DEPARTMENT of  MATHEMATICS   Pure Research Analysis and Noncommutative Geometry

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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.
 
Research Interests   |   Staff Members


Our Research Interests

Noncommutative Geometry

Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, like the commutative algebras associated to affine algebraic varieties, smooth manifolds, topological spaces, and measure spaces.

It is an old idea that noncommutative algebras should be studied as if they were the function algebras of  “noncommutative spaces”.   The remarkable work of Alain Connes has transformed this metaphor into a powerful and substantial mathematical theory.   Connes' inspiration came from functional analysis, but the theory he constructed has reached out to homological algebra, differential geometry, and quantum theory for its basic constructions. Connes' well-known book on Noncommutative Geometry and the spirited discussion it has engendered provide an ample introduction to the subject.

Recent work has brought noncommutative geometry into lively contact with number theory, dimension theory, and renormalization in quantum field theory, to name some striking examples. Many of these developments have taken noncommutative geometry beyond the plan laid out in Connes' book.

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: Baum-Connes conjecture, and 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

Dr. M. McCrudden
Prof. R.J. Plymen