Derived A-infinity algebras

This research lies in pure mathematics, in algebraic topology and homotopical algebra. The general theme involves the interaction between the topological idea of continuous deformation and algebraic operations like addition or multiplication. In this context, one has to consider "up to homotopy'' versions of familiar algebraic structures. For example, A-infinity structures arise from the homotopy invariant version of
associativity. These structures first arose in topology in the study of loop spaces and they have since come to be important in many other areas of mathematics, including representation theory and mathematical physics.

This project will investigate recent generalisations of these structures, called derived A-infinity algebras, and related constructions. The work will mainly be in homotopical algebra, in particular using algebraic operads.

The project aims to develop analogues for these derived generalisations of various aspects of the theory of A-infinity algebras.

In particular, one objective is to develop and study the theory of Massey products in this setting. Massey products are higher operations on the cohomology of diferential graded algebras. For example, triple Massey products capture the behaviour of the Borromean rings - three circles which are linked although each pair of them is not linked. In the usual A-infinity setting, there is a quite well understood connection between Massey products and A-infinity structures. The aim is to develop a similar theory in the derived setting.

Another objective is to further develop obstruction theory for these structures. This will involve understanding cohomology theories for these algebras.

The methods used will be those of homotopical algebra, particularly the theory of algebraic operads. The work is likely to involve extension of these techniques to a more general setting, with an extra grading. To give a setting for the theory, techniques of Quillen model categories, or perhaps infinity-categories, will be used.

The project lies within the EPSRC research areas of Geometry and Topology, and Algebra. It is a pure mathematics project and as such relates to EPSRC's strategy to maintain an excellent and effective capability in mathematics.