Descent theory is an area of mathematics, which provides a precise ( categorical) framework in which to formulate ideas associated with the description and classification of objects "locally isomorphic" to ones of a given type. The situation of fibre bundles and their analogues in various algebraic contexts is typical.  (This area is a speciality research area of the Coimbra group.)  Higher dimensional versions of descent theory relate to the problem of replacing "locally isomorphic" by "locally equivalent".  They form part of the Grothendieck programme "Pursuing Stacks" that aims to find higher dimensional forms of Galois theory, and relate to work on non-abelian cohomology and topological quantum field theory.  They use versions of higher dimensional algebra, algebraic homotopy theory and homotopy coherent category theory, as developed at Bangor.   The project aims to study the interaction between the two areas especially in low dimensions.

Detailed Objectives:
(i)   To examine the relationship between higher dimensional descent and the descent problem in the categories of algebraic homotopy theory (that is for various  algebraic models for homotopy types);
(ii)   To investigate the descent aspects of the 2-dimensional Galois theory of R.Brown and G.Janelidze and related local-to-global results such as the higher order van Kampen theorems;
(iii)   To study the place of localic and quantalic structures in higher dimensional descent theory problems;
(iv)   To study the relationships between (homotopy) factorisation systems and higher dimensional descent theory.
(v)   To seek a unified setting for the various generalised forms of van Kampen's theorem and an interpretation of these forms in terms of geometric / algebraic descent.
(vi)   To seek methods of calculating descent related invariants (such as non-abelian cohomology) in a variety of contexts using methods from computational group theory and related areas of general symbolic algebra.

Members of the Research Groups:
Bangor team:
T. Porter, R. Brown, C. D. Wensley.

Coimbra Team:
M.Sobral, M. M Clemintino, J. Picardo, G. Gutierres, Diana Rodelo.