# Forschungsinitiative "Real Algebraic Geometry and Emerging Applications"

Real algebraic geometry is concerned with specifically real questions in geometry and algebra. One studies objects that arise from modelling the "real" world. Traditional algebraic geometry is working over the complex field, which allows an easier access to many questions but comes at the cost of losing some ties to reality. Typically, the traditional approach ignores aspects which are crucial for many "real" problems, like questions of positivity.

In the 19th century, algebraic geometers had a well-developed sense for real questions. But for a long time during the 20th century, real algebraic geometry has been neglected. This began to change towards the end of the century, when the discovery of new algebraic and geometric methods made problems accessible which formerly had been beyond hope. Today, real algebraic geometry represents a very active and innovative area of research, rapidly developing at many ends, and with close ties to manifold other fields in- and outside of mathematics.

The Forschungsinitiative started its activities in October 2010. They are bundled into three projects:

Project 1:

Positive polynomials, sums of squares and applications

Project 2:

Linear matrix inequalities, semidefinite representations, hyperbolic polynomials and Lax-type questions

Project 3:

Noncommutative real algebraic geometry and related problems

These projects are linked with each other in many ways, and the full extent of these connections is not yet understood sufficiently well. The Forschungsinitiative is trying to bring together expertise from these areas in a more systematic way than has happened so far. It is also aiming at intensifying the ties to discrete and convex geometry and to optimization.

The Forsschungsinitiative "Real algebraic Geometrie and Emerging Applications" is part of the Forschungsschwerpunkt reelle Geometrie und Algebra.