RUI: Homotopy Theory of Commutative Algebras and its Applications

Calvin University, Grand Rapids MI

Investigators

Abstract

DMS-0206647 James M. Turner An 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.