Real algebraic geometry, convexity and topology

实代数几何、凸性和拓扑

• 批准号: 502861109
• 负责人: Professor Dr. Mario Kummer
• 金额: --
• 依托单位: Institut für Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/502861109?language=en
• 关键词: Real; algebraic; geometry; convexity; topology

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.

该项目旨在推进真实的代数几何在两个不同的战线：代数表示的真实的多项式或半代数集;和拓扑和几何的真实的代数品种。这两个分支来与自己的方法和进展在一个战线将影响到另一个。除了方法是具体的reals，我们将利用现代代数几何的潜力在很大程度上。找到某些代数表示的真实的代数或几何对象是特别感兴趣的复杂性或优化理论的观点。这包括以一种易于计算或最小化它的方式来表达多项式，例如作为行列式或平方和，或者以一种使其适用于半定规划（SDP）等优化方法的方式来表示集合。基本的开放问题是广义Lax猜想和双曲规划是否比SDP更一般。对真实的代数簇拓扑的研究可以追溯到世纪。从那时起，在平面和最近的空间曲线和三维空间中的曲面方面取得了很大的进展。对于品种/在更高的维度少得多的是已知的，我们计划通过检查品种极值几何性质来阐明这一点。乌尔里希层理论在这个项目中起着突出的作用。真实的代数簇上的乌尔里希层，如果配备了一些额外的真实的结构，会对拓扑结构以及这个簇的定义多项式的代数表示产生影响。