Operads in algebraic geometry and their realizations

代数几何中的运算及其实现

• 批准号: 269680815
• 负责人: Professor Dr. Jens Hornbostel
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/269680815?language=en
• 关键词: Operads; algebraic; geometry; their; realizations

A major objective in algebraic topology is to investigate and classify topological spaces via algebraic invariants. Topological spaces which are equivalent up to deformation (homotopy) usually have the same algebraic invariants. The fundamental group, consisting of homotopy classes of based loops in a pointed topological space, is a basic such invariant. By definition, it coincides with the path components of the associated loop space, the topological space of based maps from the circle to the given space. Special properties of the circle provide these loops with a multiplication which is associative up to homotopy. In the loop space of a loop space, the multiplication is even commutative up to homotopy. Further iterations improve this multiplication, and infinite loop spaces are equivalent to connective spectra, highly structured and somewhat manageable invariants.So-called recognition principles characterize the additional structure a topological space has to possess in order to be an iterated loop space. These principles can be phrased via actions of special collections of topological spaces dubbed operads. Despite their topological origin, operads abound in algebra, geometry, and mathematical physics. Spectacular work that Morel and Voevodsky accomplished in the 90s transferred homotopical methods to the realm of algebraic geometry, whose objects of interest are rather rigid, being defined via polynomials. Connections with Grothendieck‘s vision of universal invariants (Motifs) for these algebraic varieties coined the term „motivic homotopy theory“.The major objective of this project is the investigation, construction and modification of explicit operads, closely connected to moduli spaces of algebraic curves of genus zero, in algebraic geometry. Topological realizations of various flavors will be our preferred investigation device. Only those algebraic operads whose topological realizations act on usual loop spaces will stand a chance of acting on loop spaces in motivic homotopy. These loop spaces inherit amazing complexity from an additional "circle". This circle produces certain transfer maps. Incorporating suitable transfer maps in the aforementioned algebraic operads will be one modification, as well as completion of partial operads.

代数拓扑的一个主要目标是通过代数不变性调查和对拓扑空间进行分类。等效于变形（同型）的拓扑空间通常具有相同的代数不变性。基本群体由尖锐的拓扑空间中的基于基的循环组成，是一个基本的不变。根据定义，它与相关环空间的路径成分相吻合，这是从圆到给定空间的基于地图的拓扑空间。圆的特性为这些循环提供了乘法，该循环与同质性均匀相关。在循环空间的环空间中，乘法甚至是均匀的。进一步的迭代改善了这种乘法，而无限的环路空间等同于结缔组织，高度结构化且有些可管理的不变性。如此被称为式识别原则是拓扑空间必须具有的额外结构以成为迭代的环路空间。这些原则可以通过称为运营商的拓扑空间的特殊集合的作用来表达。尽管它们的拓扑起源，但代数，几何和数学物理学中的运营商充斥着。莫雷尔（Morel）和沃沃德斯基（Voevodsky）在90年代完成的壮观工作将同位方法转移到了代数几何学领域，其感兴趣的对象相当刚性，是通过多项式定义的。这些代数品种与Grothendieck对普遍不变的愿景（主题）的联系创造了“动机同型理论”一词。该项目的主要目的是对显式操作员的投资，构建和修改，与Algebraic Geometry in Algebraic Geometry in Algebraic Geometry of Algebraice semery nar of Speact Operation的投资，建设和修改与Moduli space of Moduli空间密切相关。各种口味的拓扑实现将是我们首选的投资设备。只有那些拓扑实现的代数运算符对通常的循环空间作用，才有机会在图案同拷贝以循环空间进行行动。这些循环空间从附加的“圆”继承了惊人的复杂性。该圆圈产生某些传输图。在预测代数运算符中合并合适的传输图将是一个修改，以及部分操作员的完成。