Real Algebraic Geometry and Optimization

Real algebraic geometry is concerned with the real solutions of equations and inequalities defined over the real numbers and has both a distinguished history (for example, Hilbert’s 17th problem) and many applications in the natural sciences and engineering. An important modern branch connects real algebraic geometry with techniques from optimization. Prominent roots of these developments go back to N.Z. Shor (in the 1980’s) and were substantially advanced by Parrilo and Lasserre (around the year 2000) through their insights of how to use semidefinite programming within polynomial optimization. An important link between real algebraic geometry and optimization has been provided through sums of squares. While some fundamental theorems connecting nonnegative polynomials with sums of squares already date back to Hilbert, the modern developments have recognized that sums of squares of polynomials can be computationally handled much better than nonnegative polynomials and that this general idea can also be effectively applied to rather general constrained polynomial optimization problems. With the strategic goal to strengthen the academic relations between Germany and Norway, the main scientific purpose of the project is to advance various lines of research concerning real algebraic geometry and optimization. The focus of the project will be on some lines of research which have raised new challenges in the last years. These subtopics both allow complementary cooperative research work on a leading international level as well as the training of young researchers in an international framework.