discipline. However, these were lessons in spatial geometry – literally – that allowed students to better understand their place in the universe. But the arrival of “modern mathematics”, which privileges abstraction and rejects any link with natural sciences, put an end to the teaching of cosmography as early as 1968, without anyone being bothered by it. It must be said, to justify the abandonment of these subjects in mathematics courses, that astronomy was gradually transformed into astrophysics, which is more similar to physics.

Today, school geometry programs ignore astronomy. What a shame to disconnect mathematics and nature in this way! Didn’t Galileo say that nature is written in mathematical language? What a shame to forget the etymology of the word “geometry”, which reminds us that it was first and foremost about measuring the Earth. Let’s hope that the two hours of science instruction, which are now part of the core curriculum of the general high school curriculum, will allow students to (carefully!) look up at the Sun and understand its apparent movement in the sky.

# The mathematician Jacques Tits has died

https://www.lemonde.fr/disparitions/article/2021/12/08/le-mathematicien-jacques-tits-est-mort_6105202_3382.html

The immense work of the mathematician Jacques Tits, who died on December 5 in Paris at the age of 91, profoundly transformed geometry in the 20th century. Born on August 12, 1930 in Uccle (Belgium), this child prodigy defended a doctorate in Brussels at the age of 20. After a stint in Germany, he spent most of his career at the Collège de France, where he held the Group Theory chair from 1973 to 1999. Among the many prizes he has received are the Wolf Prize, in 1993, and the Abel Prize, in 2008.

The concept of groups is central to contemporary mathematics. Didn’t Henri Poincaré say that “mathematics is only a story of groups”? Here is how Jacques Tits described his research theme in the introduction to the notice presenting his work to the Academy of Sciences: “The theory of groups can be summarily defined as a theory of symmetry, indistinguishability and homogeneity; the link between these notions is clear: an object possesses a certain symmetry if different angles of view give indistinguishable images of it, a medium is homogeneous if
its points are indistinguishable. The idea already appeared in Greek mathematics, where figures with a high degree of symmetry played an essential role.

He geometrized algebra

Tits has in fact devoted his scientific life to a long reflection on symmetries in a very general sense. Groups appeared in science at the beginning of the 19th century thanks to the imagination of Evariste Galois. At that time, they were purely algebraic ideas: equations were manipulated and symmetries were sought. Towards the end of the century, Felix Klein published his “Erlangen program”, which asserted that the study of geometry was the same as the study of groups. Geometry was thus subordinated to algebra. Jacques Tits worked in the other direction: he geometrized algebra.

In order to realize his program, he invented what we now call “Tits buildings”, which are geometric objects that embody algebraic groups. It must be said that mathematicians often use words that have very little to do with the meaning given to them in everyday language, which often contributes to the fact that we do not understand them. These Tits buildings have apartments, rooms and walls, but the analogy ends there, as a room can be located in two different apartments at the same time.

To tell the truth, the terminology initially proposed by Tits was in very bad taste: there were cemeteries, ossuaries and skeletons! And yet his buildings are concrete, made of segments, triangles or tetrahedrons assembled together, in the manner of Plato’s polyhedra. At a conference in his honor in 2000, he explained that he preferred “palpable” mathematics, which might surprise a neophyte who dares to read one of his articles. To caricature in the extreme, one can indeed say that algebra is the domain of abstraction while geometry deals with more manipulable objects. Geometers and algebraists have very different approaches to mathematical activity. Jacques Tits was above all a geometer. He was always joking and good-humored, and he gave me a black look one day when I dared to suggest that he was also an algebraist.

Tits attached great importance to his role as editor of the Mathematical Publications of the Institut des hautes études scientifiques, which he held for twenty years. While proofreading an article of mine that was to appear in this journal, I noticed to my surprise that a letter “i” inside a word had been printed upside down, with the dot underneath rather than above. When I asked Tits about this, he told me with a broad smile that this magazine was still printed in the old fashioned way, on a Linotype, with lead type, and sometimes a typeface would turn over. He then showed me that by closing your eyes and stroking the paper, you could feel the mathematical content. The mathematics of Tits was really palpable.

Etienne Ghys (perpetual secretary of the Academy of Sciences, director of research (CNRS) at the ENS Lyon)

# Good at Math: Beware of Gender Stereotypes in Elementary School

https://www.lemonde.fr/sciences/article/2021/11/03/bons-en-maths-gare-aux-stereotypes-de-genre-a-l-ecole-primaire_6100739_1650684.html

Since 2018, all elementary school students are assessed three times: in September when they enter CP, at the end of January in the middle of the school year, and in September when they enter CE1. These are standardized and relatively short assessments in French and mathematics, identical in all schools in France. We therefore have precise information on the skills of 700,000 children.

A recent memo from the National Education Scientific Council, entitled “What do we learn from the CP-CE1 assessments?” draws some conclusions from this mass of data. Some of them will not surprise school teachers. For example, in the same class, children born in January do significantly better than those born in December of the same year. This is to be expected: when you are six years old, one year more or less is a huge difference. It goes without saying that teachers know this and take it into account. Another observation, unfortunately obvious, is that schools in disadvantaged areas have lower results. To reassure oneself about the role of the school, we fortunately verify that at the beginning of the school year in CE1 this deficit has been partially reduced, even if it remains.

A startling finding is the extreme speed with which a difference is developing between boys and girls in mathematics. The numbers are truly alarming. Upon entering first grade, boys and girls have exactly the same math skills. Just five months later, boys have significantly better results, and a year later, when they enter second grade, the gap has widened. This does not depend on the type of school, social position, or age of the students.

Seeing beyond calculation

How can we understand this distressing phenomenon? In many countries, it is girls who perform better. International surveys show that many countries have managed to close the gender gap in mathematics. We will have to uncover the gender stereotypes that lurk in schools and in our society. Many of us unconsciously think that mathematics is more masculine than feminine. We do not behave the same way towards a little girl and a little boy when it comes to mathematics. We also need to be careful that these mathematical assessments are primarily about numbers. However, mathematics, especially in school, goes far beyond calculation: there are also the rudiments of logic or the manipulation of shapes, which are not part of the assessments and for which one should not draw hasty conclusions.

En français, à l’entrée en CP, il existe un avantage pour les filles qui se réduit en janvier, pour réapparaître de manière plus faible en CE1

Surprisingly, this phenomenon does not occur in French, or more precisely in language skills. At the beginning of the first grade, there is an advantage for girls that diminishes in January, only to reappear in a weaker way in the second grade. These are other gender stereotypes that need to be deconstructed.

The main purpose of these assessments in elementary school is not only to determine the state of the art of children’s skills at the national level: a simple survey would suffice. It is above all to offer teachers a tool that allows them to measure as objectively as possible the progress of each of their students, and to detect in time any difficulties that may arise in their learning. For the moment, only a small half of school teachers say that these assessments have allowed them to detect problems in their students. It is not surprising that they are wary of these national assessments, which are a little too prescriptive: who knows the students better than the teachers?

# Math in your sanitary pass

https://www.lemonde.fr/sciences/article/2021/09/15/des-maths-dans-votre-passe-sanitaire_6094702_1650684.html

QR codes are loaded with math. The health pass consists of 7,225 little white or black squares, arranged in 85 rows and 85 columns, that encode vaccination status, or test result, or certificate of recovery. This poses some very interesting mathematical and computer problems.

The first problem is geometric. The optical reader that scans the QR code sees the square in perspective, as any quadrilateral. So the perspective must be straightened: this is fairly easy. You also have to recognize the top and bottom, the right and left. This is also quite easy because three of the four corners are decorated with small 7 × 7 squares that are easily recognizable. Sometimes the QR code is presented on a sheet of paper that has been folded or crumpled and the rows and columns are not straight: you have to rectify them. Thirteen 5 × 5 squares, also recognizable, are spread out in the large square to help the software get it all straight.

The second problem comes from the fact that the reader can get confused because some of the little squares can be damaged. You have to use error-correcting codes that produce deliberately redundant messages, to make sure you get back what you need. Airplane pilots have known this for a long time by saying “Papa, Tango, Charlie” instead of “PTC”. QR codes use a more elaborate method, invented by Irving Reed and Gustave Solomon in 1960 and based on deep arithmetic theorems. The result is remarkable, since the reading can be done correctly even if 30% of the small squares are unreadable. Try making an ink stain (not too big) in the middle of your health pass and you will see that it is still valid.

Finally, the authenticity of the document must be guaranteed. Here again, very subtle mathematical and computer methods are used. Anyone can read the content of the certificate (provided they know a little about computers) but it is accompanied by an encrypted and unforgeable “digital signature” produced from the content of the message using an asymmetric secret code. The idea is that some operations are easy to do and almost impossible to undo.

Isn’t it said that it is easier to get toothpaste out of the tube than to get it in? The tube in question is still mathematical, based on 19th century arithmetic, greatly improved by 20th century computer scientists. Thanks to these methods, the TousAntiCovid Verif application can guarantee authenticity: we can verify a signature that a forger could not have produced.

Possible malpractice

However, not everything is perfect and malfeasance is possible. Access codes to Medicare servers can be stolen, or a dishonest caregiver could make a fake vaccination certificate. On the other hand, TousAntiCovid Verif only guarantees the validity of the pass, and does not provide any information other than name and date of birth. However, the QR code contains other data, such as the date of vaccination, type of vaccine, etc., which are intended for border crossings and which should not be accessible to everyone. Even if it is not legal, many websites allow to read and store the complete content of health passes.

Two centuries of mathematics have passed since the pioneering work of Carl Friedrich Gauss and Evariste Galois led to the emergence of modern cryptography. They would have been the first to be surprised to see that they are at the origin of these small black and white squares. Science takes its time and reserves surprises.

# The Sun is a kind of gigantic photon pinball machine

https://www.lemonde.fr/sciences/article/2021/06/16/le-soleil-est-une-sorte-de-gigantesque-flipper-a-photons_6084325_1650684.html

Les grands prix des fondations de l’Institut de France ont été remis le 2 juin à douze lauréats : écrivains, artistes, ou scientifiques. Josselin Garnier, professeur de mathématiques à l’Ecole polytechnique, a remporté le prix de la Fondation Simone et Cino Del Duca sur le thème « Phénomènes de diffusion : théorie mathématique et applications ». Cette distinction permettra à son équipe d’étudier la propagation des ondes dans des milieux aléatoires. Les phénomènes ondulatoires sont omniprésents : le son, la lumière, les ondes radio, la houle dans les océans, les tsunamis, une corde de guitare, les tremblements de terre, sans oublier les fameuses vagues successives apportées par la pandémie. Les mathématiciens comme les physiciens s’y intéressent au moins depuis le XVIIe siècle. Aujourd’hui, notre compréhension des ondes s’est considérablement enrichie mais il reste encore beaucoup à faire.

Lors de la remise des prix, Josselin Garnier a donné un exemple pour illustrer son propos. La vitesse de la lumière dans le vide est de 300 000 kilomètres par seconde et la distance entre le Soleil et la Terre est de 150 millions de kilomètres, si bien que la lumière met environ huit minutes pour venir jusqu’à nous. Mais bien entendu, cette lumière est créée quelque part à l’intérieur du Soleil et il faut d’abord que les photons qui portent la lumière trouvent un chemin qui les mène à la surface. Cette recherche de la « sortie » nécessite quant à elle quelques dizaines… de milliers d’années.

Enchaînement frénétique

L’intérieur du Soleil est un plasma incroyablement dense et chaud. Des noyaux d’hydrogène fusionnent en permanence pour former de l’hélium. Des photons sont absorbés et d’autres sont créés et repartent dans une direction aléatoire. C’est cet enchaînement frénétique d’absorptions et d’émissions qui met quelques dizaines de milliers d’années pour produire un photon à la surface, qui peut alors s’échapper et arriver chez nous huit minutes plus tard. On peut penser à la bille d’un flipper gigantesque qui percute toutes sortes de bumpers avant de finir par sortir du plateau. On peut aussi penser à la marche d’un ivrogne qui fait des pas dans des directions aléatoires. Dans ces conditions, il faut bien sûr beaucoup de temps pour trouver la sortie. De grands progrès ont été réalisés ces dernières années dans la théorie de la diffusion des ondes dans les milieux aléatoires, à la frontière entre la théorie des probabilités et l’analyse mathématique.

En 1946, le physicien Léon Brillouin affirmait que « toutes les ondes ont des comportements analogues ». Des idées et des méthodes communes s’appliquent en effet à des ondes de natures très différentes et constituent aujourd’hui un chapitre important de la physique mathématique. Josselin Garnier s’intéresse principalement à la propagation des ondes sismiques, pour mieux décrire la structure interne de la Terre. La première difficulté provient du fait que l’intérieur du globe n’est pas homogène et contient des irrégularités réparties de manière aléatoire : les ondes ont donc tendance à se comporter un peu comme la lumière dans le Soleil. C’est surtout l’existence de la croûte terrestre qui complique le problème, car les ondes de volume, qui traversent la Terre, interagissent avec les ondes de surface.

Les recherches du lauréat se concentreront donc sur la propagation des ondes à l’intérieur d’un corps hétérogène et limité par une surface. Nul doute que, selon l’adage de Léon Brillouin, les progrès réalisés pour les ondes sismiques permettront de mieux comprendre la propagation des ondes dans des tissus biologiques, comme par exemple dans un corps humain, lui aussi irrégulier et possédant également une surface. De belles applications concrètes en perspective.

# Napoléon Bonaparte and science

“If I had not become general-in-chief and the instrument of the fate of a great people, […] I would have thrown myself into the study of the exact sciences. I would have made my way along the path of the Galileos and the Newtons. And since I succeeded constantly in my great undertakings, well, I would have distinguished myself highly also by scientific works. I would have left the memory of beautiful discoveries. No other glory could have tempted my ambition. ”

These words of Bonaparte, reported by Arago, confirm to us that he did not lack ambition. But it is much more interesting that his ambition also turned to science, suggesting that he could even surpass Newton, even though Lagrange had declared – naively – that this was impossible! In the history of France, some of our kings, emperors, or presidents have supported science, but Napoleon Bonaparte is probably the only one who would have dreamed of being a scientist… if he had not been “the instrument of the fate of a great people”.

Bonaparte loved science but he understood very quickly that he could use scientists to develop his political project. In return, scientists loved him and supported him, sometimes slavishly. Monge, the mathematician, and Berthollet, the chemist, were literally fascinated by the young general during the Italian campaign. They managed to get Bonaparte elected to the National Institute in 1797 when he was only 28 years old and his scientific contributions were non-existent, and… will remain so. The general took the chair of Lazare Carnot, who was a much better scientist than he was, but who had just been expelled from the Institute following the coup d’état of Fructidor, of which Bonaparte was one of the instigators. The Institute showed a self-serving foresight in securing the favors of the man who would later become its protector. Bonaparte often used the prestige of his new status and signed his letters “Member of the Institute, General-in-Chief, Bonaparte”.

It is said that on December 11, 1797 Bonaparte dined with some influential members of the Institute to ensure his election, which was to take place two weeks later. To show off his mathematical skills, he explained to Laplace – the so-called French Newton – how to find the center of a circle if you only had a compass and no ruler. Laplace would have exclaimed “We expected everything from you, general, except lessons in geometry”. Did Bonaparte mention that this geometrical construction was in a way a war prize, since he had obtained it from a Milanese mathematician, named Mascheroni, whom he had just met during the Italian campaign? It is – perhaps – what convinced Laplace to vote for Bonaparte.

Then came the Egyptian campaign, which ended in a military defeat but a remarkable scientific success. Do we know that Bonaparte was sufficiently convincing for 160 scientists to accept to embark in Toulon with 50,000 soldiers, without having any idea of their final destination? The only information given to the geologist Dolomieu was that “where we go, there are mountains and stones”. Had we ever seen in history an army of invaders joined by mathematicians, naturalists, archaeologists and philologists? War and science sometimes make alliances. On the deck of the ship that took him to Alexandria, Bonaparte educated himself and organized scientific conversations, to the great displeasure of the soldiers who found it all useless. Science conferences on board a warship! As soon as he arrived in Egypt, after the victory of the Pyramids (“forty centuries contemplate you”), the Institute of Cairo was founded in the image of the National Institute: president Monge, permanent secretary Fourier, vice-president Bonaparte. Behind the troops trampling in the desert in pursuit of the Mamelukes, Monge wrote articles explaining the phenomenon of mirages and Berthollet understood the nature of chemical equilibrium by observing lakes of natron.

Bonaparte fled Egypt in a hurry at the end of 1799, before the military disaster, abandoning his army and most of the scientists of the expedition. But his lifelong friends, Berthollet and Monge, were on the return trip to Paris. A few days later, the coup d’état of 18 brumaire, the end of the Directory, the beginning of the Consulate, which will then lead to the Empire and the absolute power of Napoleon Bonaparte, until Waterloo, in 1815.

The period of the Consulate and the Empire was probably the most glorious in the history of science in France. Here are a few names that sound like a list of streets in Paris: the mathematicians Fourier, Lacroix, Lagrange, Laplace, Legendre, Monge, Poisson, the astronomers Arago, Cassini, Lalande, the physicists Ampère, Biot, Borda, Carnot, Coulomb, Fresnel, Haüy, Malus, the chemists Berthollet, Chaptal, Charles, Fourcroy, Gay-Lussac, the naturalists Cuvier, Geoffroy Saint-Hilaire, Lamarck, the Jussieu brothers, the doctors Laennec or Sabatier, and I forget many!

Napoleon was a great supporter of science during this period. A support not only of principle, but especially financial. Scientists have probably never been so well paid in our history: enough to make contemporary scientists dream. Very generous prizes were distributed by the Institute. For example, impressed by Volta’s experiments, the emperor offered a considerable sum of money to advance the nascent theory of electricity.

Napoleon Bonaparte was convinced that scientists should play a major role in political life and he placed some of them in the highest positions. Never has the French political world been so aware of the latest advances in science. Should we be inspired by it today? Certainly, the first attempt was a failure. Three days after the 18 brumaire, Laplace was named minister of the Interior. The First Consul dismissed him six weeks later, and justified himself by writing, “A first-rate geometer, Laplace soon proved to be a more than mediocre administrator; from his first work we immediately understood that we had been mistaken. Laplace did not treat any question from a good point of view: he looked for subtleties everywhere, he had only problematic ideas and finally he carried the spirit of the infinitely small into the administration. “But Napoleon knew how to make remarkable choices of great servants of the State among the best scientists, heirs of the Enlightenment. I will cite only two emblematic examples, Fourcroy and Chaptal.

Fourcroy, a chemist, was the author of an overhaul of the French educational system, with the creation in particular of the famous Napoleonic lycées in 1802. These were boarding schools for boys with a quasi-military discipline that trained the elite that the centralized power needed to maintain order. Precise programs were imposed by law. All this is not very conducive to individual creativity and we still feel the deleterious effects today. At the same time, science was finally given the place it deserved: a real revolution compared to the Ancien Régime. Latin, history and geography were taught, of course, but also, on an equal footing with the humanities, mathematics, physics, chemistry, natural history and mineralogy, throughout a six-year curriculum ending with studies in Latin and French belles lettres and so-called transcendental mathematics. Alas, the implementation of this system was laborious and from 1809, with the creation of the Imperial University, the beautiful equality was to regress, and scientific education was to virtually disappear during the Restoration. Science was then reproached for distracting from religion. During the nineteenth century, the teaching of science will experience ups and downs and it will be necessary to wait for the great pedagogical reform of 1902 to see a very partial rebirth of science in high school. Today, science is still the poor relation of the elementary school.

As for Chaptal, his contribution goes far beyond the production of sugar from sugar beet, when the continental blockade prevented the importation of sugar from cane. He was an excellent Minister of the Interior, giving an impulse to the industrialization of France that would continue throughout the century. He updated the way the medical professions functioned and reformed the hospitals. He promoted vaccination with enthusiasm, without making it compulsory, as it is today. He organized the road network, re-established the chambers of commerce, and set up the first public statistical services, important for a good national administration. He never hesitated to oppose the emperor, who did not hold it against him.

Napoleon protected the Institut de France, sometimes excessively so: in the law of 11 Floréal of the year X, we read “that no establishment may henceforth take the name of Institute. The National Institute will be the only public institution that will bear this name”. This law has not been repealed to this day and seems to be little applied! In return, the Institut de France did not fail to show its affection for the emperor, for example by inaugurating with great pomp a majestic statue in the Palais Conti. Napoleon is represented in imperial costume and his right hand rests on a small column on which is engraved a Minerva, symbol of the Institute. During the ceremony, a very obsequious lyrical song was performed. Scientific and political circles know flattery.

Of course, such intimate ties based on mutual seduction can only lead to crises when trust is called into question. From Elba, during the first Restoration, Napoleon noted with bitterness the eagerness with which the Institute had disowned him. Had not the president of the Institute written, the day after the abdication of the emperor: “With liberty, we find the king that our wishes called for”? After the flight of the eagle, back in Paris, the emperor expressed his irritation through Lazare Carnot, who had become his minister of the interior. He no longer wished to be a member of the Institute, he was no longer one of their colleagues but he was their superior and the title that should be given to him from now on was that of protector of the Institute.

Napoleon’s love for science was not feigned. After Waterloo, he believed he could escape to America without difficulty. He said to Monge: “Idleness would be the cruelest torture for me. Condemned to no longer command armies, I see only the sciences that can strongly seize my soul and my mind. Learning what others have done is not enough for me. I want to leave in this new career, works, discoveries, worthy of me. I need a companion who will first of all bring me quickly up to date on the current state of science. Then we will travel together across the new continent, from Canada to Cape Horn, and in this immense journey we will study all the great phenomena of the physics of the globe, on which the learned world has not yet pronounced itself. “Monge exclaimed: ‘Sire, your collaborator is found: I accompany you! “. Napoleon replied that his friend Monge was too old to embark on the adventure. Sire,” replied Monge, “I have your business with the person of one of my young colleagues, Arago. “The young Arago did not accept the offer. It is understandable, he had much better things to do in France. Later, on St. Helena, Napoleon would say of Monge: “He loved me like a mistress, and I returned him well. As for Monge, he would confess around the same time, “I had four passions: geometry, the Polytechnic, Berthollet and Bonaparte. ”

Indeed, Napoleon and science were passionately in love.

# To understand the risks of a vaccine, let’s listen to psychologists

https://www.lemonde.fr/sciences/article/2021/04/26/vaccins-pour-apprehender-les-facteurs-de-risque-il-faut-ecouter-les-medecins-et-les-mathematiciens-mais-aussi-les-psychologues_6078055_1650684.html

How can we understand the public’s distrust of AstraZeneca’s vaccine? On the one hand, one in 700 French people has died from Covid-19 in the last year. On the other hand, one case of thrombosis per 100,000 eliminated. Calculating the probabilities will not be enough. A smartphone application, called Risk Navigator, assesses the risks involved in common activities. The unit of measurement is the “micromort”: a probability of 1 in 1 million of dying. Thus, 1,000 km in a car costs 3 micromorts. But humans almost never perceive risks in terms of chi#ers or micromorts. Fortunately, we are not calculating machines. Our behavior is often irrational, and that’s good. vaccinations. The balance seems clear: the risk of thrombosis is 140 times lower than that of Covid-19. And yet, mistrust has set in and will be difficult to eliminate. Calculating probabilities won’t be enough. A smartphone application called Risk Navigator assesses the risks
in common activities. The unit of measurement is the “micromort”: a probability of 1 in 1 million of dying. Thus, 1,000 km in a car costs 3 micromorts. But humans almost never perceive risks in terms of chi#ers or micromorts. Fortunately, we are not calculating machines. Our behavior is often irrational, and that’s good.

The debate is not new. Smallpox inoculation – the voluntary transmission of an attenuated form of the disease – dates back to the 18th century in Europe. An inoculated child had a 1 in 200 “chance” of dying within a month, but if he or she survived, he or she would not be contaminated for life, at a time when 1/8 of the population died of smallpox. How can we compare these fractions 1/200 and 1/8? Are they of the same nature? Is it legitimate to risk someone’s death to protect them from a disease they might never catch? The Swiss mathematician Daniel Bernoulli published a remarkable work in 1766 in which he compared two populations, depending on whether they used inoculation or not. Using the statistical data at his disposal, he showed that, while 1/200 of the children died quickly when everyone was inoculated, life expectancy increased by three years. He concluded that it was necessary to inoculate.

The discussion that followed was fascinating in this century of the Enlightenment where the value of human life was being questioned. The mathematician D’Alembert was convinced of the advantages of inoculation, but he thought that these “are not of a nature to be appreciated mathematically.” He opposed many arguments, such as the fact that one cannot compare an immediate death with another in an indeterminate future.

Instinctive decisions

For the past few decades, psychologists have been studying how we perceive risk. They have described and measured a large number of systematic biases. For example, we accept risks that are much greater when we choose them (such as driving a car) than when we cannot (such as a nuclear accident). Similarly, we minimize risks if they threaten us only in the indefinite future (like smoking). And we exaggerate a risk that is widely reported in the media (like thrombosis). These biases are universal and we cannot get rid of them with mathematics courses. They are part of human nature. Even experts are subject to them as soon as they leave their field of expertise. On the other hand, the good news is that these biases are now well understood by psychologists and can be explained to the public, something that schools and the media unfortunately do very little about. It’s not about making calculations but about understanding our behavior and controlling our risk-taking. We make most decisions instinctively, but when things get serious, we must learn to think and analyze our irrational reactions. Listen to the doctors and mathematicians, of course, but also to the psychologists. You can accept your uncontrolled fear of spiders, but for the risks that really threaten you, take the time to educate yourself and think before you make a decision!

# The index theorem, at the top

American mathematician Isadore Singer died on February 11, at the age of 96. With his collaborator Michael Atiyah, who died in 2019, he had proved the index theorem, famous among mathematicians, which earned them the Abel Prize in 2004. The exceptional importance of this theorem is attested by the fact that it establishes an unsuspected link between two parts of mathematics that were previously distant, analysis and topology, but also by its consequences in theoretical physics. We often think, wrongly, that the role of the mathematician consists in solving equations. In fact, there are all kinds of equations. Many of those encountered in physics involve unknowns that are functions rather than numbers. These are called differential equations and their study is part of “mathematical analysis”. It is rare that we know how to solve this type of equation, but the index theorem allows us to count the number of their solutions, which is often sufficient for applications. Atiyah and Singer associate to the equation an object called a “fibered”, the study of which is part of topology, and on which one can directly read the number of solutions. A bridge is thus established between analysis and topology.
The theorem was proved in 1963 but Atiyah and Singer did not publish a proof until 1968. In fact, they waited until they had three different proofs, a bit like a summit that one reaches by several ways, each one bringing a new perspective. All this did not suddenly appear in their minds. For more than twenty years, they developed their ideas based on many previous theorems that did not seem to be related. The most important advances in mathematics are often syntheses: heterogeneous results suddenly appear as mere special cases of a much more powerful theory.

The external and the internal

A few years later, the link with physics became clear. The “gauge theory” of physicists was very close to the “fibers” of mathematicians. The index theorem became a crucial tool in quantum physics. It can be seen as an example of the “unreasonable efficiency of mathematics in the natural sciences”, to use a famous expression of the physicist Eugene Wigner.

The links between physics and mathematics are as old as science, and opinions differ. The mathematician Vladimir Arnold asserted that mathematics is only a chapter of physics. Others insist instead on the importance of mathematics as an abstract and autonomous discipline. The view of Atiyah and Singer is intermediate. According to them, almost all mathematics was born from external reality, for example what concerns numbers, but then it turned to internal questions, such as the theory of prime numbers. Other parts of mathematics, on the other hand, are closer to the external world and physics plays a crucial motivating role. The strength of mathematics lies in these two complementary components: external and internal. In 1900, David Hilbert stated that “a mathematical theory can only be considered complete if it is so clear that you can explain it to the first person you meet in the street”. Alas, we will have to wait a little longer before we can clearly explain the index theorem to the readers of Le Monde!

# Vaccine effectiveness in four concepts

https://www.lemonde.fr/sciences/article/2021/01/06/covid-19-l-efficacite-vaccinale-en-quatre-notions-distinctes_6065332_1650684.html

Pfizer’s vaccine efficacy is estimated at 95 percent. Does this mean, as is sometimes heard, that five out of every 100 people vaccinated will get sick from Covid? Thankfully, this is not the way this figure should be understood. A few definitions may be helpful to avoid such misunderstandings. Worldwide, the laboratory has selected 43,000 volunteers. Half of them, chosen at random, were vaccinated. The other half were “vaccinated” with a placebo: salt water. The volunteers could not know if they had really been vaccinated. It was then waited for 170 of them to experience symptoms of Covid and for their test results to be positive. Of these, eight had been vaccinated and 162 had received the placebo. Thus, the number of vaccinated patients was twenty times less than those who were not vaccinated. The risk of getting sick if you are vaccinated is therefore 5% of the risk of getting sick if you are not vaccinated. In other words, the risk of getting sick has been reduced by 95%, which is expressed as 95% clinical efficacy. This clinical trial has to be done before the vaccine is released, because an efficacy of more than 50 percent is required to obtain market authorization: 95 percent is therefore a very good score.
We are much more interested in real-world efficacy: the question now is how much the risk of disease is reduced in the real world for a vaccinated person. It’s quite different from a clinical trial, which mainly measures an individual’s degree of protection. The real effectiveness depends on the number of people vaccinated in the population: the more people vaccinated, the less the virus circulates, and the fewer infections and therefore the fewer people who get sick. Moreover, the duration of protection provided by the vaccine, which is still poorly known, is very important in reality, whereas it is of little importance in a clinical trial, which lasts only a short time. Actual efficacy can only be assessed after the vaccine has been released, through sensitive epidemiological investigations: it will take time to know it in the case of covid vaccines.

Benefits for all

There are two other kinds of efficiency to be added. Let’s not forget that vaccination is first and foremost a public health measure, which aims not only to limit the risk of disease for the vaccinated individual, but also for society as a whole, a significant proportion of which is not vaccinated (sometimes for good reasons). We can then estimate the indirect effectiveness, i.e. the reduction in risk that unvaccinated individuals benefit from those who are vaccinated and who do not contaminate them. Finally, there is overall effectiveness, perhaps the most important and most difficult to estimate: the decrease in average risk in the total population (vaccinated or not) compared to what that risk would be if no one were vaccinated. So these are four different notions of efficacy. In all cases, Covid vaccines will be extremely useful even if their
overall efficiency will likely be less than 95%. Even a value of 50% would prevent half of all diseases, lead to a significant decrease in the circulation of the virus in the population and save a large number of lives. As always, one must be careful with numbers. Let’s imagine that in a population there are ten times as many vaccinated people as unvaccinated people. Imagine that the risk of disease for a vaccinated person is five times less than for a non-vaccinated person. Since there are ten times as many people who are vaccinated, the number of vaccinated patients will be twice as many as the number of non-vaccinated patients. This does not mean that vaccination is ineffective.

Do not hesitate! As soon as you have the opportunity, vaccinate yourself!

Etienne Ghys

# The Academy of Sciences opens its Comptes Rendus in free access

Paris, December 14, 2020

Historic publication of the Academy of Sciences, the journal Les Comptes Rendus de l’Académie des Sciences is now available online under the “free diamond access” formula. This publication model makes all articles permanently available worldwide, without any financial burden, neither for readers nor for authors. In addition, the Academy allows the deposit of preprints in open archives. Faithful to its mission of encouraging scientific life and transmitting knowledge, the Academy of Sciences is thus evolving the publishing of its scientific journals, in order to bring it in line with the principles of open science, in collaboration with the National Museum of Natural History, the CNRS and the University Grenoble Alpes.

In 2020, the Academy of Sciences has completely overhauled its scientific journals: the seven series of the Comptes Rendus de l’Académie des sciences are now available free of charge on the https://comptes-rendus.academie-sciences.fr/ website.

This fundamental evolution was made possible by the Academy through the establishment of two founding partnerships:

– The Mersenne Center for Open Scientific Publishing (CNRS – Grenoble Alpes University), a pioneering publishing platform in open science, was chosen to publish the journals Mathematics, Physics, Mechanics, Chemistry, Geoscience and Biology.  This partnership is part of a memorandum of understanding signed with the CNRS on October 28, 2020, which aims to set up a mechanism for consultation and cooperation, particularly in the area of scientific publishing.

– The publication of the journal Palévol has been entrusted to the National Museum of Natural History, whose expertise is an undisputed reference in the international community of taxonomic and naturalist palaeontologists. This partnership between the Academy and the Museum reflects the historical convergence of the two institutions’ missions of general interest.

“We are delighted with the fruitful collaborations established between the Academy and its prestigious partners. Thanks to them, this complex project, which was particularly close to our hearts, was able to see the light of day. In the perspective of the strategic objectives that the Academy has set itself for the coming years, it aims to lay the foundations for a renewal of French scientific publishing,” emphasizes Etienne Ghys, Permanent Secretary of the Academy of Sciences.

“Thanks to the strong support, particularly financial, of the CNRS and the exceptional mobilization of its team, the Centre Mersenne has succeeded in enthusiastically taking up the challenge proposed by the Academy of Sciences. This prefigures a strengthened partnership that will make the Academy of Sciences, the CNRS and the University of Grenoble Alpes major players in open science,” said Evelyne Miot, scientific director of the Centre Mersenne.

“I can only salute with enthusiasm and pride the collaboration between the Academy of Sciences and the National Museum of Natural History,” said Bruno David, President of the National Museum of Natural History. “Paleontology has always been a discipline at the heart of the research conducted at the Museum, a discipline that has greatly contributed to its international reputation. The arrival of Palévol in this new partnership framework follows in the footsteps of such prestigious personalities as Lamarck, Cuvier, d’Orbigny, Gaudry and many others. I wish the same success to the beautiful magazine that is Palévol”.

The archives of the articles published between January 1, 2000 and December 31, 2019 in the 7 Comptes Rendus journals remain available for free access on the Elsevier website. Previous archives, up to 1835, are available on Gallica and soon on Persée.

The Comptes Rendus are a set of 7 peer-reviewed electronic journals.

In 2020, the editorial line of some of them has been reoriented.

Comptes Rendus – Mathematics. A new impetus has just been brought by the enrichment of the editorial board and the broadening of the editorial objectives. CR-Mathématique now welcomes different types of publications, and particularly encourages: original and significant research work; articles presenting in a non-technical or synthetic way important or topical mathematical developments; texts presenting important mathematical works in a global way; thematic issues taking stock of various approaches to the same problem (for example to report on colloquia or working days); texts of historical, philosophical or didactic reflection closely related to mathematics. Editors: Jean-Michel Coron, Jean-Pierre Demailly, Étienne Ghys, Laure Saint-Raymond.

Comptes Rendus – Physique covers all areas of physics and astrophysics and mainly proposes dossiers. Thanks to this formula, which has become a reference in the field, readers will find in each issue the presentation of a particularly fast-developing subject. The authors are chosen among the most active researchers and the coordination of each thematic issue is ensured by a guest editor, guaranteeing that the most recent and significant results are taken into account. CR-Physique also allows space for new results (on the recommendation of an academician), editing, and presentation of the work of the Academy’s award winners. Editors: Denis Gratias, Jacques Villain.

Comptes Rendus – Mécanique publishes original research papers, journal articles, thematic issues and articles reflecting the history of the discipline. The journal covers all the fields of mechanics: dynamic systems / solid mechanics / fluid mechanics / acoustics, waves, vibrations / automation, signal processing. The articles are proposed in the form of original notes relating briefly an important discovery. The publication of the results is fast. The thematic issues present the most up to date dossiers in the treated fields. Editor: Jean-Baptiste Leblond.

Comptes Rendus – Chimie aims to maintain high-level scientific exchanges between the different sub-disciplines of chemistry. The journal publishes original research works (notes, short memoirs) and review articles (reviews, historical chronicles) in all fields of chemistry. Preliminary papers should describe new and important results, while full papers should provide a detailed view of new results. In all cases, the work must be of high general interest or exceptional specialized interest. The journal also places great emphasis on thematic issues, bringing together the best specialists in the field around a guest editor. Editor-in-chief: Pierre Braunstein.

Comptes Rendus – Géoscience, which traditionally covers all fields of Earth sciences (geophysics, geomaterials, geochemistry, surface geosciences, oceanography, stratigraphy, tectonics, geodynamics…), is now broadening its editorial policy by encouraging the publication of articles dealing with the “sciences of the Planet” in the broadest sense. The journal is more open to scientific themes at the heart of current societal and environmental issues: natural hazards, energy and metal-material supply, water resources, pollution, climate change, both in the continental and oceanic/atmospheric domains. The submission of interdisciplinary papers is encouraged, to better understand the global effects of human activities on the functioning of the “Earth system”. Editors: Ghislain de Marsily and François Chabaux.

Comptes Rendus – Biologies sees in 2020 its objectives profoundly modified. True to the spirit of its title, the journal focuses its articles on the scientific activities of members or winners of the Academy’s awards, which are very rich . It only receives submissions of research articles by invitation only, but solicits mostly the biggest names in biology for articles divided into several sections: “C’est paru dans la presse/ News and views”, “Articles et revues”, “Notices biographiques”, “Opinions et perspectives”. This last section allows discussions and hypotheses on various subjects. Thematic issues on topical issues will be regularly scheduled, such as the one on COVID 19, which is currently being prepared. The articles are fully bilingual (English/French) and publication is fast. Editors: Jean-François Bach, Pascale Cossart, Bernard Dujon, Jean-Dominique Lebreton.

Comptes Rendus – Palévol is a continuous flow journal, dedicated to research in paleontology, prehistory and evolutionary science. It publishes original research results in systematics, human paleotonology, prehistory, evolutionary biology, and macroevolution. The journal also publishes thematic issues under the responsibility of guest editors. The co-publishing partnership agreement with the Academy allows CR-Palévol to benefit from the rigorous publication standards in force for the Museum’s journals, from the respect of the different codes of nomenclature and from direct compatibility with the major international databases. Editors: Philippe Taquet and Michel Laurin.

Created by Colbert in 1666, the Academy of Sciences is an assembly of scientists, chosen from among the most eminent French and foreign specialists. The reflections and debates that it conducts have the role of providing everyone with a framework of expertise, advice and alert, with regard to the political, ethical and societal challenges posed by science. By virtue of this mission, it works for the sharing of science as a common good in order to inform citizens’ choices, and formulates recommendations on which government authorities can base their decisions. It also supports research, is committed to the quality of science education and promotes scientific life on an international level.

Find us on our website and social networks

Delegation for Information and Communication of the Academy of Sciences

Sandra Lanfranchi – +33 (0)1 44 41 43 35

# To decide between two candidates, long live the simple majority!

In his carte blanche, the mathematician Etienne Ghys goes back over the different ways, from the most to the least fair, of electing a representative among two competitors.

https://www.lemonde.fr/sciences/article/2020/11/18/pour-departager-deux-candidats-vive-la-majorite-simple_6060148_1650684.html

By Etienne Ghys (perpetual secretary of the Academy of Sciences, director of research (CNRS) at ENS Lyon)

Carte blanche. Can mathematics shed some light on the American election soap opera? Let’s imagine a population voting for two candidates and assume that voters flip a coin to choose one or the other. At the end of the ballot, the ballots are counted and the candidate with the most votes is elected. Now suppose that, during the counting, the scrutineers make a few mistakes (or fraud), for example, by being wrong once out of 10,000. What is the likelihood that these small errors will distort the overall result and the other candidate will be elected? It turns out that this probability is of the order of 6 out of 1,000 (for the curious, it is 2/π times the square root of 1/10,000). Is this an acceptable risk in a democracy?

American elections are two-tiered. Each state elects its representatives by a majority and these in turn elect the president. Assuming one more reading error out of 10,000 (which is reasonable when looking at the American ballots), what is the probability of distorting the final result? The existence of this second level makes the probability much worse: one election in 20 would be distorted! This is far too much.

Noise Sensitivity

Of course, all of this depends on very unrealistic assumptions and does not in any way substantiate Donald Trump’s allegations of fraud! Assuming that voters flip a coin is obviously meaningless, even if one can be amazed by the near-equal results in Georgia, for example. However, this illustrates a phenomenon highlighted by mathematicians some twenty years ago: the “noise sensitivity” of various decision-making processes, which go far beyond elections. This concerns computer science, combinatorics, statistical physics and social sciences. When a large number of “agents”, who can be human beings or neurons for example, have “opinions”, what are the right processes that allow a global decision to be made in a stable manner? This stability means that we want the decision to be as insensitive as possible to noise, i.e. to small errors that we cannot control.

One can imagine many electoral processes. For example, each neighborhood could elect its representative who would then elect the city representative, who would elect its representative in the canton, then the department, and so on. It would be a sort of sports tournament, in successive stages, a bit like the American elections but with many more levels. This method happens to be extremely sensitive to noise, and it must absolutely be avoided. The slightest proportion of errors in the count would result in a very high probability of being wrong about the final result. This is unacceptable for a vote, but it is part of the charm of sports tournaments: it is not always the best who wins, and that’s just as well.

What is then the best method, the one that is the most stable? The answer is a bit distressing and shows that the question is badly asked. It is enough to ask a dictator to decide alone. This “method” is indeed very stable because, to change the result, you need an error on the only ballot that counts, which happens once out of 10,000. The question must therefore be rephrased by looking for equitable methods that give the same power to all voters. About ten years ago, three mathematicians demonstrated a difficult theorem in this context, which is ultimately only common sense. To decide between two candidates, simple majority voting is the most stable of all fair methods. Long live the majority!

Some references :

https://arxiv.org/pdf/math/0412377.pdf

https://gilkalai.files.wordpress.com/2018/01/18-kalaix-7.pdf

# Vaughan Jones, knotter and ultra-creative mathematician

https://www.lemonde.fr/sciences/article/2020/09/30/vaughan-jones-faiseur-de-n-uds-et-mathematicien-ultracreatif_6054222_1650684.html

The 1990 Fields Medal winner, the New Zealander passed away on September 6, 2020. Etienne Ghys pays tribute to him in his column in “Le Monde”.

Carte blanche. The mathematician Vaughan Jones died on September 6, 2020 in Tennessee, USA. He had received the Fields Medal in Kyoto in 1990. Sometimes a mathematician builds bridges between fields that were thought to be completely independent. These are moments of grace in the development of mathematics, reserved for the most creative, like Vaughan. However, it should not be thought that it is eureka! that suddenly appears. It almost always takes a long maturation, hardly compatible with the demand for immediacy of our current university system. The University of Geneva allowed Vaughan Jones to blossom and give the best of himself.
Vaughan arrived in Switzerland in 1974 from New Zealand to do a doctorate in physics. One day, with his thesis almost finished, he passes the door of the mathematics department and is fascinated by André Haefliger’s course: he abandons physics to do a thesis in mathematics (although, of course, his training as a physicist will remain fundamental). He works on “von Neumann algebras”, a field so abstract that the spaces studied have non-integer dimensions. Imagine for example a space whose dimension is 3.14 ! Haefliger – his thesis supervisor – is not a specialist in this subject, which is a sign of the great originality of the student and the open-mindedness of his master.
The Swiss Pierre de la Harpe, who knows the subject well, will become a friend and a “big mathematical brother” of Vaughan. At that time, the small department of Geneva was a breeding ground animated by a few exceptional senior mathematicians who fought against any form of exaggerated specialization. Algebra, geometry and analysis were spoken about a lot, very often in the small Italian bistro on the first floor. On the day of Vaughan’s defense in 1979, he was dressed in a tuxedo, which contrasted with the way the jury was dressed. In 1990, during the Fields Medal ceremony, in the presence of very formal Japanese authorities, he had insisted on wearing the All Blacks jersey, out of attachment to his New Zealand origins.

Sideration of the specialists

After his thesis, he settled in the United States but he often returned to Geneva. One day, after one of his lectures, someone pointed out to him, perhaps at the Italian bistro, an analogy between a relationship he wrote on the board and what is called the “group of braids”, which Vaughan did not know. That was all it took to glimpse a link between the subject of his thesis and a theme that was new to him: the theory of knots. All this led to a major discovery in 1984: the “Jones polynomial” associated with a knot. Knots, in mathematics, are those we imagine, like those of sailors. The mathematical theory of knots dates back to the 19th century and had a priori nothing to do with von Neumann’s algebras. Vaughan’s announcement of an important application of these algebras in the field of nodes will generate a kind of astonishment among topology specialists. He was awarded the Fields Medal but was also elected Life Vice-President of the International Knotmakers Guild, something he was very proud of.
The rest of his career was admirable. For about twenty years, the Ecole normale supérieure de Lyon has organized a mathematics weekend for about fifty students and an experienced mathematician. In 2012, Vaughan Jones literally charmed the young students. We have not only lost a brilliant mathematician, but also a model of generosity and openness for young people.

# Does the Covid-19 pandemic herald the end of the mathematics laboratory concept?

https://www.lemonde.fr/sciences/article/2020/07/01/la-pandemie-de-covid-19-annonce-t-elle-la-fin-du-concept-de-laboratoire-de-mathematiques_6044863_1650684.html

In his column in “Le Monde”, the mathematician Etienne Ghys notes that the confinement has brutally accelerated, with the imposed teleconferences, a process of reduction of physical interactions between researchers.

Carte blanche. The months of confinement that we have just experienced will probably permanently change the working methods of scientific researchers, including those with no connection to biology. Mathematicians, for example, do not use experimental equipment, and their physical presence in the laboratory may not seem indispensable. They have been among those for whom teleworking has been the easiest to set up.
The Researchseminars.org site lists 739 mathematics presentations that can be participated in via the Internet, being able to interact live with the lecturers on all subjects, at any time of the day or night, taking advantage of the time difference. This opens up unprecedented possibilities for communication between researchers and abruptly accelerates a slowly evolving process. The consequences that this will have on the social life of the mathematical community are unknown.
Mathematicians usually work alone, but of course they need to exchange ideas with other colleagues. For a century, a major communication tool has been the laboratory seminar. These are meetings, usually weekly, during which a new result is presented to the members of a team. In France, the first seminar was created in 1920 by Jacques Hadamard, a professor at the Collège de France. At the beginning of the academic year, he would invite a few mathematicians to his home and distribute recently published research articles to be studied. He would then draw up an annual program.

The seminar, a Sunday mass

At the time, the Hadamard seminar was unique in France, but today, all the teams in the mathematics laboratories are organized around their seminars. Their role goes far beyond the transmission of knowledge: they are social events that unite the teams. They are sometimes compared to Sunday mass. Sometimes one attends out of obligation, or to see friends and colleagues. It must be said that it is not always easy to follow a mathematics conference and that one is often lost, sometimes from the very first sentences.
Over the last twenty years or so, the Internet has, of course, made these modes of communication evolve. First of all, all scientific journals are now available online. In the past, mathematicians used to go to their laboratories to be close to their library, which was their real working tool. This is still the case, but libraries have become virtual. E-mail, which is abused, has replaced the letters that were carefully written by thinking about each word. It’s not uncommon to see researchers, with a helmet on their head, collaborating via Skype with someone on the other side of the world, and forgetting to go and chat with their close colleagues in the common room of the laboratory.
This gradual evolution has great advantages, of course, but also obvious disadvantages. The weekly “face-to-face” seminars remained, however, and made it possible to preserve the human link within the teams. The pandemic suddenly accelerated this evolution: the seminars had to meet by videoconference, and participants no longer had to be members of the same laboratory. Lists of “global web-seminars” emerged, offering impressive amounts of live conferencing, each more appealing than the last. This evolution is probably irreversible. Does it herald the end of the mathematics laboratory concept? That would be a pity.
This summer I’m going to take part in a conference in Russia… without leaving home.

# Percolation theory or the art of modeling a pandemic

https://www.lemonde.fr/sciences/article/2020/05/12/la-theorie-de-la-percolation-ou-l-art-de-modeliser-d-une-pandemie_6039452_1650684.html

The mathematician Etienne Ghys details the theory established by two British researchers in 1957 to understand the propagation of a fluid in a random environment. Like any modeling, it requires juggling with a lot of unknowns.

Carte blanche. Many articles have described the development of an epidemic over time, with an exponential growth in the number of new cases at first, then the famous peak, and finally the long-awaited decrease. There has been less discussion of contagion across a territory.
The mathematical theory of percolation is interested in this kind of problem. The word comes from the Latin percolatio meaning “filtration” and of course it evokes the coffee percolator: boiling water under pressure finds its way through the ground coffee particles, just as a virus finds its way into a population.
The theory originated in 1957 in an article by two British researchers, John Michael Hammersley and Simon Ralph Broadbent. Their initial motivation was for the much-talked-about breathing masks. In their case, these were protective masks for coal miners. The porous filter is likened to a regular network of very fine interconnected tubes, a number of which are randomly plugged, and the question is to understand whether a gas can pass through such a maze.

Determining the critical probability

More generally, these researchers study the propagation of a fluid in a random environment. One of their examples is a very simple model of an epidemic. It involves a huge orchard in which fruit trees are planted regularly in a square network. It is assumed that at some point in time one of the trees has a disease that it can potentially transmit to its neighbors. Each diseased tree can infect each of its four neighbors with a certain probability p (the lower the probability, the lower the trees respect the “social distancing”).
How will the epidemic spread? Hammersley and Broadbent show that if p does not exceed a certain critical value, the epidemic remains localized: these are clusters in which the contamination reaches only a small group of trees. When this critical value is exceeded, the disease suddenly invades a large part of the orchard (infinite if the orchard is infinite) and it is the pandemic.
Of course, this theorem is of interest only if this critical probability can be determined. Numerical simulations suggested that the cluster-pandemic transition occurs for p = 0.5, and it was not until 1980 that this was rigorously established. Unfortunately, this kind of precise result is only known in very simple cases, such as that of a regularly planted orchard. As soon as the trees are more or less in disorder, the phenomenon is less well understood.

Very partial information

In this case, the trees are flesh and blood individuals that fortunately are not planted regularly and are moving around. Moreover, the number of contacts of an individual, i.e. the number of people he meets in a day, and that he can potentially contaminate, is extremely variable from one individual to another. It depends on where he lives, his age, and many other parameters.
Only very partial information is available on the statistics of these contacts. A final problem arises: when a sick person meets a healthy person, the probability of contamination is also variable, and not well known.
In order to do this properly, a large number of parameters should be precisely known, many of which are inaccessible. The modeler must select a small number of them that seem most relevant to him, and of which he has a reasonable knowledge. He must then determine whether the other parameters – which he knows little about – could have a significant influence on the outcome of his predictions. This is not an easy task. Mathematical modeling is an art.

# Epidemics: flattening the exponentials

Carte blanche. These last days will have at least allowed the French to understand in their flesh what an exponential is. We have all become aware that the powers of 2 grow really fast: 1, 2, 4, 8, 16, 32, 64, etc., to exceed one billion in just 30 steps. What is less well known is that while the number of new infections in an epidemic doubles every three days, half of those infected since the beginning of the epidemic have been infected for less than three days. The exponential function has terrifying aspects.
The first scientist to highlight this type of growth was Leonhard Euler, in 1760, in an important article entitled “General Research on the Mortality and Multiplication of the Human Race”. In 1798, Thomas Malthus understood that exponential growth is a threat to humanity. Fortunately, in 1840, Pierre-François Verhulst discovered “logistic growth”, which allowed him to understand why the exponential growth must eventually calm down. This is the curve that was presented so clearly on a television set by our Minister of Health.
In a purely exponential growth, the number of new cases of contamination is proportional to the number of people contaminated. In formula, the derivative y’ of the number of cases y is proportional to y, which translates into a diabolically simple equation y’ = ay, whose exponential solution y = exp (at) may bring back memories to the reader. The coefficient ‘a’ depends on the average number of contacts we have: the larger it is, the faster the exponential explodes.

Bell curve

In a logistic growth, the number of new cases of contamination is proportional to the number of people already contaminated, but also to the number of people who are contaminable, i.e. who have not already been contaminated. Fortunately, the number of contagious people decreases as the epidemic progresses, and the evolution is reversed.
In the formula, y’ = ay (1-y/b) where b denotes the total population. In this model, the number of new cases follows the bell curve drawn by the minister. There is an exponential growth at the beginning (when the number of cases is still small), then a maximum, and finally a decrease. The only parameter we can act on is this seemingly innocuous coefficient “a”, which is related to the average number of our contacts. When we decrease “a”, the curve keeps the same speed, but it flattens. Certainly the peak comes later, but it will be lower. The epidemic lasts longer, but it is less deadly. That’s why you have to stay home!
In the 18th century, the question was raised as to the value of inoculation in the fight against smallpox, which had decimated nearly half of Europeans. It was a very primitive version of vaccination, but one that presented dangers for inoculated patients (unlike vaccination). Mathematician Daniel Bernoulli will write an article entitled “Testing a new analysis of smallpox mortality and the benefits of inoculation to prevent it” which mathematically demonstrates that inoculation is beneficial. Alas, it will not be listened to.
A few years later, the article “Inoculation” in Diderot and d’Alembert’s encyclopedia stated: “When it is a question of the public good, it is the duty of the thinking part of the nation to enlighten those who are susceptible to light, and to drag along by the weight of authority this crowd over whom the evidence has no hold. »
This may be true, but it is even truer when “the thinking party” clearly explains its choices by drawing a curve on a TV set.

https://www.lemonde.fr/sciences/article/2020/02/05/faites-plutot-vacciner-vos-amis-c-est-mathematique_6028500_1650684.html

Mathematician Etienne Ghys evokes the implications that the “paradox of friendship” could have in strategies to fight pandemics.

Carte blanche. To understand how a virus spreads in a population, biology is of course very important, but it is not enough: mathematics is needed. Once a number of parameters – the transmission rate, incubation time, etc. – are known, the virus can be transmitted to the population. Once a certain number of parameters are known – transmission rate, incubation time, etc. – formidable mathematical problems still need to be solved. In the simplest epidemiological model, the population is broken down into three compartments: healthy people, infected people and people who are immunized after the disease. Healthy people can be infected with a certain probability when they meet a person who is already infected. An infected person becomes immune after a certain period of time. This leads to relatively simple differential equations.
It is clear that this model (developed a century ago) is very naïve. Many others, increasingly complex, have been imagined and work in many situations. The major difficulty is that most of these models are based on an assumption of population homogeneity, whereby individuals come into contact at random and the probability of infection does not depend on the individuals who meet. The population would have to be broken down into a multitude of compartments, taking into account, for example, their age, where they live, etc. The main difficulty is that most of these models are based on an assumption of population homogeneity, whereby individuals come into contact randomly and the probability of infection does not depend on which individuals meet. This becomes extremely complicated.
The problem is to understand the “network of contacts”. Draw 7 billion dots on a sheet of paper, one per human being, and join 2 dots with a line every time the 2 corresponding individuals met last week. Since this “drawing” is impossible to do in practice, we try instead to describe its global properties. For example, it is thought to be a “small world”: any two human beings can be connected by a very short series of individuals such that each is a friend of the next. It is even said that a string of length 6 should be enough, which can be worrisome if the virus is transmitted between friends.

Large network theory

On a much smaller scale, a group of researchers carried out an experiment in a high school in the United States: for one day, a thousand students wore small detectors around their necks, and it was possible to obtain a complete list of all the encounters between them (within three meters, for at least one minute). The researchers were then able to analyze in detail the properties of this network of encounters and then how an infectious disease could spread in this high school.
The theory of very large networks is currently in full expansion, both in mathematics and computer science. Here is a very simple but surprising theorem: “A majority of individuals have fewer friends than their friends”. Let’s take the following example: Mr. X has 100 friends who are friends only with him. So, of these 101 people, all but one of them have only one friend, but their (only) friend has 100 friends. It turns out that this phenomenon always happens, regardless of the nature of the friendship network.
As an application, let’s imagine that there are only a small number of vaccines available, and that it is a matter of choosing which people should be vaccinated. We could vaccinate randomly selected people, but a much better idea would be to randomly select one person and ask them to name one of their friends, and vaccinate that friend. If the friend has more friends, more people are likely to become infected and it would be better to vaccinate that friend. In the previous example, it is Mr. X.
The paradox of friendship goes further. Not only do your friends (in general) have more friends than you, but they are said to be happier than you!

# The mutual attractions of the abbot Sigorgne

This ecclesiastic, who popularized Newton’s ideas, is an example of the spirit of the Enlightenment that deserves to be brought out of oblivion, according to the mathematician Etienne Ghys.

Carte blanche. You probably don’t know Abbot Sigorgne. However, he was the subject of a fascinating symposium on October 4 and 5 in Mâcon, bringing together specialists in the history of science and literature. Born in 1719 and died in 1809, in Mâcon, it is difficult to classify him: mathematician, physicist, writer, man of the church? In our society of immediacy, we must always remember the importance of historical research to better understand our contemporary world, which owes so much to the Enlightenment.
In the 18th century, the battle raged between the English, supporters of Newton’s theory of gravitation, and the French, supporters of Descartes’ theory. According to Descartes, space is filled with an unknown fluid, forming whirlpools of all sizes that drag the planets in their course. According to Newton, space is empty and the bodies are subjected to mysterious forces of mutual attraction that act instantaneously, even if the distances between them are considerable.

As we know, Newtonians will win the battle against the Cartesians (while waiting for Einstein’s arrival with his theory of general relativity). Voltaire will play an important role by writing his wonderful Elements of Newton’s Philosophy (1738) in an almost journalistic tone. Newton will penetrate scientific France thanks to the translations and commentaries of Emilie du Châtelet. But it was Abbot Sigorgne who allowed Newton to enter university teaching by writing his Newtonian Institutions in 1747. Of course, Sigorgne is not as well known as Condorcet, d’Alembert, Voltaire or Rousseau, but history is not reduced to celebrities, and it is important to look at a less well-known Mâconnais than Antoine Griezmann.

Reconciling Descartes and Newton

Our abbot is a man of the Enlightenment, open to dialogue. He will exchange about a hundred letters with Georges-Louis Le Sage, a Geneva physicist, who will try to convince him that it is possible to reconcile Descartes and Newton. According to Le Sage’s theory, space is filled with microscopic particles that partially penetrate bodies by bouncing off atoms. That made it possible to explain the mysterious force of gravitation whose origin Newton himself admitted not to understand. However, this beautiful theory of The Wise Man was not successful.
Sigorgne is also a teacher. Several letters from Turgot show that he had not forgotten his teacher and that he could seriously discuss the Newtonian attraction and the geometry of ellipses or hyperbolas. Happy times when rulers knew geometry! On the other hand, fifty years later, it seems that Lamartine did not really benefit from his mathematics lessons.
Of course, all this is mixed with intense theological debates: how to reconcile Reason and Faith? The abbot, for example, violently attacked Rousseau’s Lettres écrites de la montagne (1764) by publishing the Lettres écrites de la plaine (1764), or the defense of miracles against the philosopher of Neuf-Châtel (1766).
At the end of his life, Sigorgne judged that “high scientific works no longer suited his age”, and wrote a collection containing a large number of fables, in the manner of La Fontaine. The manuscript was recently found in the archives of Mâcon. A literary historian made a detailed analysis of it and came up with a very nice idea: in collaboration with a school teacher, she worked on some of these fables in a class of CM1-CM2 in a neighboring village. A video maker staged the whole thing and produced a nice film. What an emotion to see in 2019 children declaiming forgotten texts, as if echoing the Age of Enlightenment!

# In Shanghai, an obsession for the square root

The mathematician Etienne Ghys comes back on the strange formula which presides over the establishment of the Shanghai ranking.

Carte blanche. The famous Shanghai ranking list of universities was published as every year in August. We learn that the top trio is made up, as always, of Harvard, Stanford and Cambridge, and that the universities of Paris-Sud and the Sorbonne occupy the 37th and 44th positions. This ranking is criticized from all sides, except of course by the universities that are well placed. It is perhaps useful to explain how it is constructed, to show how little sense it makes.
First, the ARWU (Academic Ranking of World Universities) assesses five “indicators” for each university. These are the number of Nobel Prize or Fields Medal winners who work there, the number of alumni who have received these same honors, the total number of published articles, those published in the two journals Nature and Science, and finally the number of “highly cited” researchers.
Each of these indicators is problematic. For example, the list of most cited researchers includes 90 mathematicians, 16 of whom sign their papers … in Saudi Arabia. On the other hand, there are no French mathematicians in this list. Without being chauvinistic, this makes no sense.
Of course, these five indicators favor the big institutions and leave little chance to the small ones, even if they are excellent. To try to remedy this, a sixth indicator is used, which is a sort of average of the previous ones, divided by the total number of researchers in the university.

As in the decathlon

The icing on the cake is the formula used to aggregate all this and make an overall ranking. The “score” assigned to a university is an average of the square roots of the six indicators, assigned certain coefficients. You read that right: it is an average of the square roots. To understand the idea, we can refer to the decathlon. How do you aggregate a sportsman’s results in ten disciplines as different as high jump and shot put? The solution is to first transform each of the ten performances in a certain way, specific to each discipline, before calculating averages. An improvement of 1 cm in the high jump will earn you many more points if you jump 2.45 m (world record) than if you jump “only” 1.50 m. For a university that already has a lot of Nobel Prize winners, however, it is easier to recruit one more than it is for a university that has none. In order to take this into account, the ARWU did not look very far and decided to transform all indicators in the same way and to use the square root.
There are at least two differences between academics and decathletes. Firstly, there has been much debate among athletes in the past about what a good formula should be. Nothing of the sort has occurred among academics, and the arbitrary choice of the square root is puzzling. On the other hand, a decathlete participates in a competition that he has freely chosen and for which he knows the rules. This is not the case of universities, which do not have the mission of following rules imposed unilaterally by a Chinese institute that promotes square roots.
The ARWU also establishes world rankings by discipline. I have of course consulted the one concerning mathematics. There you learn that Princeton is first, the Sorbonne is second, Paris-Sud is in fifth place, and the French department of mathematics that follows, in a very honorable 27th place in the world, is my laboratory at the Ecole normale supérieure de Lyon. In the end, these rankings are not so bad…

# Those women who counted in the shadows

“Carte blanche”. One of my favorite scientific articles was written by Edward Lorenz in 1963 and is entitled “Deterministic Nonperiodic Flow”. It is one of the founding texts of chaos theory. Its content will be passed on to the general public a little later through the beautiful image of the butterfly effect: a flapping of a butterfly’s wings in Brazil could create a hurricane in Texas. This publication is an extraordinary blend of physics, meteorology, mathematics and numerical simulations. I have read and reread it many, many times and thought I knew it until last week.
An article by Joshua Sokol in Quanta Magazine told me that I should have read the last paragraph in which the author thanks “Miss Ellen Fetter who took care of the many calculations and graphs”. How? It was not Edward Lorenz who did the calculations, but an assistant? It must be understood that simulating the movement of the atmosphere on a computer was an essential component of the article. In 1963, computers were primitive and “taking care of the calculations” would probably have deserved a little more than a discreet thank you.

This is not the first time that scientists have used “female calculators”, whose names appear at best in the acknowledgements. Ten years earlier, Enrico Fermi, John Pasta and Stanislaw Ulam published the first numerical simulation of a complex physical system. This article can be considered the birth of a new discipline in mathematical physics. It involved studying, on a computer, the vibrations of a chain made up of about sixty “non-linear” springs.
Here again, two discrete lines in the publication thank Miss Mary Tsingou for “the efficient programming of the problem and for having carried out the calculations on the Maniac de Los Alamos computer”, which represents a very important part of the work. It is only in 2008 that the physicist Thierry Dauxois will read these two lines and will propose to call Fermi-Pasta-Ulam-Tsingou this numerical simulation. I would have even proposed to respect the alphabetical order…
Going back in time again, we arrive at a period when calculations were done by hand, and the hand in question was often female. In the 1940s, a member of an institute of applied mathematics dared to talk about the kilogirl (kilofille): the amount of calculations a woman can produce in a thousand hours! Around 1880, the astronomer Edward Charles Pickering recruited a team of more than 80 female calculators at Harvard, Massachusetts, known as the “Pickering Harem” and paid less than a labourer.
Halley’s Comet is known to be visible in the sky about every 76 years. Its trajectory is disturbed by the attraction of Jupiter and Saturn. In the middle of the 18th century, some scientists still had doubts about Newton’s theory of gravitation. The calculation of the date of the comet’s return was a great moment in the history of science. In November 1758, the academician Alexis Clairaut announced a return “around April of next year”. It was a triumph when his prediction came true. The theory was indeed due to Clairaut, but the monstrous calculations were performed by Joseph Lalande and Nicole-Reine Lepaute who “calculated from morning till night, sometimes even at the table. Clairaut “forgot” to thank her collaborator. The City of Paris will do Nicole-Reine partial justice in 2007 by naming a street after her.

In 2017, Google engineer James Damore was fired after claiming that the lack of female computer scientists was of biological origin.

# Some idea of math lessons

https://www.lemonde.fr/sciences/article/2019/04/15/une-certaine-idee-des-lecons-de-mathematiques_5450426_1650684.html

In his column, Etienne Ghys takes us to the challenges organized by the association MATh.en.JEANS, for students of all school levels. A nice way to give a taste of maths.

“Carte blanche”. The 2018-2019 season of the association MATh.en.JEANS ends. Since March, ten mathematical congresses have been held throughout France, and two more will be held in May, abroad. These congresses are very unusual: the participants and speakers are students of all school levels, from primary to high school. In 2018, 4,500 students participated (almost half of whom were girls) and 680 mathematical topics were discussed in 300 “workshops”.
The principle is as follows: teachers propose to students (volunteers) to reflect on a topic that has been suggested by a referent researcher. Small groups are formed, often straddling several schools, and the students meet once a week to reflect together on their problem. The big moment is the congress during which the students present their results in front of their classmates, but also in front of the teachers present in the amphitheater. These moments of exchange are magical; it is so rare to see a student on the blackboard explaining to a teacher what he or she has discovered! Some of these presentations are written and published by the association.

The themes covered are surprisingly diverse. Sometimes it is about number theory. For example: if I multiply all the integers from 1 to 1,000, how many 0’s will there be at the end of the result of my calculation? Other times, it is combinatorics that is in the spotlight: how can I place a certain number of points in the plane so that the line joining any two of them contains at least one more? Or again: if I place an even number of points in the plane, can they be joined two by two by segments that do not meet?
Other themes are much more “useful”. I remember, for example, a group of students who could no longer stand the long queues at the canteen at noon. They tried to optimize the timetable by suggesting to the principal to slightly modify the hours of classes so that the students did not all go out at the same time. Optimization is not as simple as one might think. There are also groups working on magic tricks or winning strategies in a (very) simplified version of poker.

Nice exponential behavior

The association was founded in 1989 and its growth shows a beautiful exponential behavior, a tripling every ten years or so: we should exceed one million students involved in… fifty years! All the surveys show a drop in the level of French students in mathematics. Should the number of hours of classes be increased? What should we think about the future disappearance of mathematics from the common core in the first grade? Shouldn’t we support more firmly initiatives like MATh.en.JEANS by moving to much higher orders of magnitude?
This would require massive financial support from the national education system, which is largely insufficient. Today, 600 teachers and 200 researchers are involved in the association, all volunteers. It should be considered that this kind of activity is an integral part of the mathematical training of students. Voluntary work and volunteering have their limits .
It’s an opportunity to do a little advertising. “The purpose of the André Parent Prize is to recognize research work, supervised or not, carried out by a group of young people (primary, middle or high school) during the school year, on a scientific subject in which mathematics plays a key role. “The prize will be awarded during the 20th Salon Culture et Jeux Mathématiques, to be held on May 23, 24, 25 and 26, Place Saint-Sulpice, Paris.