All posts by ghys

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.

Get your friends vaccinated instead, it’s mathematical

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.

Handmade calculations

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

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.