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
