Research in Analysis and Noncommutative Geometry

The Analysis group at Manchester currently has 2 members, whose combined research interests cover a wide range of topics in analysis. We have regular K-theory Days, shared between Oxford , Manchester and Southampton .

Reseach Interests

Operator algebras, K-Theory, Noncommutative Geometry: Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces.

It is an old idea that noncommutative algebras should be studied as if they were the function algebras of "noncommutative spaces".The recent inspiration for this viewpoint is due to Alain Connes.

The subject of topology is dominated by homology and cohomology theories, which satisfy a small number of simple axioms. From the present point of view, all this has to be transported to the noncommutative world.So it comes about that new theories have been created in order to study noncommutative algebras from this point of view.

Prominent among these new theories are K-theory of C*-algebras, analytic K-homology, periodic cyclic homology and cohomology.

Current work includes: structure of C*-algebras attached to p-adic groups; the relation between K-theory, group representations and algebraic number theory.

Probabilities on Groups: Dr Mick McCrudden is interested in the theory of probabilities on locally compact groups, here the overall objective is to extend the classical theory of real-valued random variables to goup-valued random variables. A major attraction of this area is that it calls for the interplay of different parts of mathematics including algebra, analysis, measure theory and geometry. McCrudden is particularly interested in embedding theorems for infinitely divisible probabilities, especially on Lie groups, and in support and density behaviour of non-symmetric Gauss measures on Lie groups. McCrudden regularly collaborates in research with other mathematicians, including S.G. Dani, D. Kelly-Lyth and S. Walker .

Members of staff involved: