The work of the Konstanz group in real algebraic geometry is concerned with specifically real questions in geometry and algebra. One studies objects that arise from modelling the "real" world. Typically, they are built up using the real numbers as the basis, corresponding to a line or to the time axis. Traditionally, one works over the complex numbers in algebraic
geometry. This allows easier access to many questions, but comes at the cost of losing some ties to reality. Also, the traditional approach typically ignores aspects which are crucial for many "real" problems, like questions of positivity.
In the 19th century, classical algebraic geometers had a well-developed sense for real questions. But for a long period in the 20th century, real algebraic geometry has been neglected. This began to change in the 70s and 80s, when the discovery of new algebraic and geometric methods made problems accessible which formerly had been beyond hope. Currently, real algebraic geometry is a field in the middle of a rapid development along many lines.
The Konstanz group in real algebraic geometry has existed for more than ten years. The research work done in Konstanz centers around the following subjects:
- Algebraic geometry of real varieties
- positive polynomials and sums of squares
- connections to analysis and optimization
- quadratic forms
- algebras with involution and linear algebraic groups
- logic and model theory
The real algebraic geometry group cooperates with the group in differential geometry.
The old websites of the Research Group, the former Forschungsinitiative Real Algebraic Geometry and Emerging Applications and the Model Theory Working Group are archived, but still available.