22 May 2006

INDEX:

- The Hour of the Industries

Every evening during the ICM2006, companies at the cutting edge will explain the role played by mathematics in their activities - Interview with Avner Friedman, Director of the Mathematical Biosciences Institute

“Biology is the new frontier of mathematics” - Plenary Lecture: Ronald DeVore

The World of Zeros and Ones - The ICM2006 Section by Section

Topology - Satellite Symposium: Cantabria

Software for Proving Theorems - Applications

A Fascination for Fluids

Every evening during the ICM2006, companies at the cutting edge will explain the role played by mathematics in their activities

Few things define a company better than the rosary of figures making up their balance sheets, but figures have a habit of extending beyond the accounts department to become major players in their own productive activity. Mathematics constitute the tool for shaping many of the developments and innovations carried out in industries at the cutting edge, especially those concerning information and communication technologies.

With this in mind, the ICM2006 has included in its programme a new and original formula; a series of evening sessions devoted to increasing knowledge and encouraging mutual dialogue among a select group of companies and mathematicians attending the congress. These “Technological Evenings” will be held on the 23rd, 24th, 25th, 26th and 28th of August between 18.00 and 19.30, after the normal sessions of the congress. A different industry will feature under the spotlight on each evening (except on the 28th, when a double session will be held). The participants will explain how mathematics are used in their respective companies, and which new tools are required of researchers to meet current and future company needs in terms of innovation. A debate will be held with the audience after each talk to discuss the questions raised.

Discussions are currently in progress with a select group of companies, Vodafone, Intel, MathWorks (devoted to the development of mathematical software) and Indra among them, and with whom negotiations for their participation have already been, or are about to be, concluded. However, in spite of negotiations conducted with two other leading companies, IBM and Telefónica, no agreement has yet been reached.

Although on this occasion the aim has been centered on companies working in data processing and communication, the possibilities are in fact much greater. As stated in the report “Mathematics and Industry”, drawn up by the ICM2006 Executive Committee, society’s needs are becoming increasingly sophisticated, and the response to this demand must inevitably come from the world of mathematics. In the report we may read the following: "Consider the example of air passenger transport. Society not only requires better aircraft, but also an improved organization to optimize fuel consumption and connection times, and to prevent air traffic congestion… These are problems typical of complex systems, control theory, decision-taking theory, etc.. The way they are dealt with cannot but involve mathematics, and furthermore mathematics not in the least trivial, requiring considerable effort in research and development”.

Examples are given of concrete applications to productive sectors of different branches of mathematics, such as Lie Groups in robotics and automotion, statistical methods in quality control, fractals in the management of data traffic, number theory in cryptography, stochastic process techniques in financial management, quasiconformal applications in laser eye surgery, integral geometry in images by magnetic resonance, among many others included in the report “Mathematics: Giving Industry the Edge”, drawn up in 2002 by the Smith Institute for Industrial Mathematics and System Engineering.

Precisely in order to pave the way to this mutually beneficial contact, a scheme was created in 1968 in Oxford under the name of “European Study Group Mathematics With Industry”, which has enabled a fruitful and open dialogue to be conducted between mathematicians and industry. This dialogue has taken shape in a series of regular meetings held in different venues. The last such meeting, the 55th in the series, took place at the beginning of 2006 in Eindhoven (Holland).

Thanks to these mechanisms, and other similar initiatives, many companies have been able to provide concrete proof of their interest in the form of agreements for collaboration with research groups. Motorola, National Air Traffic Services, Satra, Unilever, Acordis, DuPont Electronics, Deutsche Bank, QinetiQ, Nomura, HSBC, Trikon, Goldman Sachs, and the Royal Bank of Scotland are just some of the companies involved in these schemes. The hour of the industries has arrived, and will certainly not be exhausted during the “Technological Evenings” to be held as part of the ICM2006. The future of such companies depends on their level of awareness of the need to draw on and benefit from mathematics. It’s all a question of numbers…

Smith Institute for Industrial mathematics and system engineering

http://www.smithinst.ac.uk/

55th Meeting of the European Study Group Mathematics With Industry

http://www.win.tue.nl/swi2006/

Interview with Avner Friedman, Director of the Mathematical Biosciences Institute

Although Avner Friedman talks about cardiovascular diseases, diabetes or the communication among neurons in the brain, he is in fact a mathematician, the Director of the Mathematical Biosciences Institute (MBI) in Ohio (United States). This centre was created only four years ago following a proposal by Friedman to the National Science Foundation. After devoting most of his career to industrial mathematics - as the Director of the Institute for Mathematics and its Applications (IMA) at the University of Minnesota - he now sees biology as “the new frontier of mathematics”.

**You have said that biology is the future of mathematics. Why?**

Biological problems are very complex. There are lots of data and it’s difficult to get knowledge from them without mathematical modelling.

**Does it have to do with the fact that biology has evolved very quickly in recent years?**

Yes. Biology has evolved very quickly because of technology. Computers, imaging… all this machinery provides a lot of data, and biologist just don’t know how to draw knowledge from them. It is a great opportunity for mathematics. There is no urgency in proving a theorem in pure mathematics: civilisation is going to be around for at least a thousand years. But with disease the public wants to see a solution. And at the same time, history suggests that mathematics itself benefits by looking at new problems. So I think a new mathematics is going to emerge out of all these new problems. Biology is the new frontier of mathematics.

**¿Can you mention some of the problems?**

Genome analysis is one type of problem. But then there are problems related with tissues, there is ecology, how viruses interact with the environment… There is a huge number of new areas and problems.

**What about cancer?**

Cancer is one of the most challenging, because it’s a phenomenon that occurs at different scales. The heart is also very interesting because a lot of people die from heart diseases; it’s the big killer. Imaging is also important, because the better images you can get, the more things you can detect… And there are also more basic questions: how do muscles work? How does the brain process things? And personalised medicine, in which you go to the hospital, they take your DNA and tell you which drug you should use. There are a lot of fascinating questions.

**Do you believe this personalised medicine is really going to happen?**

Oh yes, it’s going to come! There is a lot of money in it.

**How did you enter this field?**

After being at the Institute for Mathematics and its Applications for more than ten years I was looking for something else interesting to do! I’d had enough of industrial mathematics. I was looking for something else and thought that biology would be the thing. I prepared a proposal for the National Science Foundation and I have been learning biology for the last four years.

**Is the ‘marriage’ between maths and biology well accepted by both sides?**

Biologists are realizing more and more the value in mathematics. If just 5 % of biologists believe in it, it is already a lot of population. As for mathematicians, there are still a small number of them interested in biology. Part of our mission is to increase that number, that is why we put half of our budget into post-docs. And our post-docs are getting multiple offers as soon as they finish.

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Personal web of Avner Friedman

http://www.math.ohio-state.edu/~afriedman/

Plenary Lecture: Ronald DeVore

When we download the photographs taken with our digital camera at the last party onto the computer, we usually only see how flattering or unflattering the camera has been to us. But there are others, such as in some well-known science fiction films, who see a series of zeros, ones and a mass of calculations that determine the quality of the images. This is the case of Ronald DeVore, who will give one of the plenary lectures at the International Congress of Mathematicians in Madrid. This North American mathematician says that “for a mathematician, an image is a function, and the values of its pixels are the values of the function”.

Professor DeVore is attempting to find a better numerical solution to problems arising in the field of computation. In his lecture he will provide different examples of “optimal computation”. One of these examples concerns images stored in a computer or sent via Internet. The total number of bits that can be used is generally determined by the device, the CD or DVD, or by the communication channel. If the number of bits is fixed, then the problem resides in providing the most faithful representation of the image, given the number of bits. However, it is not a question of compressing a single image, but rather of repeating this process millions of times every day. In order to do this, different types of images are studied and a method of compression is sought that gives the best results for an entire type.

The optimization of solutions is not solely confined to the question of pixels. A further example provided by professor DeVore is taken from the sphere of communications. Thousands of signals – from a cell phone or a radio, for example – are emitted every second. How can the most relevant signals be identified in the shortest possible time? For instance, when it concerns an emergency call for help? In this case, DeVore compares his work to a game; he says it would be like determining the fewest number of questions one could ask in order to guess any number between 1 and 128 that someone has chosen at random.

Although the ability to see life in terms of zeros and ones is not a gift available to all, if we had a miraculous pair of spectacles that enabled us to do so, we would see this pair of numbers fluttering about all over the place.

Ronald DeVore was born in Detroit in 1941. He graduated in Mathematics in 1964 from the University of Eastern Michigan and gained his doctorate at the State University of Ohio in 1967. He worked at the University of Oakland for eight years, and in 1977 went on to become professor at the University of South Carolina, where in 1999 he founded the Industrial Mathematics Institute (IMI), of which he was director until his retirement in 2005. After retiring, he was made professor emeritus of the University of South Carolina. He has received many prizes in recognition of his work, such as the Humboldt award for research in 2002. In 2001 he was elected a member of the American Academy of Arts and Sciences.

**Lecturer: Ronald DeVore**

Optimal Computation

Date: Tuesday, August 29th, 11:45-12:45

ICM2006 Scientific Programme

/scientificprogram/plenarylectures/

Ronald DeVore personal web page:

http://www.math.sc.edu/~devore/

The ICM2006 Section by Section

**Topology**

The legend of Alexander the Great and the Gordian Knot provide an example of what topology is. During his victorious campaign in Asia Minor, the Macedonian king was passing through a Persian city when in a temple he saw a curious sight; a knot that had been tied by a wise man from the city of Gordium. According to tradition, whoever was able to untie the knot would become emperor of Asia. Alexander the Great cut through the knot with a blow from his sword.

Topology, and knot theory in particular, accounts for why Alexander Magnus needed his sword. Topologists see knots as a lengths of elastic cord whose ends are joined. In topology we are not able to cut or tear the cord, but since it is elastic we can deform or stretch it as much as we want. If we take a knot and apply these continuous transformations, it is necessary always to have the same knot (providing we do not cut it and join it again).

In order to distinguish between different knots, mathematicians use algebraic invariants. These are algebraic objects which do not change when subjected to continuous deformations, providing the knot is not cut or torn. For example, motivated by mathematical physics, V. Jones was awarded the Fields Medal in 1990 for discovering the polynomial that now bears his name. When the Jones polynomial of a knot is non-trivial, topologists know that it cannot be untied without cutting it.

Knots are one-dimensional spaces, but topology also deals with spaces in any dimension and even with more abstract spaces. As with knots, topology can be interpreted as an elastic geometry in which distance has no relevance; it can be deformed but neither cut nor stuck together. The recent solutions to two classical problems in three-dimensional spaces will be explained at this ICM: Poincaré’s Conjecture (first posed in 1904), and the so-called P Property (dating from the 1970s). Techniques of differential geometry are used in the solution to both problems, a further demonstration of the fruitful interaction between the different sections of the ICM.

Joan Portí

Universidad Autónoma de Barcelona

Satellite Symposium: Cantabria

It is a well-known fact that mathematics provide one of the essential foundations for computer science. What is not so well known is that this is also an inverse phenomenon; revolutionary advances in computer science are contributing to the progress of mathematics. Indeed, the appearance of programmes designed for constructing examples, proving theorems and discovering new mathematical phenomena is now one of the most significant developments in the recent history of this discipline.

Research in pure mathematics frequently provides the ímpetus for the development of new algorithms and computational systems arising from co-operation between mathematicians, computer programmers and algorithm designers. The interaction between both disciplines has reached such a high level that it is becoming increasingly difficult to conceive of mathematics without such programmes.

The aim of the talks to be given from September 1st to 3rd in the Cantabrian town of Castro Urdiales is precisely to expound the “state of the art” in this field. Among the subjects to be addressed will be; computational algebra, mathematical visualization, free software for computational algebra, and programmes for the optimization of computational geometry as well as access to mathematical contents and resources via the Internet.

As on the first occasion of this type of conference held in Beijing in 2002, the meeting in Cantabria seeks to bring together not only the designers of such systems, but also mathematicians and algorithm specialists who are interested in the subject as a whole, providing both groups with the opportunity of exchanging ideas on the development of mathematical software.

International Congress of Mathematical Software 2006

Castro Urdiales (Cantabria)

**Person to contact:** Jaime Gutierrez

**e-mail:** jaime.gutierrez@unican.es

**web:** http://www.icms2006.unican.es/

How does a fluid move? Today, thanks to mathematics and computers, this question, which for centuries has fascinated scientists, is yielding new discoveries and important applications in the real world. According to Ana María Mancho, researcher in Applied Mathematics at the *Consejo Superior de Investigaciones Científicas* (CSIC – Higher Council of Scientific Research), finding out how a particle moves in a fluid is useful for answering a broad range of questions; for example, for determining the movement of plankton carried on certain sea currents and deducing for which populations it will provide food; for studying how oil spills from damaged tankers are dispersed in the ocean; or for gauging how brine pumped into the sea from a desalination plant will be dissolved. The process of mixing and transportation of particles suspended in a fluid can be addressed by means of advection equations.

Within the study of fluids, the sphere of microfluids is a science undergoing a rapid process of development. Microfluidic devices are devices smaller than a millimetre containing fluids that move in a laminar way. These microdevices possess many applications, from ink-jet printers to the aerospace industry. One field providing great possibilities is the bio-medical industry and the “lab-on*-*a-chip”, which enable new, mobile, clinical analysis systems to be developed. In order to function correctly, these microdevices require the elements suspended in the fluid to be mixed, a difficult task due to the fact that the dynamics of the fluid is not turbulent. Concepts and ideas belonging to pure mathematics, such as ergodicity, mixing and Linked Twisted Map Theorems have proved to be highly useful for studying the mixing conditions for these elements.

For further information:

Ana María Mancho: a.m.mancho@mat.csic.es

58th Annual Meeting of the Division of Fluid Dynamics (2005):

http://meetings.aps.org/Meeting/DFD05/sessionindex2

Research in Fluid Mechanics (2006)

http://www7.nationalacademies.org/usnctam/Fluid_Mechanics_II.html

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