9th May 2006
Articles on InfoICM2006 are written by the ICM2006 Press Office (Ignacio Fernández Bayo, Mónica G. Salomone, Clemente Álvarez, Pablo Francescutti and Laura Sánchez), and by Spanish mathematicians (their names on each article). InfoICM2006 is translated into English by Jeff Palmer.
Mathematics are becoming more and more crucial, and not only in physics and engineering or for financial success. The life sciences are increasingly coming to depend on mathematics, whether it be for understanding the effects of climate change on an ecosystem, for interpreting the information contained in the genome, or for designing new therapy in the treatment of cancer. This is one of the conclusions to emerge from “Mathematics for the 21st Century”, the symposium that has just been held at the Ramón Areces Foundation in Madrid, which among other speakers was attended by two winners of the Fields Medals, - the highest award in mathematics – and the president of the International Mathematics Union.
“Nanotechnology; biomedicine, with all the new information from genomics and proteomics; the neurosciences; the study of biodiversity and environmental protection in general… They are the emerging sciences in the 21st century, and mathematics has a key role to play in their rise to prominence. Indeed, there are many who consider mathematics to be an emerging technology in itself”. These are the words of Manuel de León, chairperson of the 2006 International Congress of Mathematicians (ICM2006, Madrid, August 2006) and organizer of the symposium at the Ramón Areces Foundation.
Some of the areas with the greatest “growth potential” for mathematics were dealt with during the symposium. Avner Friedman, from the University of Ohio and director of the Mathematical Institute of Biosciences, created in 2002, status that “the future of mathematics is in biology”, and goes on to explain: “Biology is replete with very complicated problems and has advanced very rapidly in recent years. This has given rise to such a huge quantity of data that it’s difficult to know what to do with it all. That’s why mathematics is so important; it is very difficult to extract information from biological data without the help of mathematics. This is a great opportunity for us”. For mathematicians, however, this is a very new field. Thus one of the aims of Friedman’s newly created institute, financed by public funds, is precisely to promote a “marriage of convenience” between these two sciences.
Furthermore, for Friedman this is an “urgent” challenge, since in fact there are many lives at stake. In his talk at the symposium he provided various examples of mathematical-biological models: for describing how part of a neurone functions; for finding the best possible treatment for a certain type of brain tumour. The list of applications is long. For Friedman, mathematics hold the key to questions concerning the struggle against diabetes and obesity, AIDS and cardiovascular diseases.
“Mathematics are today what the microscope was a century ago”
Jordi Bascompte, researcher with the Consejo Superior de Investigaciones Científicas (CSIC – Higher Council of Scientific Research) at the Biological Station of Doñana, agrees with Friedman about the importance of mathematics when it comes to making sense out of the swelling tide of biological data. “Until now, biology has been an eminently descriptive science, but we have not been particularly good at building a theoretical structure capable of helping us to make predictions. For example, we know how to count species and how populations fluctuate, but we don’t know to what extent a fishing ground can be exploited before all the resources are exhausted, or how logging can be carried out in a tropical forest without undermining the entire ecosystem. These are questions that cannot be tackled descriptively, because it would take years, and anyway it would be too late”, says Bascompte. “We need to look for short cuts”.
Mathematics are already providing these short cuts. One of them, a concept “which is very important and has already brought about a paradigm change” for ecologists, explains Bascompte, is the ‘extinction threshold’. “Formerly it was thought that if 20% of a habitat was destroyed, 80% would still remain. But today we know that there is a threshold [of habitat destruction], which when crossed necessarily results in the extinction of a species. These thresholds are critical points, not unlike pushing a glass across a tabletop. Everything seems fine … until the glass falls”. These same critical points exist, for example, when pollutants are emptied into a lake; after a certain quantity a ‘point of no return’ is reached. After this point, the ecosystem cannot be recovered, no matter how much cleaning is carried out or how many nutrients are added.
It is also thanks to mathematics that ecologists can model the real loss of biodiversity – for many, it is currently undergoing its ‘sixth extinction’, caused by human action and comparable to what brought about the disappearance of the earthbound dinosaurs. Predictions are not optimistic: “It has been determined that the main effects will not be felt for another few centuries”, says Bascompte. An idea that leads to the concept of ‘ecological ghosts”, species that still exist, but whose days are numbered.
“We have some interesting problems”, continues Bascompte, “and mathematicians have the tools. Mathematics today are what the microscope was a century ago: they enable us to see wonderful things, to go deeper and discover general patterns”.
Modelling the Smallest
The nanosciences study what happens at scales of millionths of millimetres and are considered the emerging science par excellence. They were represented at the symposium by Jesús Ildefonso Díaz Díaz of the Complutense University of Madrid. This mathematician has managed to model a phenomenon that occurs in nano- technology; the drastic change of properties undergone by a system according to the scale at which it is manipulated. Nanotechnologists know, for example, that gold becomes a conductor when it is fragmented into nanoparticles. Díaz has developed a mathematical model that describes how adsorption, the process by which molecules are captured and retained on the surface of a material, varies according to the scale of the particles involved. And this model enables predictions to be made about how to improve the process.
Alain Connes, Fields Medal award winner in1982, carried out a mathematical description of the ‘standard model’, which describes the first moments of the life of the Universe – the ‘Big Bang’. The standard model is a development of physics which mathematicians, as Connes himself explained, have yet to study very deeply.
Other subjects addressed were finance – how to predict losses and evaluate risk – mathematical aspects of image processing; the ‘limits’ of artificial intelligence; and the efficiency of combustion processes, which are responsible for eighty percent of the energy currently consumed by humanity.
Protecting the Unity of Mathematics
The Englishman John Ball, president of the International Mathematical Union, and Efim Zelmanov, Fields Medal winner, spoke about the importance of the International Congress to be held in Madrid in August. In Ball’s opinion “it will be a unique opportunity of listening to the best mathematicians in the world addressing a broad range of topics”, while for Zelmanov, it will help to “maintain the unity of mathematics. Although in recent years the volume of mathematical publications has grown greatly, the best mathematical work is interdisciplinary and covers many fields. That’s why it’s essential to do everything possible to keep mathematics unified, and why this Congress is so important”.
(go to “Actos programados en el primer semestre” and then to “Matemáticas para el siglo XXI”).
Francisco Santos Leal (Valladolid, 1968) is one of the Spanish mathematicians invited to give a lecture in the different sections of the ICM2006. Santos Leal is professor of Geometry and Topology at the University of Cantabria. His lecture will deal with “Triangulations of polytopes”, a subject in which he is an expert of international renown. Although the title of his talk may not mean very much to the uninitiated, the matter becomes clearer if we know that a polytope is a geometrical shape equivalent to, although in any number of dimensions, the polygon in the plane (2-D) and the polygon in space (3-D).
What are triangulations of polytopes?
Triangulating a polygon consists in dividing it into triangles, and in the case of a polyhedron it means dividing it into tetrahedrons. In the same way, triangulating a polytope is dividing it simplices, equivalent in whatever dimension is concerned.
What’s your most relevant contribution in this field?
In 2000 I published a description of a certain triangulation of a polytope with six dimensions and more than 300 vertices. This triangulation possessed some surprising properties – properties which somebody had conjectured could not possibly belong to any triangulation. Two years later I achieved a triangulation in five dimensions with only thirty vertices.
Why in five and six dimensions?
The reason is quite simply because I wasn’t able to do it with fewer. As strange as it might seem, it’s usually easier to do things with more rather than fewer dimensions; with six dimensions instead of the three we’re used to in daily life. To draw a comparison, a gymnast’s job is more complicated if, in addition to the difficulty of performing a pirouette, she has to do it in a confined space. My frustration was not being able to make the corresponding triangulation in a space of three dimensions.
What applications do these researches have?
The main motivation for most mathematicians is purely aesthetic, but the triangulation of polytopes has applications in topography, the numerical solution of equations with several variables, and in geometric design… In particular, algorithms are used in certain problems in computational geometry for optimal triangulation by means of combinatorics. Furthermore, it has other surprising applications in pure mathematics. In particular, part of my work has been devoted to exploring the connections with algebraic geometry and combinatorial topology, but this is such an abstract question that it would be difficult for the average reader to grasp it. I also intend to describe these applications during my lecture at the ICM.
You mentioned combinatorics, which is the name of the section in which you are participating. What does it consist of?
Combinatorics is essentially the art of counting. Or rather, of knowing how many objects there are without having to count them. For instance, we can determine that there are 13,983,816 possible results in the lottery by saying that this is the ‘combinatorial number 49 over 6’. And this can also be applied in a geometrical context.
Through the relations between the typical elements of geometrical shapes, such as vertices. One of the best-known theorems of geometrical combinatorics is Euler’s formula, which is studied by everybody at secondary school, and which states that in every polyhedron the sum of the number of vertices and faces is equal to the number of edges plus two. Another example is “Kepler’s Conjecture”, which took almost 400 years to prove, until Thomas Hales came up with the proof a few years ago. This conjecture states that the best way of packing equal spheres is by compact cubic packing, which is the method normally used by grocers to display their oranges. Although the result may be obvious, proving it was a very complex process requiring an enormous amount of calculations by computer.
Nuclear reactions, living organisms, sociological studies… For many years physicists have used predictions to describe the behavior of these systems. Only recently have mathematical proofs been found for these predictions. In his plenary lecture at the ICM2006, Oded Schramm will discuss various open problems, some of which have been selected to illustrate achievements in physics that have yet to be understood mathematically.
Schramm will speak about random systems in the plane, and about how to understand their behaviour. One example of a random system is the so-called percolation. Percolation analyses the movement of a fluid in a porous material. For oil and natural gas, for instance, theoretical results in this area have helped to improve productivity in the exploitation of both these resources.
Although some of the systems studied by professor Schramm are well defined mathematically, he points out that "often arguments provided by physics cannot be translated into mathematical arguments that make sense".
Oded Schramm was born in Jerusalem in 1961. After graduating in mathematics at the Hebrew University, he moved to the United States to do his doctorate at Princeton University. Since 1999 he has been employed at Microsoft Research. Schramm has received such important prizes as the Clay Research Award in 2002, or the Henri Poincaré Prize in 2003.
Speaker: Oded Schramm
Title: Random, Conformally Invariant Scaling Limits in 2 Dimensions
Date: Wednesday, 30 of August, 11:45-12:45
“The order of factors does not alter the product” is an expression we hear quite frequently. Nevertheless, many phenomena in life are “non-commutative”; that is, their result depends on the order in which they appear. Imagine for example an aeroplane approaching an airport that receives an instruction to fly ten miles to the north (manoeuvre A) and then turn 180º around the control tower (manoeuvre B). The final position of the aircraft after performing either manoeuvres AB or manoeuvres BA would be completely different.
One of the most fruitful mathematical ideas of the 19th century was the realization that this type of “operation” with movement or displacement, and with more complicated physical transformations, showed a formal resemblance with operations we perform with numbers (except commutativity), which enables them to be studied systematically. It also deals with “non-discrete” phenomena (rotations, transferings, contractions, relativist transformations), which lead us to the use of not only algebraic but also geometrical techniques.
This theory of “continuous transformation groups” was initiated by the Norwegian mathematician Sophus Lie (1842-1899) and is at present an essential tool both in robotics (control theory) and theoretical physics (quantum mechanics). This is the theory which in pure mathematics has enabled all the non-Euclidian geometries to be unified.
Now providing the description of a physical system and its movements or transformations is quite a complex business (imagine a top that spins, lurches and moves). However, it’s possible to codify such a system in a few algebraic equations and variables, in a way analogous to plotting the orbit of a planet giving only its velocity and direction at each point instead of its complete trajectory. This algebraic object that codifies and simplifies the description of a continuous transformation group is what we call its “Lie algebra”.
At present, the theory of Lie algebras is well established and is used in such varied fields as the study of differential equations (mathematical models) and nuclear spectroscopy. What’s more, the so-called Lie superalgebras, related to particle physics, provide a highly active field of research for physicists and mathematicians.
Enrique Macias Virgós
Universidad of Santiago de Compostela
The Digital Era has drastically transformed the way in which researchers seek, produce, publish and disseminate their scientific work. This phenomenon involves mathematicians in a twofold way: 1) as web users, just like their colleagues in other disciplines; and 2) as specialists with the responsibility of providing the Information Sciences and the technologies of Computer Science with the paradigms and tools required for the design of digital libraries, electronic publications, Internet search engines, scientific resource catalogues and redigitalization (the digitalization of texts published originally on paper), to mention just some of the aspects of utmost interest for the dissemination of knowledge.
These are just some of the subjects that will be dealt with on the extensive agenda of the meeting to be held in Aveiro (Portugal) in August. It is one of the ten satellite symposia that will take place in Portugal from June onwards, during which subjects ranging from mathematical solutions to problems in telecommunication, turbulence and hydrodynamics, and the Lie super-algebra will addressed.
XVth Oporto Meeting on Geometry, Topology and Physics
Oporto, 20-23 July
Persons to contact: Miguel Costa / Roger Picken
The Summer School "Statistical Tools in Knowledge Building"
Coimbra, 23-29 July
Person to contact: Dinis Pestana
”New Trends in Viscosity Solutions and Nonlinear PDE”
Lisboa, 24-28 July
Person to contact: Diogo Gomes
International Summer School and Workshop on Operator Algebras, Operator Theory and Applications
ITS-Lisboa (Portugal),1-5 September
Person to contact: Amélia Bastos
Geometric Aspects of Integrable Systems
University of Coimbra,17 -19 July
Person to contact: Joana Nunes da Costa
Summer School on Calculus of Variations and Applications, Ponta Delgada (Azores)
Person to contact: Margarida Baía
The film: “Lord of the Rings”; part three: “Return of the King”. In one of the last scenes, Gollum falls into a sea of lava clutching the ring in his hand. He sinks slowly beneath the surface of the viscous, incandescent liquid holding the treasured ring aloft. The scene lasts only a few seconds, but it took at least two people a month’s work to make, and many, many equations to simulate the movement of the lava by means of computer. Curiously, the firm responsible for this scene is Spanish; “Next Limit”, a company of engineers devoted to research and development in many fields, not only in cinema. As the founder and general manager of the company, Víctor González, explains, to reproduce lava realistically by computer they used a tool they have developed themselves, RealFlow, which is based on the Langrangian mathematical method or fluid simulation by particles. This entails calculating the movement of a particle in a particular fluid in terms of its properties and the interaction with the rest of its surroundings. In this way they can reproduce the behaviour of a fluid such as lava, a material whose viscosity changes when hot to flow more quickly and which slows down as it cools. The complexity of the simulation is governed by the number of particles; before developing the scene for that film, the highest number of particles that Next Limit had worked with was 500,000. However, for the scenes in “Lord of the Rings” they had to carry out calculations for 2,000,000 particles. After reproducing the real movement of lava on a computer by means of dots, the next step involved joining the dots together in order to create a uniform volume, and finally to give this volume the definite appearance of volcanic material, with all its colours and reflections. This is the magic of many of the special effects that are the delight of audiences in today’s movie theatres.
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