Covariant Spectral Theory
Families of invertible transformations of geometric sets are rich sources of interesting and important groups. Properties of geometric objects, which are invariant under such transformations, are subject of geometry according to the Erlangen Programme of Felix Klein. Transformations of sets can be naturally extended to actions on linear functional spaces defined on those sets, so we obtain linear representations of groups. There are many questions in analytic function theory which are greatly simplified by a consideration of an appropriate group representation.
Finally, we can consider actions of the same groups on operators or, more generally, on Banach algebras and other non-commutative sets. There are oftenly intertwining operators linking group actions on non-commutative spaces and linear spaces of functions. Depending on the direction they act those intertwining operators are known as functional calculi or functional models. A study of covariant properties of functional calculi provide valuable characterisation of operator spaces in geometrical terms. Thus it is a natural extension of the Erlangen programme to non-commutative sets.
Analysis group at Leeds
The Analysis group at Leeds is one of the strongest in the UK in abstract analysis. The group comprises 5 members of staff (Jonathan Partington, Matthew Daws and Charles Read are also interested in supervising PhD students) together with a research Professor and a number of visiting Fellows. The group currently has six Graduate Students. Researchers at Leeds have interests ranging from the interplay between analysis and algebra (Daws and Read) to links with complex analysis and control theory (Partington) through to applications to Mathematical Physics (Kisil). Leeds hosts a weekly seminar series (which has close links with York University) and is currently the organising node for the North British Functional Analysis Seminar series.
This project is eligible for School of Mathematics EPSRC Doctoral Training Grant funding - please contact us for more information.