Mathematics and computers get together at INRIA

Version française

By Philippe Flajolet

Most often, the general public is only aware of the technological aspects of computer science: networks that are linking together users who employ cameras connected to the Internet to exchange pictures, while huge amounts of textual or visual information are circulating freely. In industry; data of all kinds are being analyzed automatically and new prototypes, perhaps even your next car, are designed, analyzed, optimized and drawn on computer as a matter of routine. As a matter of fact, all these achievements are the result of the evolution of a science in full bloom, computer science. The purpose of the following articles is to display those aspects of computer science that are related to the queen of sciences, mathematics.

Classical Greek science invented logic and it is quite amazing to see that logic is now playing an essential role in computer science. Logic also enters into play in the design of secure software, certified programs or robust cryptographic protocols. Geometry, another offspring of Greek science, is playing a key role in image processing, medical imaging, computer aided design and robotics.

The 17th century, in particular with Newton and Leibniz, saw the birth of mathematical analysis, the science of continua. Modern mathematical analysis is a multifaceted field that shows up in the modeling of all physical phenomena, and is, in this capacity, crucial for the computer processing of the continuum: identification, signal processing, control of electrical or mechanical systems, resource optimization and so on. Analysis is also the foundation of most of probability theory, which makes it possible to optimally manage the randomness that is inherent in all complex environments.

Finally, in the 20th century, discrete concepts came back in full force, which is to say finite objects. In fact, finite objects are the only ones that can be processed in an exact fashion by a computer. Programs handle finite objects following finite rules, the algorithms, to which the science of finite games, combinatorics, applies. And so it is that, at the core of computer systems, programming often means cleverly discretizing. The most recent offshoot of this line of thought is probably symbolic computation, whose goal is precisely to treat in a finite, thus automatable way, some of the most complex models and equations. In some sense, computers are then giving back to the other sciences what they had borrowed earlier.

At the dawn of the third millenium, computer science is gradually assuming its place among all sciences. It is in complex relationship with mathematics in particular, through exchanges of methods and results. The goal of the following articles is to give a brief highlight of this situation.

Philippe Flajolet, researcher at INRIA,
project Algo at Rocquencourt
Tel.: +33 1 39 63 56 26,
Philippe.Flajolet@inria.fr

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