Bulletin -09

Bulletin Nº 9
June 26th, 2006

INDEX:

When is a sphere not a sphere

A New Geometry for Footballs Makes its Debut at the World Cup In Germany

It is common to hear football commentators in Spain using the world “sphere” to refer to the ball used in league matches, but the truth of the matter is that it is virtually impossible to achieve a perfect sphere. An officially approved football is used in every World Cup Competition, although until now the changes introduced in the ball were not what one might call “mathematical”. However, the official ball used for this year’s World Cup in Germany has undergone some changes in its geometry. It consists of a ball made of 14 wing- or butterfly-shaped panels, which according to its designers makes it lighter and more spherical. What follows is a mathematical take on the greatest global sporting event taking place this summer, prompted by the celebration of another “World” event, the ICM2006 International Congress of Mathematicians in Madrid this August, which will be attended by 5,000 mathematicians from all over the globe.  
Mathematically, the soccer ball is an Archimedean Solid called a "truncated icosahedron" - a polygon with 60 vertices and 32 faces, 12 of which are pentagons (5-sided objects) and 20 of which are hexagons (6-sided objects).
The combination of pentagons and hexagons is obtained by truncating an icosahedron, which is a shape occurring in nature and is the one found in fullerenes, molecular structures of carbon with a more or less spherical shape.
Up until the 1970 World Cup in Mexico, professional football casings were made of rectangular strips. After 1970, the footballs used in top flight championships were made of a combination of 12 pentagonal and 20 hexagonal panels, a design that was successively reduced throughout subsequent World Cups. However, the World Cup this year in Germany marks a break in this tradition with an official ball made of 14 wing-shaped or butterfly-shaped panels, similar to those on a tennis ball casing. The designers of this football, called the Adidas +Teamgeist (teamspirit), maintain that they have reduced the surface area of contact, making the ball lighter and more spherical. However, the sphericity of this ball has been called into question.

A web page on soccer and science:
http://www.oceansiderevolution.com/EINSTEIN_1.htm

Interview with Terence Tao

“At age two I tried teaching other kinds to count using number blocks”

Terence Tao (Adelaide, 1975) was just 13 years old when he won the gold medal at the International Mathematical Olympiad. In the two previous editions he had won the bronze and silver medal. He is currently full professor at the University of California in Los Angeles. He has received prestigious prizes, such as the Salem prize in 2000 and the Clay Foundation award in 2003. In this interview he encourages the reader “to play with mathematics”, and comments on the ‘public image’ of mathematics; the way they are portrayed in the movies, for instance: “Very few of them give anything close to an accurate perception of what mathematics is, and what it is like to do it”, says Tao.


You were a very young winner of the International Mathematical Olympiads. How did you get interested in Mathematics? Would you say it was something innate or did it also have to do with a particularly good teacher, for instance?
My parents tell me I was fascinated by numbers even at age two, when I tried teaching other kinds to count using number blocks. I remember as a child being fascinated with the patterns and puzzles of mathematical symbol manipulation. It wasn’t until somewhat later, in college, that I also began to appreciate the meaning and purpose behind mathematics, and how it connects with the real world and with one's own intuition. Actually, I enjoy this deeper level of mathematics now much more than the problem-solving or symbolic aspects.
I think the most important thing for developing an interest in mathematics is to have the ability and the freedom to play with mathematics – to set little challenges for oneself, to devise little games, and so on. Having good mentors was very important for me, because it gave me the chance to discuss these sorts of mathematical recreations; the formal classroom environment is of course best for learning theory and applications, and for appreciating the subject as a whole, but it isn’t a good place to learn how to experiment. Perhaps one character trait which does help is the ability to focus, and perhaps to be a little stubborn. If I learned something in class that I only partly understood, I wasn't satisfied until I was able to work the whole thing out; it would bother me that the explanation wasn't clicking together like it should. So I’d often spend a lot of time on very simple things until I could understand them backwards and forwards, which really helps when one then moves on to more advanced parts of the subject.

How do you look for new problems to work with? And how do you know a particular problem will be really interesting?
I pick up a lot of problems (and collaborators) by talking to other mathematicians. I was perhaps lucky that my original field, harmonic analysis, has so many connections and applications to other areas of mathematics (PDE, applied mathematics, number theory, combinatorics, ergodic theory, etc.), so there was never any shortage of problems to work on. Sometimes I can stumble across an interesting problem by systematically surveying a certain field and then discovering a gap in the literature; for instance, by taking an analogy between two different objects (e.g. two different PDE) and comparing the known positive and negative results for both.
There are some vague and general questions which I would like to pursue (e.g. "How to control the long-time dynamics of evolution equations?"; "What is the best way to separate structure from randomness in combinatorial problems?"). I’m drawn to problems which, while offering some promise of progress in one of these questions, preferably by forcing one to develop a new technique, is also placed in as simple a setting as possible (a "toy model"), where all but one of the difficulties has been "turned off". Of course, it is often not obvious a priori what the difficulties will be, although this seems to be easier to work out with experience. I’m also a great fan of interdisciplinary research - taking ideas and insights from one field and applying them to another. For instance, my work with Ben Green on progressions in the primes came in part from my trying to understand the ideas behind Furstenberg's ergodic theory proof of Szemeredi's theorem, which turned out to be very compatible with the number-theoretic and Fourier-analytic arguments that Ben had in mind for this problem.

Are there such things as 'hot topics' in Mathematics? If so, which would you say are the hot topics now?
I am only really familiar with the areas of mathematics that I actively work on, so I cannot say what are the 'hot' things in other fields. But
in my own fields, it seems that nonlinear geometric PDE is taking off right now (most dramatically in Perelman's use of the Ricci flow to solve
the Poincaré conjecture) - there is an increasingly exciting synthesis between geometric, analytic, topological, dynamical, and algebraic methods here. The combinatorial approach to number theory, in which one develops results on specific sets (such as primes) by first establishing results involving much more arbitrary sets (e.g. sets of integers of positive density), also is rather active right now, and promises to offer a rather different set of tools (including ergodic theory!) to the other methods we currently have in analytic number theory.

What would you say the relationship is between Mathematics and the general public? How should it be ideally?
It probably varies quite a bit from country to country. In the United States, there seems to be a vague consensus among the public that
mathematics is somehow "important" for various high-technology industries, but is also "hard", and best left to experts. So there is support for funding mathematical research, but not much interest in finding out exactly what it is that mathematicians do. (There have been a recent spate of films and other media involving mathematicians, but unfortunately very few of them give anything close to an accurate perception of what mathematics is, and what it is like to do it). I’d like to see mathematics demystified more, and to be made more
accessible to the public, though I am not really sure how to try to achieve these goals.

Plenary Lecture: Arkadi Nemirovski

Mathematics for Making the Right Decisions

“There is a branch of applied mathematics that seeks to respond to one of the basic human desires, making the right decisions”. This is mathematician Arkadi Nemirovski’s poetic reply when asked if he would like the public to know about his field of work. Nemirovski, who will be giving one of the plenary lectures at the next Congress of Mathematicians in Madrid, has developed his career in the field of mathematical programming focused on theory and numerical algorithms for solutions to problems of optimization. This involves taking into account various criteria among which the optimum solution to one of these criterion is sought, given the restriction of the values of the others. In his lecture he will present the advances in a particular group of problems known in mathematics as convex problems.
This mathematician who lives and works in Israel draws a distinction  between “traditional” mathematics and his own particular field, which may surprise many who associate mathematics exclusively with numbers. While early or “classical” mathematics are more descriptive and seek solutions with an analytical form, mathematics related to programming are more “operational”, since their purpose is to find a numerical solution.
As Nemirovski points out, “in mathematics there is no magic wand for solving all the problems posed by real life”.  Only solutions to quantifiable problems can be found in the field of numerical programming. However, they cover a wide range. Arkadi Nemirovski gives examples as diverse as the preparation of cattle feed containing all the necessary nutrients at a minimum cost, the design of parts for cars or electrical circuits in engineering, or the conversion of medical data into 3D image simulation.

Arkadi Nemirovski was born in Moscow in 1947. He graduated in 1970 and obtained his PhD in mathematics from Moscow State University in 1973. Since 1993 he has taught at the Israel Technological Institute and has gained many honours throughout his career. In 1982 he was awarded the Fulkerson Prize from the American Mathematical Society and the Society of Mathematical Programming, his field of specialization. In 1991 he was awarded the Dantzig Prize and in 2003 the John von Neumann Theory Prize.

Speaker: Arkadi Nemirovski
Advances in Convex Optimization: Conic Programming
Monday, August 28th: 10:15-11:15

ICM2006 Scientific Programme
/scientificprogram/plenarylectures/

Arkadi Nemirovski personal web page:
http://iew3.technion.ac.il/Home/Users/Nemirovski.phtml?YF

The ICM2006, Section by Section

The History of Mathematics

The history of mathematics has been present at all the International Congresses of Mathematics (ICMs) since these professional reunions were first held more than one hundred years ago. This year will be no exception, since the International Mathematical Union (IMU) has once again invited historians of mathematics to the ICM2006 in Madrid. Some of them are mathematicians with a special interest in the history of their discipline, others are historians of science with a predilection for the oldest of all the sciences.  
The history of mathematics provides the content for Section 20 within the structure of the ICM 2006. This section will be devoted to historical studies on all the mathematical sciences in all periods and cultures. While none of the plenary lectures deals with historical matters, the congress organizers have scheduled two talks to be given in this section by the distinguished researchers Leo Corry (Israel) and Niccolo Guicciardini (Italy), who will address subjects concerning the work of Hilbert and Newton, respectively.
The broad range of subjects to be covered in the section is also evident in the brief communications and posters that complete the programme, with contributions from all over the world. They include many aspects of Oriental and Western history from a variety of periods, whether from the point of view of the evolution of mathematical theories, the analysis of specific works, or biographical material. A genealogy of the international mathematical community is currently being assembled and will be available on the Internet. The relation between mathematics and philosophy, education and political science will also be examined, and mathematical libraries of particular historical relevance will be described, etc..  
Historians of science have their own specific international congresses organized by the International Union of the History and Philosophy of Science / Division of History of Science and Technology (IUHPS/DHST). Through the International Commission on the History of Mathematics (ICHM), both international organizations collaborate in the promotion of the history of mathematics. In the “other activities” section of the ICM 2006, the ICHM is providing support for a symposium on Ibero-American Mathematics in the 19th and 20th centuries, organized by Sergio Nobre (Brazil), Luis Saraiva (Portugal) and Elena Ausejo (Spain), which will undoubtedly be a fine complement to Section 20 of the congress.  

Luis Español González
University of La Rioja. Departament of Mathematics and Computation

To see the Section 20 programme and summaries of talks in PDF:
http://icm2006.org/web_UC.php?CodiSeccio=20&Format=UCSeccio

Satellite Conference: Catalonia

Saloon Cars like Sports Cars, thanks to Mathematics

The design of stylized aerodynamic lines would be restricted to sports cars if it were not for Computer-Aided Manufacturing or CAM, the innovation enabling these enviable shapes to be transposed to conventional saloon cars. This innovation itself would have been impossible without algebraic geometry, from which the techniques of geometric modelling are derived for the design of industrial dies and moulds.
“Straight lines, curves, dots and surfaces constitute mathematical problems,” explains Laureano González, the co-ordinator of the conference to be held at the beginning of September in Barcelona, where engineers and IT specialists interested in geometric modelling will meet. The event will focus on the search for applications for new algebraic techniques. The challenge in this field consists in integrating shape and aerodynamics in a software package aimed at industrial design. Some of the most promising items on the agenda – apart from those dealing with the automobile sector – will be those devoted to computer-assisted medicine. According to González, “in order to represent a bone on the tomographer’s screen, one has to give it a mathematical form by joining up the luminous dots”.  
Two further symposia will also be held in Catalonia at the same time as this conference: one in the Catalan capital, Barcelona, devoted to mathematical analysis, and the other in the town of Manresa, which will deal with Geometric and Asymptotic Group Theory.  

Algebraic Geometry and Geometric Modelling (AGGM 2006)
IMUB, Barcelona, 4-7 September
Person to contact: Laureano González
e-mail:  laureano.gonzalez@unican.es
web: http://www.imub.ub.es/aggm06/
Geometric and Asymptotic Group Theory with Applications
UPC Manresa, 31 August - 4 September
Person to contact: Enric Ventura
e-mail: enric.ventura@upc.edu  
web: http://www.epsem.upc.edu/~gagta/

Barcelona Analysis Conference
Barcelona, 4-8 September
Person to contact: Javier Soria
e-mail: soria@mat.ub.es
web: http://www.imub.ub.es/bac06/

Applications of Mathematics

Genes and Statistics

As in many other spheres, mathematics has also made its presence felt in medical laboratorios. Specifically, biostatistics is extremely useful for analyzing data in Life Sciences, including Medicine. How is it possible to determine the hereditary factors in an illness, or to discover if a person is genetically predisposed to suffer from a disease? As Wenceslao González Manteiga, professor of Statistics at the University of Santiago de Compostela, explains, mathematics can provide the answers to these questions. To take one simple example: relevant data is gathered on a group of people in what are called micro-matrices, which is then compared with the same information from another group of patients diagnosed with breast cancer. Statistical analysis enables the risks that healthy individuals run of contracting a disease to be calculated in order to improve prevention. Another way of studying the data is to look for which genes or other factors (sex, age, environmental) may be involved in the appearance of this type of cancer. “We observe the data in order to analyze it, “ says González Mantenga, “but the quantity and complexity of such data is enormous. We’re talking about some 30,000 genes in the human body”.
Professor González Mantenga works together with the mathematician Carmen Cadarso on the different applications of biostatistics to medicine. It is not only a question of looking for genetic risk factors; mathematics can also be used to analyze directly some physiological symptoms related with disease, as well as to calculate patients’ life expectancy. Mathematics also play a vital role in the clinical testing of new drugs to assess their effect on a particular group or population.  
Some of the main mathematical tools developed for application to the field of Medicine are: statistics in stochastic modelling, and differential equations in deterministic mathematical modelling.

For further information:

Wenceslao González Manteiga: wenceslao@usc.es

Carmen Cadarso: eicadar@usc.es