RUI: Homotopy Theory of Commutative Algebras and its Applications

DMS-0206647 James M.特纳自20世纪50年代以来，交换代数的一个活跃研究领域涉及使用同调方法来表征诺特环和它们之间的同态。在1960年代，单纯形方法被用来使同伦理论适用于交换代数，并发展了更丰富的同调技术。这个项目试图用同伦理论来描述具有诺特性质的（单纯）交换代数。这一努力的一部分将寻求解决一个猜想奎伦的刚性的同源性交换代数和提请更深层次的联系单纯的方法和方法从微分同调代数。本项目还将研究代数的上同调如何作为障碍物的宿主，从给定的代数数据中实现拓扑空间，这些代数数据用作假定空间的适当同伦不变量的值。同伦理论是一种从全局的角度研究某些数学对象及其之间关系的方法。在几何对象的情况下，可以分配某些代数不变量来研究和区分同伦类型。因此，理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联，是这一理论的重要组成部分。 本项目旨在从代数的角度进一步理解几何对象的同伦理论。这个项目有两个目标。首先是进一步发展同伦理论的代数平行的几何对象。第二个目标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属性可以提升到几何对象的相应结构和属性的程度。

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.