Real algebraic geometry, convexity and topology

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This project intends to advance real algebraic geometry on two different fronts: Algebraic representations of real polynomials or semi-algebraic sets; and topology and geometry of real algebraic varieties.Both branches come with their own methods and progress on one front will have impact on the other one. Apart from methods that are specific to the reals, we will utilize the potential of modern algebraic geometry to a large extent.Finding certain algebraic representations of real algebraic or geometric objects is of particular interest from the viewpoint of complexity or optimization theory. This includes expressing a polynomial in a way that makes it easy to evaluate or to minimize it, like as a determinant or a sum of squares, or representing a set in a way making it applicable to optimization methods like semidefinite programming (SDP). Fundamental open questions are the Generalized Lax Conjecture and whether hyperbolic programming is more general than SDP. The study of the topology of real algebraic varieties dates back to the 19th century. Since then a lot of progress has been made for planar and, more recently, spatial curves and for surfaces in three-space. For varieties of/in higher dimension much less is known and we plan to shed more light on this by examining varieties with extremal geometric properties. The theory of Ulrich sheaves plays a prominent role in this project. Ulrich sheaves on a real algebraic variety have, if equipped with some additional real structure, an impact on the topology as well as on algebraic representations of the defining polynomials of this variety.