RUI: Homotopy Theory of Commutative Algebras and its Applications

DMS-0206647 James M.Turner自20世纪50年代以来一直是交换代数研究的一个活跃领域，它涉及使用同调方法来刻画Notherian环及其之间的同态。在20世纪60年代，单纯形方法被用来使同伦理论应用于交换代数，并发展了更丰富的同调技巧。本课题旨在利用同伦理论刻画具有Noether性质的(单纯)交换代数。这项工作的一部分将试图解决Quillen关于交换代数同调刚性的猜想，并在单纯方法和微分同调代数方法之间得出更深层次的联系。这个项目还将研究代数的上同调如何充当从给定的代数数据实现拓扑空间的障碍的宿主，这些代数数据充当假定空间的适当同伦不变量的值。同伦理论是从全局的角度研究某些数学对象以及它们之间关系的一种方法。在几何对象的情况下，可以指定某些代数不变量来研究和区分同伦类型。因此，了解这些代数结构的性质以及它们如何与它们所关联的几何对象相关是这一理论的重要部分。本课题旨在从代数的角度加深对几何对象同伦理论的理解。这个项目有两个目标。第一个是进一步发展代数的同伦理论，使之与几何对象的同伦理论平行。第二个目的是利用代数的同伦及其各种性质来理解空间的同伦。这将涉及研究代数结构和性质可以在多大程度上提升到几何对象的相应结构和性质。

DMS-0206647James M. TurnerAn active area of research in commutative algebra since the 1950s has involved the use of homological methods to characterize Noetherian rings and the homomorphisms between them. In the 1960s, simplicial methods were used to enable homotopy theory to apply to commutative algebras and develop richer homological techniques. This project seeks to use homotopy theory to characterize (simplicial) commutative algebras with a Noetherian property. Part of this effort will seek to resolve a conjecture of Quillen on the rigidity of the homology of commutative algebras and draw deeper connections between simplicial methods and methods from differential homological algebra. This project will also study how the cohomology of algebras can serve as host for obstructions to realizing a topological space from given algebraic data that functions as the value of a suitable homotopy invariant of the putative space. Attention will be paid to developing methods for computing such obstructions.Homotopy theory is a method of studying certain mathematical objects and the relations between them from a global perspective. In the case of geometric objects, certain algebraic invariants can be assigned to study and distinguish between homotopy types. Understanding the properties of these algebraic structures and how they relate to the geometric objects they are associated to are therefore important parts of this theory. This project seeks to further the understanding of the homotopy theory of geometric objects from the algebraic perspective. There are two aims to this project. The first is to develop further the homotopy theory of algebras that parallels that of geometric objects. The second aim seeks to make use of the homotopy of algebras and their various properties to understand the homotopy of spaces. This would involve studying the extent to which algebraic structures and properties can be lifted to corresponding structures and properties for geometric objects.