April 10, 2006
Three years ago, a young Russian mathematician called Grisha Perelman surprised the world by stating that he had solved one of the most famous problems of 20th century mathematics, Poincaré’s Conjecture, which was first posed in 1904. The most prestigious analysts appear to corroborate that the demonstration is convincing, or at least until now they have either not found or not reported finding any holes in it.
The problem is so intricate that many brilliant mathematicians have attempted to solve it and failed. There has been no lack of incentive, since it is one of the 7 problems the solution of which carries a prize of one million dollars offered by the Clay Institute in the year 2000.
However, this was not the bait that led Grisha Perelman to tackle the problem, since he sequestered himself away for that very purpose 1994, and for the following eight years gave no signs of life until May, 2003, when he announced that he had managed to solve it. Although three years have passed since that time, the final verdict has still to be pronounced, and analysts are still at work checking the proof, which is more extensive that Poincaré’s Conjecture itself.
The expectation aroused is understandable, and everything is likely to be definitively clarified during the celebration of the ICM2006, when two of the principle lectures will be devoted to this subject and given by two of the leading experts, Richard Hamilton (who developed a tool used by Perelman in his solution of the problem) and John Morgan, a recognized specialist in topology. Although Perelman himself, reluctant at least until now to appear at public events, will be absent, the subject will undoubtedly be the highlight of the congress. The ICM2006 will in all probability go down in history as the occasion when this conjecture became a theorem, a term used by mathematicians to refer to a demonstrated hypothesis.
Henry Poincaré, who together with David Hilbert was the most famous and influential mathematician of the late 19th and early 20th century, contributed especially to the development of topology, which is according to his own definition, “what remains of geometry when one forgets the notion of distance”. According to Vicente Miquel, professor of Geometry and Topology at the University of Valencia, topology is the study of what remains alter deforming and object without breaking it, because what changes are the distances. Thus in topology, a football, a rugby ball and an orange are considered to be the same, but not a donut, which is the same as a ring or the earth’s orbit. “There are properties of nature, in particle physics, in DNA and in many other aspects”, explains Miquel, “which depend on topology alone”.
Perelman says that he has not only solved the conjecture but has also completed this classification. He has already published three papers on the subject, general agreement having been reached on the fact that the first and much of the second are correct, leaving a “technically more difficult” part still to be checked. “Everyone understands the third paper, which together with the verified parts of the first two would seem to provide a proof of the Poincaré Conjecture, and would be enough for Perelman to receive the Clay Institute million-dollar prize”, says Miquel.
Problems selected by the Clay Institute:
Article by Mark Brittenham (University of Nebraska) about the Poincaré Conjecture and the solution proposed by Perelman:
Article by John Milnor, Fields Medal 1962:
Henri Poincaré’s biography
Manuel de León, mathematician and research professor at the IMAFF Department of Mathematics (CSIC), is currently facing the most ambitious and demanding challenge of his career: the organization of the ICM2006 World Congress of Mathematics, to be held in Madrid at the end of August. He chairs the committee responsible for organizing the congress, which will be attended by around 5,000 mathematicians from all over the world.
What’s going to be pivotal theme of the Congress?
There’s no doubt that the subject arousing most interest is Poincaré’s Conjecture, because it’s a problem that’s been around for a century, and while its demonstration has yet to be officially accepted, all the signs seem to be that has indeed been solved. A good indication of this has been the two lectures given on the subject, one plenary and one outside the programme. This latter lecture was delivered by one of the leading experts in differential topology and topology, John Morgan, which suggests that the demonstration has all but been approved, and that it will be officially accepted during the congress in Madrid.
What other important subjects will be addressed?
There are several. One of them will be debated in a round table discussion, and that’s the relation been pure, basic mathematics and applied mathematics. I think they are in the process of converging again; much of the strength of mathematics lies in their unity, and a good example of that is the fact that the congress still covers the whole of mathematics. The ICM congresses are increasingly open to applications, and this one more than ever before. This is an important subject because the false separation between the two facets, this barrier, doesn’t really exist. In fact, theoretical results are the ones most frequently being applied, and whereas in the past they were arrived at with pencil and paper, now we need powerful computers to obtain them. So we really can’t speak any longer of pure and applied mathematics.
Do the solutions obtained for these historic problems mean that this is a good time for mathematics?
Yes, this is certainly a good time... three important problems have been solved in the last ten years. The first was Fermat’s Theorem, then Kepler’s Conjecture, which required an enormous amount of computation, and now Poincaré’s Conjecture... One of the classical problems still remains to be solved, and that’s Riemann’s Hypothesis, which is the great problem in mathematics, and one that is proving to be a tough nut to crack. It’s been attempted from every possible mathematical point of view, but without success. There’s no sign at the moment that it can be resolved, and it’s important because it has to do with the distribution of prime numbers, to see if there are any guidelines, not mention its many practical implications in the technological field, such as data encryption.
Is the scientific content of this ICM greater than on previous occasions? In other words, is it increasing?
There’s one more section. There’s always some advance on previous congresses, although the International Mathematical Union is very slow to make changes; it’s held back by inertia. But in general I think the scientific content is slightly greater in volume than in Beijing.
How is the programme structured?
Well, there are really two different programmes, one invited and one free, so to speak. The first is made up of 20 plenary lectures, the 169 talks included in the 20 sections, and the complementary activities, such as the four round tables, three special lectures – the Emmy Nether, the Mandelbrot and the Morgan – presentations by technological companies and cultural activities… That’s the invited programme, while the in free one there are the short communications, posters, and brief presentations of mathematical software, which is a type of talk that began some years ago. The deadline for submissions was March 30th, and as in all the congresses the selection process will last until the end of April. I think there’ll be about 1,200 talks of this kind all together.
How have the invited speakers been chosen?
A scientific committee is responsible for the main programme. Its composition is secret; only the name of the chairperson is released. There are a dozen people on this committee, usually including a member from the organizing country. We’ve managed to have two people appointed to the committee, and all the members’ names will be made public at the end of the congress.
What about the sections?
The subjects for the sections don’t vary. They are approved by the IMU executive committee and later subjected to ratification at the General Assembly. Generally speaking, there aren’t many changes between one congress and another and the classical subjects are always on the list. This time a new section has been included, mathematics applied to control and optimization. Then there are recently introduced sections of growing importance, such as education and dissemination, the history of mathematics and the applications of mathematics to the sciences, which are achieving greater prominence, and will continue to do so in the future.
“The nucleus of the universe, a very small part right at the centre, about 10-35 metres in diameter”, says the mathematician Jean-Pierre Demailly, “could be a Calabi-Yau manifold, which is a special example of Kahler compact manifolds”. These exotic names refer to the complex geometrical structures being studied by Demailly, who will give one of the ICM2006 plenary lectures, during which he will present the latest advances in the understanding of the geometrical structure of Kahler projective algebraic manifolds.
Knowledge of these structures provides potential applications in other fields of mathematics such as algebra or topology. Demailly employs analytical methods, a branch of mathematics, to solve problems in other areas of this science as well as other sciences, since in his methods he includes the solution of the Monge-Ampère equations, which are related to the general theory of relativity equations.
Manifolds are mathematical constructions that generalize the idea of curve and surface to any dimension and any body. Professor Demailly’s talk will focus on the study of Kahler manifolds in the field of complex numbers. These manifolds are a generalization of projective algebraic manifolds, which are the solution to a system of polynomial homogeneous equations.
Jean-Pierre Demailly was born in 1957 in Péronne (France). He began his mathematical studies in 1973, and graduated from the University of Paris VII in 1976. Under the supervision of Henri Skoda he wrote his doctoral thesis “On the Different Aspects of Positivity in Complex Analysis” (“Sur differents aspects de la positivité en analse complexe”, which he defended in 1982. He has been professor at the Joseph Fourier University in Grenoble since 1983. In 1994 he was elected a member of the Academy of Sciences, and in 2002 a senior member of the Institut Universitaire de France (IUF). He has gained much recognition and won several awards, the 1996 Humboldt Prize for International Collaboration from the Max Planck Society among them.
Speaker: Jean-Pierre Demailly
Title: “Compact Kähler Manifolds and Transcendental Techniques
in Algebraic Geometry”
Date: Tuesday, August 29th. 10:15-11:15
ICM2006 Scientific Programme
Jean-Pierre Demailly – personal web page:
Mathematical analysis has its origin in the formulation and solution of elementary equations of physics. The movement of a particle or a star, the temperature of point on the earth’s surface or inside an oven, or the growth of a particular population of bacteria are phenomena which, like all those which occur around us in nature, can be modelled by means of a differential equation. The solutions to these equations are functions that depend on the number of variables and implicit parameters included in the model (position, time, the materials and quantities employed in a chemical reaction, etc). Mathematical analysis studies the properties of these functions (their continuity, differentiability and integration), the spaces they assume and the transformations that act upon them.
It is considered to be a breakaway discipline from geometry and topology alter the formulation of differential and integral calculus by Newton and Leibniz. It shares with them numerous objects of study and similar techniques. Thus when we speak of geometric function theory and related applications, or of geometric measure theory, we are referring to different aspects of analysis (of real or complex variables). An important part of functional analysis also consists of the study the geometry of the spaces of functions or the operators on them. Likewise, so called global analysis is closely related to the differential geometry of variations.
Mathematical analysis is markedly interdisciplinary in character, with numerous applications to other fields of mathematics such as number theory, dynamic systems, probability and stochastic processes, particularly by jeans of harmonic analysis, with the solution of partial differential equations and applied mathematics in general.
The speakers invited to the ICM-2006 by the IMU (International Mathematical Union) Committee to give talks on analysis are a clear reflection of the variety of topics belonging to the subject: Quasi-conformal geometry in fractal sets (M. Bonk), the solvability of differential equations using techniques of harmonic analysis and the Carderón-Zygmund theory (S. Hofmann), the convergence of series of functions with respect to classical systems (S. Konyagin) and general systems (V. Temlyakov), complex dynamics, related continuous and discrete applications (l. Rothschild, S. Smirnos y E. Straube), and potential theory and analytical capacity (by the Spaniard Xavier Tolsa). Among the plenary speakers, the IMU Committee has chosen Terence Tao, a specialist in the field of harmonic analysis, who will present his work in collaboration with B. Green on the structure of prime numbers.
Professor of Analysis at the Universidad Autónoma de Madrid
Space-time modelling has become a crucial tool in research concerning the statistical analysis of natural processes, especially those related with environmental studies (the concentration of contaminating particles in the air, salinity of the oceans, the advance of deforestation, etc.). In recent years its progress has been aided by the availability of computers with an extraordinary capacity of calculation, as well as by sophisticated algorithms developed by mathematicians.
The workshop due to take place in Pamplona, the third on this subject to be held in Spain in the last six years and the first at an international level, falls within this framework and is aimed at giving impetus to the development and application of spatial, temporal and above all space-time statistical methods in fields concerning the environment.
The latest advances in theory, methods and applications will be presented during the sessions of this conference, illustrated with statistical procedures based on real data. Talks will also be given on the drawing up of epidemiological maps, one of the fields of space-time modelling that has proved to be the most productive.
Internacional Workshop on Space-Time Modelling (METMA3), Pamplona, 27-29 September 2006
Venue: Escuela Universitaria de Estudios Sanitarios
Avda. de Barañáin s/n, 31008 - Pamplona (Navarra)
A corpse, the scene of the crime, and a fingerprint on a bloodstained knife. All the forensic scientists have to do is gather the clues and check them in their data base for a suspect to come up on the screen. However, have we ever stopped to think about the enormous storage capacity required for recording digital versions of the millions of fingerprints on police files? Fingerprints on FBI files alone currently occupy 200 terabytes (200.000.000.000.000 bytes).
Mathematics can also be of great assistance in this field. Techniques for compressing information can perform miracles. If we take two digital images of the same fingerprint, they will be identical, but one of them will be composed of of 5% of data from the other. The difference resides in mathematics.
This is possible thanks to the theory of Wavelets, which comes to say that every image can be decomposed in simpler images. So what FBI does is reducing the information to the minimum amount of data needed to rebuild the image. This not only allows to store 20 fingerprints where there used to be room only for one; it is also a great help in the process of checking millions of entries in search for a suspect.
Ronald Coifman, of Yale University, an expert in this field and the autor of the algorithms used in the FBI fingerprint database, will attend the ICM2006.
On the FBI fingerprint database: