Operads in algebraic geometry and their realizations

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。在变形（同伦）范围内等价的拓扑空间通常具有相同的代数不变量。由点拓扑空间中基环的同伦类组成的基群就是这样一个基本的不变量。根据定义，它与关联的环空间的路径分量重合，即基于的拓扑空间从圆映射到给定空间。圆的特殊性质为这些循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，乘法是偶交换直至同伦的。进一步的迭代改进了这种乘法，无限循环空间相当于连接谱，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备的附加结构，以便成为迭代循环空间。这些原则可以通过称为操作符的拓扑空间的特殊集合的动作来表达。尽管它们起源于拓扑结构，但操作数在代数、几何和数学物理中大量存在。Morel和Voevodsky在90年代完成的惊人工作将同调方法转移到代数几何领域，其感兴趣的对象是相当刚性的，通过多项式定义。与格罗滕迪克对这些代数变体的普遍不变量（母元）的看法的联系创造了术语“动机同伦理论”。本课题的主要目的是研究、构造和修正代数几何中与零属代数曲线模空间密切相关的显式操作数。各种口味的拓扑实现将是我们首选的研究设备。在动机同伦中，只有拓扑实现作用于一般循环空间的代数操作才有可能作用于循环空间。这些循环空间继承了额外的“圆”的惊人复杂性。这个圆产生了一定的转移图。在上述代数操作中加入适当的转移映射将是一种修改，以及部分操作的完成。

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.