by Jouko Väänänen and Ulrich Trottenberg

Mathematics saturates everyday life more and more. It is used not only in large applications running on huge computers to predict weather or to calculate parameters for an expensive industrial process or marketing strategy: it has now become ubiquitous in the more mundane aspects of our existence. A good example is the mobile phone. Mobile phone technology depends heavily on such fundamental areas of mathematics as analysis, algebra, and number theory. Introduction to the Special Theme

Mathematics is in principle inexpensive. As the old joke says, a mathematician needs only paper, a pencil, an easy chair and a waste basket. Also, the criterion for success in mathematics is by and large universally accepted. This makes mathematics an attractive 'investment'. Moreover, a mathematical result is valid forever. It may fall out of fashion, or fall outside the current area of application, but even the oldest known mathematical formulae - such as that for solving quadratic equations, known 2400 years ago by Babylonians, Chinese and later the Greeks before being crystallized into its present form in 1100 AD by a Hindu mathematician called Baskhara - are the bread and butter of present-day elementary mathematics. Alas, the downside is that the results are usually not immediately applicable – and therein lies the risk. Who wants to 'invest' in something that may not lead to applications for several hundred years? The good news is that the distance between theory and application is becoming shorter and shorter. Mathematics can be compared to a pyramid. On the top of the pyramid are applications of mathematics to health, weather, movies and mobile phones. However the top of this pyramid would not be so high if its base were not so wide. Only by extending the width of the base can we eventually build the top higher. This special feature of mathematics derives from its internal structure. A good modern application of mathematics can typically draw from differential equations, numerical analysis and linear algebra. These may very well draw from graph theory, group theory and complex analysis. These in turn rest on the firm basis of number theory, topology and geometry. Going deeper and deeper into the roots of the mathematics, one ends up with such cornerstones of logic as model theory and set theory. It is clear that mathematics is heavily used in large industrial projects and in the ever-growing electronic infrastructure that surrounds us. However, mathematics is also increasingly infiltrating smaller scale circles, such as doctors' reception rooms, sailboat design and of course all kinds of portable devices. There has also been a change in the way mathematics penetrates our society. The oldest applications of mathematics were probably in various aspects of measurement, such as measuring area, price, length or time. This has led to tremendously successful mathematical theories of equations, dynamical systems and so on. In today's world, we already know pretty accurately for example the make-up of the human genome, yet we are just taking the first steps in understanding the mathematics behind this incredibly complex structure of three billion DNA base pairs. Our understanding of the mathematics of the whole universe of heavenly bodies, even going back in time to the first second of its existence, is better than our understanding of the mathematics of our own genes and bodies. What is the difference between the hereditary information encoded in DNA and the information we have about the movements of the heavenly bodies? Is it that we have been able to encapsulate the latter into simple equations, but not the former? Or is it perhaps that the latter has a completely different nature than the former, one that makes it susceptible to study in terms of equations, while the former comes from a world governed by chance, and algorithms, a world...