The Topic: Real Algebraic and Analytic Geometry

Real algebraic and analytic geometry (RAAG) studies geometrical objects arising in connection with the mathematical modelling of the "real world", i.e., the mathematical description of physical objects as well as of natural and man-made processes in science and technology. The geometric models are based on the real numbers, geometrically a line, and more and more complicated objects built up from this most elementary one, e.g., planes, spaces of three or more dimensions, parts of these spaces, as well as abstract algebraic structures that are needed to study the geometry. These real methods are appropriate whenever phenomena are studied that depend on continuous parameters like time or position in space.

RAAG helps to achieve a deeper understanding both of the natural world we live in and of the technological world we are shaping to an ever increasing extent. As computing permeates our society at every level, so do mathematical methods since they form the basis of most computer applications. The fundamental importance of computers in our society should be matched by an equally deep understanding of the foundations of computing, among them the methods of geometric modelling. Thus, on the one hand RAAG is an important part of the human quest for fundamental knowledge. On the other hand, the very origins of RAAG, i.e., the modelling of reality, assign to it an important role in the solution of practical mathematical problems. These two aspects of RAAG are not contradictory or mutually exclusive – rather, they supplement each other. The applications give meaning and direction to the structural studies, which, conversely, help to recognize both the limits and the possibilities of practical methods. Thus it is impossible to classify RAAG either as pure mathematics or as applied mathematics – it is a combination of both.

RAAG has developed into a major branch of mathematics only recently. Eminent mathematicians of the past were always interested in phenomena connected with the use of the real numbers. But at a first glance the behavior of real geometric objects appears to be irregular and unpredictable compared with geometric objects arising from the complex numbers. As a consequence, real geometry was overshadowed by complex geometry for a long time. Today real geometry is recognized as being equally important as complex geometry, but with quite different properties and phenomena. The real numbers have an order relation, i.e., numbers can be compared by size. By contrast, the complex numbers cannot be ordered – so the tools from complex geometry do not deal with order relations and, therefore, are not always well adapted to the real case. RAAG came into its own when mathematicians recognized and addressed the need for tools tailored specifically for the real situation. Any genuinely real tools make explicit or implicit use of order relations or some variants thereof. We have witnessed a spectacular development of the real methods in the past two decades. But considering the multitude of phenomena to be modelled and studied there will always be a demand on RAAG for new methods and also a need for deeper understanding.