Free Semialgebraic Geometry and Convexity

Classical semialgebraic geometry examines subsets of euclidean space defined by polynomial inequalities. This algebraic structure of sets allows for strong methods and results. The projection theorem for example states that the projection of a semialgebraic set is again semialgebraic. From this it also follows that the convex hull of a semialgebraic set is again semialgebraic. This is of particular importance in optimization, where the geometry can often be simplified significantly by passing to the convex hull, whereas the optimal value of the problem is unchanged. Finding easy algebraic descriptions of the convex hull is then of great interest for practical purposes. The project addresses similar question, however in a free, i.e. non-commutative context. Semialgebraic sets here consist of matrices of all sizes simultaneously, i.e. they are dimension- free. The free theory of semialgebraic sets is a recent development, triggered by applications in linear systems engineering, quantum physics, optimization and group theory. Many of the main building-blocks of the commutative theory have not been transferred successfully to the free context however. This is one of the main goals of the project. The most important questions concern the projection theorem in the free context, the correct notion of a free semialgebraic set, and how free convex hulls of semialgebraic sets can be described as simple as possible. The methods applied to solve these questions stem from operator algebra, group theory and functional analysis over real closed fields. The results are of great importance for the future developments of the field of free semialgebraic geometry. They also provide a sound basis for many of the algorithmic applications via semidefinite optimization, that have become very popular recently. Any result also opens the way for more of these applications to emerge.

The project goal was to understand so-called "non-commutative semialgebraic sets" better. These are sets of matrices that can be defined by polynomial inequalities. An important result in the classical theory is the Projection Theorem, stating that a projection of a semialgebraic set is again semialgebraic. We were able to show that this fails in the non-commutative setup, i.e. projections of non-commutative semialgebraic sets cannot be described by non-commutative polynomial inequalities again. We also discovered deep connections between the theory of non-commutative semialgebraic sets, quantum information theory and the theory of operator systems. This lead to interesting new discoveries in all of these fields, and will hopefully open the way for further progress in the near future.