Professor Marja Makarow, Chief Executive, European Science Foundation.

Mathematics Everywhere

The developed world is full of modern technology that we take for granted. Mobile phones, internet, credit cards and CD and DVD discs are only a few examples of innovations that have revolutionized everyday life in the past thirty years. Common to all these is that their functioning depends heavily on mathematics. Another thing to note is that in all these cases, the mathematics was not invented for the sake of the technological innovations - it had already been developed as pure mathematics and lay ready to be applied when the time was ripe.

In Internet and mobile phone telecommunication the message is encoded and passed from one computer to another or from one phone to another. Errors always occur and may lead to the message becoming completely obscured. Error detection and self-correction is therefore essential in all telecommunication. Modern automatic error correction is based on deep mathematics such as Galois theory developed in the early 19th century by Evariste Galois. Telecommunication, as we know it, would not be possible without Galois theory. Error correction is equally important for the proper functioning of CDs and DVDs.

Digital security is an important issue in today's society. We want to be confident that our phone calls are not eavesdropped, that we can safely shop on the internet without our credit card information being intercepted and that bank secrecy and safety is not jeopardized. The solution to the security problem is efficient encryption. Cryptography for this purpose uses modular arithmetic (a part of number theory). A standard method in public-key cryptography relies on the fact that no fast algorithm is known for factorizing a very large number. Therefore the public key used for encryption could be based on the product of two very large prime numbers, whereas the secret private code needed for decryption would be based on the prime factors themselves.

The most popular way of retrieving information from the internet is certainly to use Google. It is amazing and seems like magic that in most cases the first hit contains the information one was looking for. Again, the reason for the success is mathematics (in this case linear algebra) and an efficient algorithm. Kurt Bryan and Tanya Leise have written an article on the linear algebra behind Google aptly entitled "The 25 billion dollar eigenvector". The approximate market value of Google was indeed 25 billion USD when the company went public in 2004. The article is freely available at
http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf.

In this article I have given three examples of modern technologies that we use daily and that could not have been invented without mathematics. Good mathematics is usually created for its own sake and it will eventually find industrial applications. A general trend is that the time span between the mathematical invention and the application becomes shorter and shorter. Apollonius of Perga investigated the conic sections around the year 200 BC and Kepler used this theory in the formulation of his laws on planetary motion some 1800 years later. Galois theory had to wait only about 150 years before it found its applications in telecommunication and very recent results in number theory are used in cryptography. It is very likely that contemporary research in mathematics will influence our daily lives in the very near future perhaps in an unexpected way.

The European Science Foundation is preparing a Forward Look on Mathematical Modelling, and has received a proposal from the CNRS, France, to develop one on Mathematics and Industry. Forward Looks serve as strategic instruments, where the best researchers describe the status quo of their scientific domain, envision its evolution and impact in the next 5-10 years, and predict the needs for training, infrastructure and funding. The Forward Looks provide the national research funding and performing organizations as well as the European Commission a Europe-wide analysis to facilitate their decision making on targeting research funds.

Marja Makarow