24th April 2006
Although outwardly he might not show much inclination for football, just one look at the boy with the ball at his feet is enough to see that he possesses outstanding skill. Likewise it is not necessary to be a musician to know that a girl has a good ear and can quickly pick up a new tune. In both cases, the child could easily develop his or her innate ability by following the right courses and activities. But if such talent is of a more abstract nature, for mathematics for example: Who might notice it? And how could this aptitude be developed?
María Gaspar, secondary school teacher and lecturer in Geometry and Topology at the Complutense University of Madrid, is one of the people who eight years ago started the ball rolling in answer to these questions, with the assistance of the late lamented mathematician Miguel de Guzmán (Cartagena, 1936 - Madrid, 2004), who set up the programme for the Encouragement of Mathematical Talent (Estímulo de Talento Matemático - EsTalMat), a scheme which is currently directed by the Royal Academy of Sciences and financed by Vodafone España. “It’s about talent spotting children between the ages of 12 and 13 who have a special gift for mathematics, and then working with them for two or three years to help them develop this ability”, she explains.
The first stage in the programme is detecting the talent. Here the untutored eye may be blind to such ability. “The teacher ought to pick up on it first, because unless there’s a mathematician in the home environment it’s very difficult for the family to notice it. However, it’s not easy in the school either, because the teacher usually works with classes that are too big to devote special attention to an individual pupil. And unfortunately the activities carried out in class are very routine, which makes it difficult to assess the ability for creative or abstract thinking. Furthermore, it is not easy to distinguish the outstanding candidate, the one who always gets a 10, the one who has the ‘spark’, although they are usually good students,” says Gaspar.
In order to find these latent geniuses, letters asking for prospective candidates are sent every year to schools in the public and private sectors in the Community of Madrid (which is where the idea was developed, although there are now EsTalMat programmes in other areas: in Catalonia and Burgos since 2003, and in Andalucía and the Canary Islands since 2005). Then the initiative must come from the teacher or the parents, who have to vouch for the child. First the teacher has to draw up a detailed report, then the parents have to give their consent and commitment to the candidacy. If the child is accepted on the course, they must also agree to provide their support.
“Every year we have between 250 and 300 candidates. This figure has stabilized, although at first it was less,” says Gaspar. Then the difficult selection process begins. “It’s a tricky question, because it doesn’t just take into account I.Q. or whether the child gets good marks or shows special interest in mathematics. It’s about detecting intuitive disposition, ability for abstract thinking, how the child handles geometric shapes … a combination of factors”, explains Merche Sánchez, secondary school teacher and also lecturer in Applied Mathematics at the Complutense University. In short, selection is carried out by means of a test consisting of 6 or 7 problems (examples can be found on the EsTalMat web page). “It has nothing to do with what the children do in class. What interests us is finding out not only if they are able to solve a problem, but how they arrive at the solution. We want to see their results so we can judge their ability to generalize, their level of imagination and creativity, how they approach a problem, the strategies they use, how they devise a way of arriving at a solution…”, explains María Gaspar. “Then we conduct individual interviews with the child and also with the parents”.
In the end, only 25 children are accepted for the programme. For two years they spend 3 hours a week on Saturday mornings developing the gift that nature has endowed them with. “It’s like an extra-curricular activity”, say both these lecturers. “We approach it in an amusing and entertaining way. It does not interfere with the mathematics they do at school. It’s a question of enjoying mathematics. We’re not even trying to persuade them to devote their lives to mathematics. In fact, the first ones to complete the programme are now in their 3rd year at university, and some of them are not studying mathematics. Most of them are doing engineering, medicine and even fine arts. Firing them with a love of mathematics is enough for us, since we believe that whatever they end up doing, whether it’s running a company or working in a hospital, mathematics will be useful for them”.
The fruits of this project are still ripening. “We’ve been running the programme for only eight years now”, says Merche Sanchéz, “but I’m sure that the benefits for society will be more than enough through the efforts these children expend in whatever careers they chose”. And María Gaspar adds: “I don’t know if any geniuses have emerged, but without mentioning any names there are already some outstanding people who when they mature will undoubtedly make important contributions to society, whatever type of work they decide to do”.
It is worth remarking on the fact that the stated intention is not to produce mathematicians, but to train minds that are mathematically prepared. There are few professions that do not fall under the umbrella of mathematics, the applications of which are versatile and universally relevant.
It is said that Jean Pierre Serre (1926, Bages, France) is the typical mathematical genius who (of course) enjoys working on a stimulating problem much more than having to talk about his work or following a social life. But there are other factors that belie this simple description: Serre, described by his colleagues as a “hero” or a “maestro”, also loves sport; some of his favourite films are “Pulp Fiction” and those by the Coen brothers, and he is a devotee of the Harry Potter saga.
Where his work is concerned, however, there have been times when Serre – whether he likes it or not – has been obliged to talk about it. He already has seven scientific prizes to his credit, two of them the highest awards in mathematics: he won the Fields Medal when he was only 28 years old, and the Abel prize in 2003. Furthermore, he has been honoured with 11 doctorates honoris causa, in addition to that conferred by the Complutense University of Madrid (April 27th). Various interviewers have expressed interest in his working methods, his sources of inspiration, and his opinions about the development of mathematics.
His replies have often been as concise as these he gave for InfoICM2006:
-- I understand you’ve learned a lot by yourself.
-- Unfortunately, I don’t learn much any more.
-- Would you say that the mathematical education that children receive
today is good?
-- I know very little about it, because I have no grandchildren.
-- What would you say to a young student of mathematics?
-- A good student doesn’t need advice.
Other interviews have provided further information. One of his replies in 1985(1) was enough to make some mathematicians hot under the collar: [when asked about how to encourage young people to take up mathematics] “I have a theory on this, which is that one should first discourage young people from doing mathematics. There is no need for too many mathematicians. But if after that they still insist on studying mathematics, then one should indeed encourage and help them. As for high school students, the main point is to make them understand that mathematics exists, that it isn’t dead (they have a tendency to think that the only open questions remaining are in physics and biology). The defect in the traditional way of teaching mathematics is that the teacher never mentions these questions. That’s a pity”.
He has also said that when he was an adolescent he learned mathematics from a book of calculus belonging to his mother: “At that time I had no idea that one could make a living by being a mathematician. It was only later I discovered that one could get paid for doing mathematics”.
And this on his working methods: “Quite often you don’t really try to solve a specific question by a head-on attack. Rather you have some ideas in mind, which you feel should be useful, but you don’t know what for exactly. So you look around and try to apply them. It’s like having a bunch of keys and trying them on several doors”.
Serre prefers to speak about “thinking a lot” rather than “effort”: “It is not the conscious part of the mind that does the work,” he remarked on being awarded the Abel Prize(2). Perhaps that is why he often works at night, in bed, in the dark: “When I’m half asleep. The fact that you don’t have to write anything down makes the mind more concentrated”.
Mathematics coming together
-- How have mathematics developed over recent decades?
-- The question is too ambitious. I can’t comment on ‘the development of mathematics’. Of course, the old questions have been explored more deeply (Number Theory, for example), and new questions have been posed (by cryptography, for example, or theoretical physics). But that’s hardly surprising! On a more technical side, more and more branches of mathematics are coming together. For instance, people working in Analytic Number Theory have started using deep methods of Algebraic Geometry and Group representations. It is very satisfying, and quite in line with the old Bourbaki spirit of the unity of la mathématique”.
Nicolás Bourbaki is the pseudonym adopted by a group of French mathematicians who propounded the foundations of mathematics in the 1930s. Their impact was enormous. The names of those belonging to the Bourbaki group were kept secret for many years, although today it is known that Serre was one of them from 1949 until the early seventies.
Another important detail in the biography of this mathematician is that part of his work proved crucial for Andrew Wiles’ proof of the famous Fermat Theorem in 1994.
-- The dividing line between pure and applied mathematics seems to becoming more and more diffuse. Is this perception correct?
-- I wouldn’t say ‘diffuse’. There is still a sharp distinction between a theorem which is TRUE and statements which only give approximations. On the other hand, applied mathematics and computers can help more and more branches of pure mathematics by suggesting results and disproving wrong conjectures.
-- Have you seen some of your work being applied to fields or areas you didn’t expect in the beginning?
-- Not my own work, but some closely related to it, such as elliptic curves (or even Abelian varieties) over finite fields: they’re used in cryptography.
Pilar Bayer, professor of Algebra at the University of Barcelona and one of Serre’s collaborators, has written this about him(3): “Studying a report or a book by Serre is always a pleasure; rereading them is a necessity. The broad vision he has of mathematics, his results, his conjectures and his questions, as well as the invaluable help he has so often given to other mathematicians, have crystallized into some of the most spectacular advances in mathematics in recent years”.
An Interview With Jean-Pierre Serre (C.T. Chong and Y.K. Leong). June 1985, Mathematical Medley (a publication of the Singapore Mathematical Society). http://sps.nus.edu.sg/~limchuwe/articles/serre.html
When most people see the structure of metals they never think to ask themselves if this solid material was once a liquid. This is not the case of Robert V. Kohn, who during the ICM2006 will deliver a plenary lecture on “Energy-Driven Pattern Formations”. This North American mathematician studies what happens in samples of what are known as crystalline solids undergo a martensite transition phase. No, this is not an encrypted language, but rather technical terminology referring to the study of the moment at which a metal with certain “martensite” properties (see link) starts to change from a liquid to a solid.
Computer hard discs also fall under Kohn’s scrutiny, since his work extends to other fields. He studies other physical models connected with micromagnetism. Thus in the future, when hard discs of 100GB will already seem small, we are likely to find that a tiny magnetic device with an enormous storage capacity is in fact an offspring of Kohn’s researches.
It might seem like a joke to ask what a bridge made with the lightest and most modern materials has in common with the RAM memory of a computer. Kohn has found the answer, and it is not intended to have any double meaning designed to raise a chuckle. Rather it concerns the appearance of "geometric patterns" characterized by the repetition of certain geometric configurations whose length is much less than the macroscopic scale of the object under consideration. These geometric structures are reproduced periodically, pseudo-periodically and in a more complicated way.
The results of R. V. Kohn’s researches lead to a better understanding of the materials mentioned above and enable them to be studied in greater depth. And while they cannot yet be transformed into something real, it is quite likely that in the future we will have at our disposal a device impossible to imagine at present, which will be a direct descendent of one of this mathematician’s results.
Robert V. Kohn has studied mathematics at the most prestigious universities in the U.S.A. After graduating from Harvard, he studied for his Master’s degreeat Warwick and subsequently obtained his doctorate in 1979 at Princeton. He is currently a professor at the Courant Institute at the University of New York. In 1999, he was awarded the Ralph Kleinman Prize by the U.S.A. Society for Industry and Applied Mathematics (SIAM).
Speaker: Robert V. Kohn
Title: “Energy-Driven Pattern Formation”
Date: Saturday, August 26th. 09:00-10:00
ICM2006 Scientific Programme
Robert V. Kohn personal web page:
Both these disciplines arose in response to questions concerning a single problem: to find the optimum configuration of the mechanism or process under consideration, where optimum is equivalent to the best possible.
For instance, the "ball bearing mechanism” for the stabilization of the steam engine, which Lord Maxwell analysed at the end of the 19th century by means of the technical properties of the qualitative theory of Differential Equations, constituted the origin of Control Theory. This discipline is concerned with the control and stabilization of all kinds of processes, from the most classical engineering processes to the most modern related with biomedical sciences (“robot surgeons” are an example). This is a multidisciplinary field in which mathematics are fused with other sciences in order to study models frequently formulated in terms of discrete, differential, stochastic and partial differential equations.
The decontamination of subsoil and water supply by studying the control of diffusion processes; the control and design of aircraft; the regulation and acclimatization of large surface areas, or the control of flexible space structures are just some of the problems addressed by research into Control.
What are the developments in this field? Once again, they are determined by real situations. The interaction of subsystems are constantly giving rise to new dynamics, and it is becoming imperative to develop ever more complex models for which existing theories are no longer sufficient.
Optimization shares the same objectives as Control, although it employs different tools. In Control, given the nature of the problems being studied, techniques belonging to the sphere of dynamic systems and differential equations are normally used, while in Optimization methods belonging to combinatorics and discrete mathematics are more frequently employed, which are vital for developing algorithms useful in modern applications for Technology and Social Sciences.
The problems that can be dealt with by Optimization are many; the monitoring and control of satellites; the planning of communication and distribution networks; the location of emergency centres and/or provision of services; financial engineering; energy markets; multi-criteria decision-making; data mining; optimal classification; sequential optimization by scenarios; design of pharmaceutical products, etc..
Control and Optimization are disciplines in constant process of expansion; their importance in applications is paramount and they incorporate techniques from all the other mathematical disciplines. The recent revolution in information technology has galvanized these disciplines, which are now capable of tackling systems of a scope hitherto unthinkable, as well as implementing algorithms which until recently were only possible as mere mathematical programmes.
Control and Optimization are two disciplines that are often found together in research journals, together with other related fields such as Variational Calculus, Optimal Design, Inverse problems and Game Theory.
Professor of Applied Mathematics
Autonomous University of Madrid
The development of appropriate conceptual tools is vital for the introduction of highly complex IT systems. Some of the most valuable of these tools arise from the latest advances in the theory developed by Stefan Banach in 1930. Known as the Banach Space Theory, it specializes in the management of infinite dimensions. Stated more simply, it postulates that “sometimes more information can be obtained from the equation to be solved if, instead of concentrating exclusively on the equation itself, we pay more attention to what it has in common with other similar equations”, explains Jesús M. F. Castillo, the organizer of the talks to be given on the subject in Cáceres.
Banach Spaces constitute one of the goals of research in the field of functional analysis. Studies on this subject have lead to the discovery of some of the principles governing complex systems (uniform boundedness; concentration; infinite Ramsey theorems, etc.), together with the design of sophisticated methods of study and measurement.
In the field of IT concerned with complex systems (characterized by their numerous dimensions), this theory is making contributions whose importance is comparable to those of computers themselves. Furthermore, it is thanks to this theory that it is now known that “apparently chaotic multidimensional systems are becoming more and more regular; or at least large areas of them are”. This idea suggests that it is possible to find a way of classifying objects that includes not only order but chaos as well.
During the conference to be held in Cáceres, Extremadura, subjects concerning Benach’s Theory will be dealt with, including related or complementary fields such as applied logical methods, the geometry of convex bodies, categoric methods or infinite-dimensional topology. Particular attention will be given to the efforts to transfer the results of the theory to other fields of mathematics whose available tools are in principle minor; for instance, the theory of spaces with a metric for measuring distances or the geometry of convex sets, which are of great interest in Economics.
The Department of Mathematics of the University of Extremadura has been organizing conferences on Banach Space Theory every two years since 1996. Three conferences have so far been held in Badajoz, Jarandilla de la Vera and Cáceres. The proceedings have been published in the international journal Extracta Mathematicae and by publishers such as the Cambridge University Press.
“Banach space theory: classical topics and new directions”
Complejo Cultural San Francisco.
Ronda de San Francisco nº 1, Cáceres.
Contact: Jesús M. F. Castillo; Email: firstname.lastname@example.org tel 92 428 9563
For further information: web: http://www.banachspaces.com/
An engineer is carrying out complicated trials in an induction oven at a temperature of 1,450ºC to improve the purity of the Silicon to be used in solar cells. He does not take his eyes off the computer screen. Mathematics are vital for industry, since they enable highly complex and costly processes to be modelled and simulated before being reproduced in a factory. A further example is the Silicon used in photovoltaic panels. The Department of Applied Mathematics at the University of Santiago de Compostela has for years been working with the Ferroatlántica Company on the development of mathematical tools to improve Silicon production. As department head Alfredo Bermúdez de Castro explains, this material derived from quartz must be purified in order to be used in solar cells, a process that entails treatment with gases to maintain it in a liquid state at 1,450º in an induction oven. How can this process be controlled and the results improved? By means of partial differential equations such as electromagnetic field equations (Maxwell’s equations), or heat transfer equations, Galician mathematicians have developed models in order to determine how the material behaves inside the oven. These equations can only be solved by using numerical methods with very complicated computers and algorithms. Indeed, as Bermúdez de Castro explains, sometimes the computer is thinking for days. At the University of Santiago de Compostela the department has designed computer programmes that are able to simulate the functioning of the induction oven. This enables the engineer to experiment more easily with the oven simply by changing the different variables on the computer.
To find out more: