April 17, 2006
Some years ago, during an informal meeting, a group of students and colleagues asked Lennart Carleson, the 78 year-old Swedish mathematician who has recently been awarded the Abel Prize, how he had managed to solve a particular problem. Carleson pondered for a moment and said, “I don’t know… If I came back in a second life and had to prove those theorems again…”. He did not finish the sentence. Fernando Soria, professor of Mathematical Analysis at the Universidad Autónoma de Madrid and present on that occasion, was always curious to know how Carleson would have finished the phrase. Now he will have the opportunity to ask him, since Carleson will be at the ICM2006 International Congress of Mathematicians to be held this August in Madrid.
Carleson will take part in the ICM2006 closing round table on August 29th (18:00) titled: “Are Pure and Applied Mathematics Diverging?”. The session will be chaired by the president of the International Mathematical Union, John Ball, and besides Carleson four other prestigious mathematicians researching in pure and applied mathematics will also take part. The subject they have chosen is a classical one, but as important and up-to-date as ever. “There has always been a ‘pure-applied’ frontier, a frontier often extended by prejudices such as ‘pure’ mathematicians are wrapped up in their own thoughts and don’t do anything useful, or that ‘applied’ mathematicians are not true mathematicians”, explains Manuel de León, chairperson of the ICM2006 Executive Committee.
Carleson’s scientific career would place him with the pure mathematicians. The Abel Prize, first awarded three years ago by the Norwegian Academy of Science and Fine Arts and worth 755,000 euros – the same amount as the Nobel Prize, which is not given for mathematics – was conferred “for his profound and fundamental contributions to harmonic analysis and the theory of continuous dynamic systems", in the words of the jury. None of these contributions have directly yielded practical applications. However, Carleson’s work has opened the door to the development of applied areas, fields which lie behind something as tangible as a photograph in .jpg format.
The problem Carleson was asked about in Fernando Soria’s anecdote concerned the Fourier Series. “In 1807, the French mathematician Joseph Fourier delivered a report … in which he stated that every periodic wave could break into an infinite sum of sines and cosines. Behind this statement lies the intuitive idea that all sounds are composed of the sum of duly amplified simple harmonics”, explains Soria. “This statement was controversial at the time and could not be demonstrated. Later, in the 20th century, it was proposed as a conjecture”. This is the statement that Carleson proved. In a sense, says Soria, the award-winning Swedish mathematician ended up struggling against himself; he was convinced that the conjecture was false, and what he was really looking for was “a counterexample, a non-decomposable wave in harmonics”
So what position will Carleson adopt at the round table in August? It could be that the demonstration of Fourier’s Conjecture in itself will be of little use, but “the arguments Carleson used certainly form part of applied harmonic analysis, and are indeed very useful. They lie behind the digitalization of signals. Solutions similar to Carleson’s work are used for example in the compression of a digital image”, remarks Soria.
Scientific program ICM2006:
Abel Prize 2006
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”, says mathematician Benoit Mandelbrot (Warsaw, 1924) in the introduction of one of his best-known books, The Fractal Geometry of Nature. He refers to the fact that the artificially smooth objects of traditional Euclidean geometry –circles, squares, triangles…-- are not as useful to describe nature as ‘fractals’, objects whose shape appears similar at all scales of magnification, and therefore often referred to as "infinitely complex”. According to Mandelbrot, fractals can be used as models of many natural phenomena, be it the shape of coastlines, the structure of plants and blood vessels; the clustering of galaxies; and stock market prices.
Mandelbrot, one of the very few mathematicians whose talks are often attended by all kinds of people –even non-mathematicians—, coined the term ‘fractal’ in 1975 from the latin word fractus (‘broken’ or ‘fractured’), and opened up a new field of mathematics devoted to the study of these structures. Today, fractals are used in such diverse applications as medicine –i.e. to detect certain types of tumors— and cinema –to generate artificial landscapes--, as well as art.
Mandelbrot is currently the Sterling Professor Emeritus of Mathematical Sciences at Yale University, and IBM Fellow Emeritus (Physics) at the IBM Research Center.
Mandelbrot will give the Special Lecture The Nature of Roughness in Mathematics, Science and Art at the ICM2006 on Saturday, August 26 (19:00-20:00). Also, Mandelbrot will chair the International Contest of Fractal Art ICM2006 (www.fractalartcontests.com/2006). There will be also a Fractal Art exhibition at the Centro Cultural Conde Duque and at the ICM2006 (Links to images of fractals, below).
Fractals have become one of the most beautiful faces of modern mathematics, and a useful one to popularise mathematics among the general public. How do you feel about this?
I'm thoroughly delighted to see that fractals help bridge the abyss that separates difficult open questions of mathematics from lay persons' interests -young and old. Some mathematicians have always liked their field to be left alone but I always thought that isolation is unwise, in fact, very harmful.
What did your colleagues first say when you showed them your results with fractals?
Fractal geometry did not involve any Eureka! moment. It was not a serendipitous discovery but the end of a very long process which repeated itself for diverse forms of roughness that occur in pure mathematics and in many areas of science. Pure mathematicians had come to believe that it was no longer possible for pictures to affect their field. To the contrary, my discovery of the Mandelbrot set consisted in a number of visual observations phrased into
conjectures demanding rigorous mathematical treatment. Naturally, the very possibility of deriving conjectures from pictures was originally received with scepticism and the extreme difficulty of my conjectures came as a very big surprise. Today the situation is completely different. For example, my 4/3 conjecture about Brownian motion has inspired a large amount of highly admired purely mathematical work. And the MLC conjecture about the Mandelbrot set -that it is locally connected- has so far resisted all attempts to proof or disproof. All those conjectures can be rephrased in ways everybody can understand and provide the layperson with glimpses of active portions of the frontier.
Do fractals have practical applications now?
There is almost always a lag between theory and practical applications. For fractals, that lag lasted several decades because the change was steep. To improve roads and walls meant to make them more nearly flat, and similarly everybody had long taken for granted that engineering design must rely upon the smoothness of classical geometric figures. But in many cases, recent ingenuity yielded far preferable rough solutions, fractal ones. For example, new high-tech radio antennas are fractal, new high-tech anti-noise walls along highways are fractal also. New high-tech concrete has many fractal characteristics. Many chemical engineering operations adopted fractal features to obtain cleaner and cheaper products. The list of such examples can go on and on. The notion of "practicality" also extends well beyond engineering and today the most active innovation in finance consists in fractal models.
Why are you coming to the ICM? Why is this congress useful?
Large international congresses are useful because they bring together people from different countries. In particular, they allow persons from the developing nations to meet the leaders in the field. As to myself, I am coming because I have been invited. More seriously, I have over the years been increasingly concerned about the abyss referred to in my answer to the first question and greatly welcome the opportunity provided by ICM to help in re-establishing the unity of knowing and feeling, the unity of mathematics, and a better understanding between its creators and those who benefit ultimately from its growth.
Mandelbrot’s page at Yale University
International Contest of Fractal Art ICM2006:
“When a fluid moves in a recipient, each particle follows a trajectory, which forms a curve in the space. Sometimes it happens that this curve closes, and that is when a knot is created”. In this way Étienne Ghys seeks to provide a graphic description of the mathematical element that is the subject of his lecture, Knots and Dynamics.
However, he is quick to point out that it is “only an analogy”, since Ghys field of study is more abstract. Nevertheless, he believes that these kinds of ideas provide a clue toward explaining how three-dimensional dynamical systems could be studied by observing the characteristics of their trajectories. Ghys’ field of study is a combination of topology, geometry, dynamical systems and group theory.
The periodic orbits define knots whose topology can sometimes be extremely complicated. In his lecture, Ghys will show some advances in the quantitative and qualitative description of this type of phenomena. The French mathematician will divide the lecture into three parts. In the first two he will focus on the characteristics of the trajectories, and end by describing an important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to well-known arithmetical functions.
Étienne Ghys was born in 1954 in Lille (France). He graduated in mathematics from the university of his home town in 1979. He is at present "Directeur de Rechercher du CNRS" at l'École Normale Supérieure in Lyon, and since 2005 has been a member of the Academy of Sciences in Paris. He gave one of the invited lectures at the International Congress in Kyoto in 1990. In 1991 he was awarded the silver medal of the French National Centre for Research (CNRS).
Speaker: Étienne Ghys
Title: Knots and Dynamics
Date: Thursday, August 24th. 11:45-12:45
ICM2006 Scientific Programme
Étienne Ghys personal page:
For a meteorologist, atmospheric pressure, wind speed, temperature, air density or humidity are physical phenomena described by variables whose numerical value depends on where and when they are measured. But what if it were possible to calculate the future value of these variables by using information from the present and the past? In fact this is possible. The mathematical laws for achieving this already exist - partial differential equations (PDEs). In their different versions, partial differential equations describe a multitude of physical processes as well as their status of equilibrium.
These equations are applicable to many fields of science:
-The diffusion of heat, of chemical substances or of biological populations. The “heat equation”, for example.
-The deformation of elastic solids (equations of elasticity).
-The motion of fluids (liquids, gases and plasmas). Thus the Euler and Navier-Stokes equations are necessary for modelling ocean currents and climate, or the lift given by an aircraft wing.
-The extraction of oil or the filtration of fluids and pollutants through the ground: these are fluids that are also diffused!
-The propagation of waves, such as acoustic waves or electromagnetic fields (Maxwell’s equations). Applications in daily life: communications in general; radio, light and sound; seismic waves; waves in medicine and other sciences.
-The study of space-time (Einstein’s equations, black holes)
-The equations of quantum mechanics (Schrödinger’s equations)
-The evolution of financial assets (Black-Scholes’ equations).
-Signal and image processing, an enormous market today.
In none of these sciences is it possible to go beyond mere generalizations without entering into the analysis of their fundamental laws, which are partial differential equations.
The ICM2006 PDEs Section contains 11 talks, two of them given by Spaniards. Luis Vega, from the Universidad del País Vasco, will speak about Schrödinger’s equations, fundamental in quantum mechanics, and Juan José López Velázquez, from the Universidad Complutense de Madrid, will speak on equations that describe the phenomena of chemically based aggregations.
In addition, there will also be plenary talks on PDEs.
The ICM2006 Scientific Programme
/paginas/?pagina=Partial Differential Equations
Game Theory, a powerful tool already assiduously employed in Economics, Political Science, Philosophy and Biology, is now expanding its applications to fields connected with the environment. These fields are at the forefront of contemporary politics at both a national and international level. On many occasions, environmental measures are based on strategic reasoning. It is therefore appropriate to use Game Theory as an analytical instrument for a better understanding of the interrelations between the economy and the environment in order to provide politicians with practical indicators.
On this premise, the meeting to be held in June in Zaragoza seeks to demonstrate the utility of this theory in the conflicts arising from the assignation of basic natural resources such as water and land, among others. The meeting also seeks to stimulate the development of techniques and methods in issues such as land and water management, forests and fishery, global warming, contamination, migration, etc. More specifically, the intention is to address aspects of practical interest for researchers, policy makers and development agencies by debating pressing global issues such as climate change, trade and national and international agreements.
The Zaragoza conference forms part of a series of meetings on Game Theory held in Europe and in Spain over the last fifteen years, and will reflect the increase in the number of Spanish specialists in this field as well as their growing presence on the international scene.
Zaragoza, 10-12 July
Venue: CIHEAM- Instituto Agronómico Mediterráneo de Zaragoza
Contact: Fioravante Patrone
The organization chart of a firm describes the hierarchy of its staff; it does not necessarily reflect real power relationships within the company. Who has the best contacts within the firm? To whom should one refer to channel a piece of information efficiently? With their ability to analyse social systems, mathematics can not only give answers to this kind of question, but can even find amazing results. According to Ángel Sanchez, professor of Applied Mathematics at the Universidad Carlos III de Madrid, the e-mail records of the staff, together with the mathematical theory of Graphs, provide a way to study the informal power flow in a company. Many features of social networks can be learnt with this method: the typical number of links, the most relevant nodes…
Mathematics have a large number of applications in sociology. Not only are they able to analyse social networks; they can help in preventing the spread of an epidemic within a community, or in trying to diminish its effect. As Sanchez explains, when immunisation of the whole community is not possible mathematics enable us to determine who should be vaccinated or which measures should be taken to avoid infection. Similar tools can be applied to find out which computers in a network should be especially protected to prevent infection from computer viruses.
This field of mathematics extends even further: Sánchez is currently working in trying to understand by means of numbers how a certain culture works. This could be very interesting for the study of street gangs or immigrant communities, as well as to predict their evolution or even try to guide it.
Ángel Sánchez: firstname.lastname@example.org
Instituto Mediterráneo de Estudios Avanzados: http://www.imedea.uib.es/physdept/eng/lines/complex.html
Network on Applications of Statistical and Non-lineal Physics to Economy and Social Sciences: