2 May 2006
Leonardo da Vinci said it in the 15th century: “A science cannot be considered as such unless it is impregnated with mathematics”, a statement that is still valid today. Current science and technology must be nourished by mathematics in order to make any advances. But this is an arrow that flies in both directions at once: mathematics also grows according to the new problems constantly being thrown up by the world. So it is not out of place to ask: What will mathematics be like in the new millennium? What questions will it have to tackle? Some of the most outstanding contemporary mathematicians have agreed to share with us their reflections on the subject during the symposium “Mathematics for the 21st Century”, due to be held on the 3rd and 4th of May at the Ramón Areces Foundation in Madrid. John Ball, president of the International Mathematics Union (IMU), and two Fields Medal winners will attend what is considered to be the antechamber to the 2006 International Congress of Mathematicians (ICM2006), which will be held in August in Madrid.
The programme for the symposium at the Ramón Areces has been designed to provide a floor for both the kind of mathematics “that though focused on themselves give rise to spectacular developments, and those that are nourished by the natural world and new technological challenges” explains Manuel de León, one of the co-ordinators of the symposium and chairperson of the ICM2006 Executive Committee.
Alain Connes and Efim Zelmanov, who were awarded the Fields Medals in 1982 and 1994, respectively, belong to the basic mathematics group, but Connes' work has proved to be of great usefulness in theoretical physics, while Zelmanov, who collaborates with Spanish mathematicians, has remarked more than once on the “beauty” of some of the current mathematical problems arising directly from applications, such as cryptography.
For his part, Avner Friedman, of the Ohio State University (U.S.A.), will be representing applied mathematicians at the Areces symposium, where he will speak on a subject that is expanding rapidly: The New Mathematical Challenges Posed by Biological and Medical Sciences. Decyphering the genome; understanding how proteins are folded; predicting the development of an epidemic and even of a tumour all of them problems in which mathematics has a vital role to play.
Of equal current concern are the challenges posed by artificial intelligence. Luis Mª. Laita de la Rica, from Spain, will speak on What Machines Can and Cannot Do, while Walter Schachermayer, from Vienna, will deliver a talk on financial mathematics. There will also be talks by non-mathematicians; Jordi Bascompte, of the Estación Biológica CSIC at Doñana, will explain how mathematics can contribute to the study of biological biodiversity, and Amable Liñán, winner of the Príncipe de Asturias Prize, will deal with the problem of developing more effective combustion engines.
In short, as Manuel de León, and Manuel López Pellicer of the Royal Academy of Exact, Physical and Natural Sciences, explain, the symposium will offer an “overview” of present-day mathematics “which will range from classical problems such as Poincaré's Conjecture, probably one of the central themes of the ICM2006, to the relations between mathematics and biology, as well as with computation, engineering, industry and finance. And last but not least, the continual two-way relation with the physical sciences, which has done so much to shape both the world today and our vision of the universe”.
The closing talk will be given by the French mathematician Jean Pierre Bourguignon, director of the prestigious Institute of Higher Scientific Studies of the Centre National de Recherche Scientifiique (CNRS).
The symposium is organized in collaboration with the Spanish Royal Academy of Exact, Physical and Natural Sciences and with the ICM2006 itself.
Luis Vega González (Madrid, 1960) has been teaching at the University of the Basque Country since 1993. He graduated from the Complutense University of Madrid in 1982 and gained his doctorate six years later at the Autonomous University of Madrid. His more than 70 papers published in international journals have attracted considerable attention, and now he is one of the nine Spanish mathematicians who have been invited to speak at the ICM2006. Telecommunications and fluid mechanics are two of the fields that are benefiting from the applications arising from his researches. His talk is included in Section 11 (Partial Differential Equations) and will deal with “the problem of initial value for nonlinear Schrödinger equations”.
I'll be talking about a versatile general method for solving nonlinear Schrödinger equations. This is a universal equation which, among other things, describes how dispersive waves evolve in time. In other words, waves whose velocity depends on their frequency. Therefore, unlike sound or electromagnetic waves, their speed of propagation is not bounded. Our work has to be understood as a stability result of many mathematical models describing physical situations where these waves are present.
All my scientific work is related with Schrödinger's equation in one way or another.
Just about, yes; all my scientific work revolves around it. There are two versions, one linear and the other nonlinear, and both are of interest to me. In both cases, together with other colleagues, I've been able to make contributions that are basic in the field.
One immediate application was being able to study the stability of the soliton or wave of translation, so-called because it's a wave that propagates itself without loss of shape or velocity during very long periods of time.. This phenomenon was described for the first time by the Scottish engineer John Scott-Russell in 1834, and is involved in many other phenomena. It's used, for example, to send waves through optic fibre without distortions or interference.
It's a wave that doesn't disperse, because of the nonlinear effect. It's a rather weak, nonlinear interaction, but it prevents the wave from becoming diluted and makes it stay like all of a piece, on principle, forever. That's because of this interaction between the dispersion created by the medium, the tendency to dispersion and the interaction with the nonlinear boundary, which is like the influence of the wave on itself and creates a resonance. It re-enforces itself in a perfect balance so that it neither dilutes, concentrates nor increases. The problem with a nonlinear equation is that it can create an explosion in finite time. It concentrates to such an extent that it becomes too big, whereas the soliton remains in perfect equilibrium.
It's not clear whether tsunamis can be understood with a soliton. A tsunami is such a big wave that the Pacific Ocean is just like a bathtub, and the tsunami is of such great dimensions that it spreads everywhere. It can be several kilometres long, so that when it reaches the coast the ground rises and the water starts to accumulate, and that's when it starts to stack up. It also propagates at a great speed, so when it meets an obstacle the momentum is tremendous. It doesn't have to be a soliton to accumulate.
In many. In fluid mechanics we've proved a stability result for the lines of a whirlpool. The line of a whirlpool is a way of describing mathematically the trajectory of, for example, smoke from a cigarette. It is thought that the interaction of these lines is the basic mechanism of turbulence and the formation of singularities, which is one of the fundamental problems of fluid mechanics.
I'm exploring the use of the Schrödinger nonlinear equations in fluid mechanics. I'd like to be able to prove beyond doubt that they are a good model for describing turbulence and the formation, or not, of singularities in this field, at least in some simple but representative situations. I believe this will enable us to quantify these phenomena.
That's an unknown quantity. I still haven't given up the idea of moving towards another discipline where I can use mathematics. In fact, at the moment I feel much closer to physics, although I'd still like to be able to go more deeply into certain things.
2, 3, 5, 7, 11, 13 Prime numbers: only divisible by themselves and by one. Maybe one of the few mathematical definitions that students can repeat in a sing-along fashion by the end of secondary school. Henryk Iwaniec will speak about these numbers, which constitute one of the most essential mathematical elements, in his lecture "Prime Numbers and L-functions". “My talk,” assures Iwaniec, “can easily be understood by all those who love numbers”.
However, his words should not be misunderstood. Iwaniec's lecture will not deal with peripheral mathematical topics; on the contrary, as he himself says, “prime numbers are at the very heart of arithmetic". Some of the most fundamental problems in Number Theory are related to prime numbers. One example is the search for the zeros of the zeta function, which is the object of study of Riemann's hypothesis, a problem still awaiting proof and considered by many to be the most important open questions in mathematics..
Henryk Iwaniec will also deal with L-functions, a generalization of Riemann's zeta function in prime ideals; not a new ideology but a mathematical term referring to the subset of a ring that fulfils a series of properties, some of them shared by prime numbers.
In his talk, Iwaniec will speak about new and old results used in mathematical proofs and techniques, and he will try to convince the audience that "there is indeed a healthy life beyond Riemann's hypothesis".
Henryk Iwaniec was born in Elblag (Poland) in 1947. In 1971 he graduated in mathematics in Warsaw University, and obtained his PhD the following year. He has been teaching at Rutgers's University, New Jersey, since 1989. Iwaniec has received several prizes, the last one, in 1973, being the Cole Prize in Numbers Theory awarded by the American Mathematical Society. He has given lectures in several international congresses of mathematics and has more than one hundred publications to his credit.
Speaker: Henryk Iwaniec
Title: “Prime Numbers and L-functions”
Date: Thursday, August 24th. 10:15 -11:15
ICM2006 Scientific Programme
Algebraic Geometry combines abstract Algebra, especially Commutative Algebra, with Geometry. It first arose as the study of the set of solutions to the system of algebraic equations, which goes beyond the mere solution of equations to where an “understanding” of the space of all the solutions (typically known as algebraic variety) becomes the focal point of the study itself. When algebraic equations are defined over the body of complex numbers, tools of Differential Geometry are required in order to analyse the solutions. Reciprocally, Algebraic Geometry is a source of numerous interesting examples of geometric spaces (differential varieties).
One example of algebraic varieties are elliptic curves (whose points form a group), which were a crucial tool for the proof of Fermat's Last Theorem. Algebraic varieties over finite bodies play an important role in Cryptography and Coding Theory, with applications to problems of data storage and transmission.
Although a large part of algebraic geometry is concerned with the treatment of abstract results on varieties, methods for effective computation with particular concrete polynomials have also been developed, which in turn lead to systems of Computational Algebra.
Classical algebraic geometry was largely developed by Italian geometricians in the late 19th and early 20th centuries. The style of this group of mathematicians was highly intuitive and lacked the rigour of modern mathematics. During the 1930s and 1940s, the foundations of the discipline were relaid by Commutative Algebra (principally the study of commutative rings and their ideals), which was developed throughout this period. In the 1950s and 1960s, algebraic geometry was re-established with the use of Sheaf Theory. Towards the end of the 1960s, the idea of a scheme was introduced, together with the tools of homological algebra. After a decade in which rapid advances were made, the field stabilized in the 1970s, when applications in Number Theory and more classical geometrical questions on algebraic varieties and singularities arose. Hodge's Conjecture, one of the so-called “problems of the millennium”, is still open in this field.
The new impulse in recent decades concerns interactions with Theoretical Physics, especially String Theory and D-branes. In particular, Complex Geometry has been the centre of much attention due to its interaction with Differential Geometry, in subjects such as moduli spaces (theory of geometric invariants), and the study of differential equations belonging to mathematical physics, on varieties. A further area of activity is the extension of the theory to non-Commutative Algebra, which in turn leads to non-Commutative Algebraic Geometry.
Researcher at the CSIC Instituto de
Matemáticas y Física Fundamental – IMAFF
It is no longer possible to understand the remarkable advances in neuroscience without the help of mathematics. One hundred years after Santiago Ramón y Cajal was awarded the Nobel Prize for Medicine for his studies on neurones, the synergy between both disciplines has become fundamental for a greater understanding of how the brain works, the fundamental role of neurones in feelings and personality, the dynamics of cortical circuits, the evolution of the nervous system, and the possibilities of computational neuroscience. Among the most noteworthy fruits of this mutual collaboration are; the algorithms employed in the digitalization of images derived from genetic analysis, statistical methods applied to the enormous mass of biological data, and the differential equations used in models for predicting neuronal activity, to mention only some of the most well-known uses.
The high degree of understanding and co-operation between neuroscientists and mathematicians has its basis in fact: the first handle an overwhelming quantity of data, while the second have at their disposal a vast repertory of analytical methods and techniques. In recent years, and starting from this common basis, the development of tools adapted to the challenges posed by neuroscience has become extremely important. Such tools have become indispensable for understanding the operations of neuronal networks and cerebral functions such as perception, language, motor control or consciousness. Their application has also enabled notable progress to be made in the study of diseases such as Parkinson's disease and Alzheimer's syndrome.
The satellite conference to be held in Andorra will be devoted to an analysis of the current state of research in this field, with the purpose of discussing the challenges posed, the advances in the mathematical approach to cerebral modelling, and the fascinating questions posed by the most recent experiments. A further objective of this conference, which forms part of the European project Shaping New Directions in Mathematics for Science and Society, is to bring together experts in the field and mathematicians interested in getting to know and becoming involved in this emerging area of science.
Mathematics are not just a way of racking one's brain, they are also a means of understanding it. Numbers and geometry play a key role in representing this complex organ and identifying all the parts involved in the different cerebral functions. Which part of the brain is used for solving equations or for calculating a square root? In order to reveal the secrets of the grey matter, researchers require accurate maps of the brain, a process that entails transposing a three-dimensional organ into a two-dimensional image; something equivalent to but far more complicated than representing a sphere on a sheet of paper, since a sphere has no crevices or folds.
Such a map is rather special, since on a conventional chart the different parts and depths of the brain would always appear on the same level. Thus one must look to mathematical science if one wishes to represent the different layers of this organ accurately, in particular to topology and to hyperbolic and spherical geometry. In the same way that an explorer penetrates into unknown lands with the aid of a map, so researchers go deeper into the brain in search of its secrets.