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The mathematician Jacques Tits has died

Professor at the Collège de France, where he held the Group Theory chair from 1973 to 1999, he received the prestigious Abel Prize in 2008. He died on December 5, at the age of 91.

By Etienne Ghys (Permanent Secretary of the Academy of Sciences, Director of Research (CNRS) at the ENS Lyon)
Published on December 08, 2021 at 4:37 pm, updated at 11:38 am – Reading 3 min.

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)

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

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

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 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.

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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.

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 :

Vaughan Jones, knotter and ultra-creative mathematician

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?

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 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

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.