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.

代数拓扑学的一个主要目标是通过代数不变量来研究和分类拓扑空间。等价于变形(同伦)的拓扑空间通常具有相同的代数不变量。由点拓扑空间中的基环的同伦类组成的基本群是这样的基本不变量。根据定义，它与相关回路空间的路径分量重合，即从圆到给定空间的基映射的拓扑空间。圆的特殊性质为这些环提供了一个结合到同伦的乘法。在循环空间的循环空间中，乘法甚至可交换到同伦。进一步的迭代改进了这种乘法，无限循环空间等价于连通谱、高度结构化和某种程度上可管理的不变量。所谓的识别原理表征了拓扑空间必须具有的附加结构才能成为迭代循环空间。这些原则可以通过被称为歌剧的特殊拓扑空间集合的动作来表述。尽管它们起源于拓扑学，但歌剧在代数、几何和数学物理中比比皆是。Morel和Voevodsky在90年代完成的壮观工作将同伦方法转移到了代数几何领域，其感兴趣的对象是相当严格的，通过多项式来定义。与Grothendieck关于这些代数簇的万能不变量(模元)的设想相联系，产生了术语“模同伦理论”。这个项目的主要目的是研究、构造和修正代数几何中与亏格为零的代数曲线的模空间密切相关的显式算子。各种口味的拓扑实现将是我们首选的研究工具。只有那些其拓扑实现作用于通常环空间的代数算子才有可能作用于Motivic同伦中的环空间。这些循环空间从一个额外的“圈”继承了惊人的复杂性。此圆会生成特定的传递贴图。在前面提到的代数算子中结合适当的传递映射将是一种修改，也是部分算子的完成。