Abstract Homotopy Theory

DMS-0203936Charles W.这个项目的目标是应用同伦理论的方法来解决代数问题。 这里涉及两条思路。 一个是同伦理论的描述作为一种拓扑丰富的范畴理论，已经发展了德怀尔-坎和许多其他人。 另一个线索是在理论的交换环谱（同伦理论推广交换环理论），特别是发展的拓扑版本的代数德拉姆复杂的环。同伦理论是一个分支的拓扑;它产生的研究某些不变性质的空间，即那些保持不变的连续变形。 令人惊讶的是，空间同伦论的方法（似乎是几何和拓扑的）与代数的几个分支（如同调代数、层论和范畴论）的方法之间有着深刻的相似之处。 例如，我们可以通过给出生成元和它们之间的关系来描述一个代数对象（如一个群或环），我们可以通过给出多面体和它们连接在一起来描述一个拓扑空间。 这个项目的目标是在环论和范畴论的背景下发展某些同伦理论方法，并使它们能够应用于传统的代数问题。

DMS-0203936Charles W. RezkThe goal of this project is to apply homotopy theoretic methods to algebraic problems. There are two threads of ideas involved here. One is the description of homotopy theory as a kind of topologically enriched category theory, as has been developed by Dwyer-Kan and many others. Another thread is in the theory of commutative ring spectra (a homotopy theoretic generalization of commutative ring theory), andin particular the development of a topological version of the algebraic de Rham complex of a ring.Homotopy theory is a branch of topology; it arose as the study of certain invariant properties of spaces, namely those left unchanged by continuous deformations. It is a surprising fact that there are deep analogies between the methods of the homotopy theory of spaces (which would seem to be geometric and topological) and those of several branches of algebra (such as homological algebra, sheaf theory, and category theory). For example, there is a close analogy between the way we can describe an algebraic object (such as a group or ring) bygiving generators and the relations between them, and the way we can describe a topological space by giving polyhedra and the way they are attached together to give the space. The goal of this project is to develop certain homotopy theoretic methods in the context of ring theory and of category theory, and to allow them to be applied totraditional algebraic problems.