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Details of Grant 

EPSRC Reference: EP/M02461X/1
Title: Sheaf cohomology for C*-algebras
Principal Investigator: Mathieu, Dr M
Other Investigators:
Researcher Co-Investigators:
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Department: Sch of Mathematics and Physics
Organisation: Queen's University of Belfast
Scheme: Standard Research
Starts: 01 June 2015 Ends: 03 June 2017 Value (£): 46,505
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
03 Mar 2015 EPSRC Mathematics Prioritisation Panel March 2015 Announced
Summary on Grant Application Form
Topology is the study of topological spaces and continuous deformations, that is, an abstract notion of shape and how it can be deformed without breaking it apart. An example of a topological space is any subset of Euclidean space. In a topological space, there can be global phenomena and very different local ones, those that are only valid in the vicinity of a point in the space. Sheaf theory provides us with tools to control the passage from local to global properties. Sheaf cohomology adds additional techniques of an algebraic (computational) nature and enables us to treat invariants (i.e., properties invariant under deformation) that distinguish between topological spaces which may otherwise be difficult to tell apart from each other. It also connects other cohomology theories with each other and is a highly sophisticated methodology drawing a lot of its strength from Category Theory, a very abstract field of Pure Mathematics.

Non-commutative Topology has been in use as the adequate mathematical language for Quantum Physics for some time and has lately found manifold, sometimes unexpected applications in numerous other areas of mathematics, such as Number Theory. The concept of a topological space is replaced by a C*-algebra (a self-adjoint closed subalgebra of the bounded linear operators on Hilbert space), the connections between C*-algebras (the "deformations") are *-homomorphisms or sometimes mappings preserving related structure. Open subsets are replaced by ideals; therefore a sheaf of C*-algebras is well suited to handle the differences between local and global phenomena in this more general setting. Based on the theory of local multipliers, which we developed in collaboration with Pere Ara (Barcelona), stalks of some fundamental examples of these sheaves are by now well understood, the section functors are available, and various important results have been published.

The next, natural step will be to develop a sheaf cohomology theory for C*-algebras which will put us in a position to employ the powerful algebraic tools from Homology Theory. After basic difficulties which arise from the (somewhat typical unpleasant) behaviour of categories of analytic objects have been overcome, we shall obtain new invariants for C*-algebras that, once again, may tell those apart that previously could not be handled (Elliott's programme for non-simple C*-algebras).
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Organisation Website: http://www.qub.ac.uk