15 May 2006

INDEX:

- Are mathematics gender-biased?

Emmy Noether’s Special Lecture at the ICM2006: A Reminder of the Long March to Gender Equality in the World of Numbers - Interview with Efim Zelmanov, Fields medallist

“It will be strange if Perelman doesn’t get a Fields Medal” - Plenary Lecture: Terence Tao

Playing with Numbers - The ICM2006 Section by Section

Number Theory - Satellite Symposium: Toledo

Stronger Bridges and More Profitable Shares - Applications

Banking Risks

Emmy Noether’s Special Lecture at the ICM2006: A Reminder of the

Long March to Gender Equality in the World of Numbers

Only a handful of women figure in the long history of mathematics from antiquity to the beginning of the 20th century. From Hypatia, who taught in Alexandria in the 4th century, to Sofía Kovalevskaya, who reinvented the whole of trigonometry when she was unable to attend university in the second half of the the 19th century, although later she was to become the first female professor of mathematics in history. And in between we have Madame de Chatelet, who translated Newton’s Works into French; Maria Gaetanna Agnesi, author of the treatise on integral and differential calculus, considered by Euler to be the best of its kind; Sophie Germain, who made invaluable contributions to elasticity and Fermat’s Theorem; Ada Lovelace, who developed the first *computer *programmes in history, and Florence Nightingale, an eminent statistician...

Their outstanding contributions are particularly remarkable when we consider the limitations and the adverse conditions they were obliged to confront if they wished to gain access to higher education, occupy the positions they deserved, or participate on an equal footing in the scientific debates of their times. Today, the situation has improved, although equality of treatment and opportunity still remains a distant goal.

According to Edith Padrón, professor of Geometry and Topology at the University of La Laguna and president of the “Women and Mathematics” Comission of the *Real Sociedad* *Matemática Española* (RSME – Spanish Royal Society of Mathematics), a good example of how far there is still left to go is the fact that no woman has yet been awarded a Fields Medal, the highest distinction in the discipline, and the number of women invited to speak at the successive ICMs is still extremely low. Efforts have been made to redress the situation, such as the introduction in 1994 of a special lecture named after Emmy Noether, in memory of the eminent early 20th century mathematician who achieved wide recognition and was the star of the World Congress of Mathematicians in Zurich in 1932.

The comission over which Edith Padrón presides is currently carrying out a study of the situation in Spain over the period 1990-2003, which has already yielded findings that speak for themselves: “Gender distribution in terms of students enrolling at and graduating from university is almost on a par, a situation that is also reflected in applications for pre-doctoral grants. Nevertheless, after this link in the chain of research careers, women are in a minority”, explains Padrón. Of all university lecturers, only 31 out of every 100 are women, and when it comes to professors, the figure hardly reaches 8%.

The Emmy Noether lectures were created in 1980 and are given every year. So far, 27 have been delivered, as well as those included in the ICM. All have been given by women mathematicians carrying out important work in their respective fields. These lectures arose from the activities of the Association for Women in Mathematics (AWM), founded in 1971 and with a current membership of some 4.100.

With regard to the ICM2006, the Emmy Noether Special Lecture will be given by the French mathematician Yvonne Choquet Bruhat (Lille, 1923), who has carried out mathematical research at different centres and universities in France, as well as at the prestigious Institute of Advanced Studies at Princeton, U.S.A. In 1979 she became the first woman to be elected to the French Academy of Sciences in the three-hundred year history of this institution. Her talk will deal with “Mathematical Problems of General Relativity”, one of the subjects in which Emmy Noether herself figured prominently. This will be Choquet Bruhat’s second Noether Lecture, her first having been given in 1986.

For further information:

http://www.awm-math.org/noetherlectures.html

http://www.awm-math.org/

http://www.agnesscott.edu/lriddle/women/prizes.htm

http://www.rsme.es/comis/mujmat/

**“It would be very unlikely for Perelman not to be awared the Fields Medal at ICM2006”**

Russian mathematician Efim Zelmanov is currently a professor of mathematics at the University of California in San Diego. He received a Fields Medal in 1994 for solving the Restricted Burnside Problem, a fundamental algebraic conjecture that mathematicians specializing in group theory had been working on throughout the 20th century. The Fields Medal is considered the world's most prestigious prize for mathematics and is often compared to the Nobel Prize.

**How do you work, by yourself, or in collaboration?**

Both, actually. Discussions with people are very important.

**Are you referring to colleagues in the same department or those close to you?**

Not necessarily. But it is true that nothing can replace personal communication. That is why the International Congress of Mathematicians is so important to us.

**What’s impressive is the fact that so many mathematicians can gather at a Congress covering all fields of Mathematics. Do you understand each other? How wide is the frontier between different areas of mathematics?**

In recent years the volume of mathematical publications has grown enormously, and that inevitably leads to specialisation. But the best mathematical works are cross-disciplinary; they involve many ideas. And every effort should be made to keep mathematics unified. The Congress is very important for this. If you look at the geniuses of the past, which area did they belong to? They belonged to mathematics.

**Who are your heroes in mathematics? **

Uh! The whole pantheon! But you also asked about the Congress being so large and diverse. I hope the Organising Committee did a good job at selecting speakers who can communicate. Not every great mathematician can communicate.

**So numbers and equations are not enough for mathematicians.**

Well, I would say that a lot of people will be curious to see the best mathematicians in the world, but it will help a lot if they also have something to say.

**Apparently, the ‘hot topic’ in ICM2006 will be Grisha Perelman’s proof of the Poincaré Conjecture. What do you think of it? **

Well, he has done something very, very important; his work really stands out.

**What about the process of evaluating his work?**

The work is evaluated by experts and we can only wait to hear what they say. I have yet to hear a formal announcement that everything has been checked up to the last page, but what has already been checked is enormous.

**What are your bets for Fields medallists this ICM?**

To tell the truth, I really don’t know. I can think of two people, although I shouldn’t give names because I will offend other people. But it will be strange if Perelman doesn’t get it.

**In your case, were you surprised when you heard about your medal?**

Yes, very surprised. I didn’t believe it at first. I got a mail from professor Jacob Palis’ secretary in Rio de Janeiro. And well, nowadays you can receive an email from the Pope… It could be a joke, so I called him to get a confirmation. And his secretary said ‘yes’. I was asked not to tell people; it was a secret.

**Did you manage to keep it from your wife?**

No, of course not!

**At what time of the year did you find out?**

I got the news in May.

**So the Fields medallists of the ICM2006 may know already or must be about to know.**

Yes.

More information:

Efim Zelmanov at ‘The mathematics genealogy project”:

http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=52166

**Playing with Numbers**

"Is it useful?": a familiar question when speaking about scientific results. However, not every scientist has an application in mind when carrying out research. Sometimes mathematics are simply a challenge to the mind, and for some mathematicians this kind of problem is a paradise in which to work. This is true of Terence Tao, who will give a plenary lecture at the 2006 International Congress of Mathematicians. He is the first to admit that "Prime numbers are a beautiful subject, although they may not have any concrete applications for non-mathematicians".

Professor Tao and the British mathematician Ben Green have proved a result that is easy to understand for a secondary school student, since its elements consist of arithmetic progressions and prime numbers. The challenge first arose some years ago. In 1939, the Dutch mathematician Johannes van der Corput proved the existence of infinite arithmetic progressions of three prime numbers; for example, 3, 5 and 7, or 31, 37, 43. Tao and Green have not only proved the existence of infinite arithmetic progressions of four prime numbers; they have even generalized the result for infinite arithmetic progressions of prime numbers of all lengths.

Some readers may feel like looking for these prime progressions as if they belonged to a sudoku-style numerical puzzle. But Tao's result is more than a game; it constitutes a breakthrough in number theory. Furthermore, as Tao himself adds, "the techniques we are developing may well be useful in future application of greater practical importance; it's hard to predict these things ".

Terence Tao was born in Adelaide (Australia) in 1975. He was the youngest participant in the International Mathematical Olympiad. He participated in the 1986, 1987 and 1988 editions, where he won the bronze, silver and gold medals, respectively. He was just 13 years old when he won the last. In 1991 he graduated in mathematics at the University of Flinders, and in 1996, before his twentieth birthday, he gained his PhD at Princeton University. Since 1996 he has been associated with the UCLA, where he became full professor in 2000. He has received prestigious prizes, such as the Salem prize in 2000, and the Clay Foundation award in 2003 for his work on Kakeya`s Conjecture and wave maps.

**Speaker: **Terence Tao

Long Arithmetic Progressions in the Primes

**Date**: Wednesday, 23 of August, 09:00-10:00

ICM2006 scientific programme

/scientificprogram/plenarylectures/

Terence Tao’s web site

http://www.math.ucla.edu/~tao/

**Number Theory**

“Mathematics is the queen of sciences and number theory the queen of mathematics”. This statement by K. F. Gauss, himself considered the prince of mathematics, may explain why number theory will be represented at the ICM in Madrid by three plenary speakers; Henryk Iwaniek, Terence Tao and Kazuya Kato.

Number theory deals with problems connected with integers both in essence and in origin. Typical examples are how prime numbers are distributed or how many integer solutions exist for the equation x127+y127=z127.

This type of question cannot normally be answered by working only with integers. One is bound immediately to work with rational numbers, with complex numbers, and also with more advanced tools: diophantic approximations, representations of Galois groups, L-functions, modular forms… All of this will be dealt with in Section 3 of the ICM: “Number Theory”.

Before continuing, it is necessary to mention that in recent years the availability of powerful computers has increased the importance of computational aspects of number theory and its applications; for example, in cryptology and in the theory of error-correcting codes. The discovery of the AKS algorithm for determining whether a number is prime or not is the most celebrated computational breakthrough to have been made in recent years.

Riemann’s Hypothesis, which describes with great precision how prime numbers are distributed, is one of the most important problems, not only with regard to number theory but in mathematics as a whole. It is in fact one of the “Millennium Problems” whose solution merits a prize of one million dollars from the Clay Foundation.

While the definitive solution to this problem is still awaited, spectacular results have been obtained in recent years with regard to prime numbers, the most outstanding being B. Green and T. Tao’s proof that the primes contain arithmetical progressions of any length, a result that was believed to be beyond the reach of mathematics at this time.

Our second example concerns Fermat’s Theorem, successfully proved by A. Wiles in 1994: the equation xn+yn=zn has no solutions in positive integers if n>2. This is the paradigm for equations in integers dealt with in number theory, not only because of its legendary status but also for the mathematical wealth it has released. As an example we might mention Serre’s Conjecture, which in cases relevant to Fermat’s last theorem has been recently proved by C. Khare and J. P. Wintenberger, and independently by L. Dieulefait. Dieulefait, from the University of Barcelona, has just published a proof of Serre’s Conjecture practically without restrictions.

A further recent result has a history going back to 1770, when Lagrange demonstrated that every natural number can be expressed as the sum of four squares. The young Indian mathematician M**anjul Bhargava****has surprised the mathematical community by finding all the expressions for any specific set of integers. **

*Javier Cilleruelo Mateo y Adolfo Quirós Gracián *

*Autonmous University of Madrid*

**Stronger Bridges and More Profitable Shares**

That the shortest distance between two points is a straight line constitutes a classical example of the solutions offered by the calculus of variations. This is the point of departure for one of the oldest disciplines in mathematical analysis, which is capable of solving challenging questions posed by both engineering and economics.

The usefulness of this tool can be appreciated every time a tunnel connecting two cities has to be built; it is necessary to know the variations in the subsoil in order to decide on the best direction to drill. Perhaps one of the best-known contributions in this field is the calculation of the routes for the space vehicles sent to the Moon during the 1960s, with the purpose of saving as much fuel as possible. Its suitability for determining the optimum design of materials and structures has also been amply demonstrated. With the help of this tool it is possible to develop a structure for a bridge providing the greatest rigidity; the best distribution for parts of an elastic material guaranteeing maximum elasticity; and the optimum path for energy through a network.

The ability to find the optimum state of a process or a material, or to analyze its inexistence in those cases where the problem has no solution, is of enormous use in both the physical and social sciences. The truth of this is such that applications for the calculus of variations have been found even in urban planning, where it is necessary to determine the best distribution of the industrial areas in and around a city in terms of the distances workers and suppliers have to travel in order to reach these places. The calculus of variations also plays a vital role in economics; for example, when it comes to choosing the best investment strategies, or how to determine a priori the best price for a company’s shares (an optimum value obtained on the basis of criteria such as previous market performance and other conditioning factors).

Recognition of the current importance of variational calculus is reflected by the many international prizes awarded to some of the most outstanding specialists in the discipline (the first Fields Medal, for instance, was awarded to Jesse Douglas for his valuable contributions in this field).

The analysis of the new challenges thrown up in this area of mathematics, together with its emerging applications, is the aim of this symposium to be held between August 16th and 19th in Toledo.

Trends and Challenges in the Calculus of Variations

and its Applications

Venue: Convento Madre de Dios (Toledo)

**Person to contact:** José Carlos Bellido

**e-mail: ****JoseCarlos.Bellido@uclm.es**

web**:**** ****http://matematicas.uclm.es/toledo2006/ **

**Banking Risks**

It seems obvious to state that mathematics play a fundamental role in the financial world. What is perhaps not so obvious is the importance of numbers and statistics in the prevention of large-scale economic losses that banks can incur due to unforeseen circumstances. Those who believe that bank vaults with security systems are enough to ensure the protection of money could not be more mistaken. As Santiago Carrillo, professor at the *Estadística e Investigación Operativa* of the Autonomous University of Madrid (UAM), explains, according to a study carried out by the Committee of Basle into cases of losses involving more than 10,000 euros due to simple errors, unexpected damage or fraud in banking operations, 81 out of the 89 banks consulted admitted knowledge of more than 41,000 such cases, involving sums amounting to approximately 8,000 million euros. Errors due to malfunctioning cash-dispensing machines, an extra zero added by a slip of the finger, or a hacker misappropriating customers’ secret codes… These are just some of the incidents that cause banks to lose large amounts of money, although they could be much worse, since other risks such as sudden fluctuations on the stock market or the non-payment of bonds also exist. “Perhaps it doesn’t seem so serious if a few customers fail to meet their mortgage repayments”, says Carrillo, “but what happens if a company like Enron collapses?”

So where do mathematics fit into the picture? The Basle agreements 1 and 2 require banks to keep part of their capital in reserve to cover 99.9% of such losses. This obliges the banks to estimate their distribution of possible losses in such a way that they have a minimum amount of capital on hold, but enough in cases of necessity to avoid problems so serious that they could even lead to bankruptcy. “Calculating the amount of this statutory capital is one of the new spheres of application for mathematics”, explains UAM’s Carrillo, and it is for this reason that Probability Calculus, Applied Statistics and numerical methods are vital for estimating these distributions. Furthermore, as in other financial applications of mathematics, such as Options Calculus, Stochastic Calculus and the Montecarlo Simulation are also used.

For further information:

Santiago Carrillo: santiago.carrillo@uam.es

Bank for International Settlements (BIS): http://www.bis.org

2004 © Copyright by Asociación International Congress of Mathematicians, ICM
2006 Madrid. All Rights Reserved.