Real algebraic geometry, convexity and topology

这个项目的目的是在两个不同的方面推进实代数几何：实多项式或半代数集的代数表示；以及实代数簇的拓扑和几何。这两个分支都有自己的方法，一个方面的进展会对另一个方面产生影响。除了特定于实数的方法外，我们还将在很大程度上利用现代代数几何的潜力。从复杂性或最优化理论的观点来看，寻找实代数或几何对象的某些代数表示是特别感兴趣的。这包括以便于计算或最小化多项式的方式来表示多项式，例如行列式或平方和，或者以使其适用于半定规划(SDP)等优化方法的方式来表示集合。基本的公开问题是广义Lax猜想，以及双曲规划是否比SDP更一般。实代数簇的拓扑学研究可以追溯到19世纪。从那时起，平面曲线和最近的空间曲线以及三维空间中的曲面都取得了许多进展。对于高维的簇，我们知道的要少得多，我们计划通过研究具有极端几何性质的簇来更好地阐明这一点。乌尔里希滑轮理论在这个项目中起着突出的作用。实代数簇上的Ulrich层如果具有某些附加的实结构，则会对该簇的定义多项式的拓扑和代数表示产生影响。

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.