My principle research interests lie at the intersection of geometry, topology, analysis and group theory. Recent projects include new homological characterisations of amenability and of property A generalising results of Johnson and Block and Weinberger. These results exposed an unexpected relationship between the classical results for amenability and led to new characterisations of generalisations such as the notion of a topologically amenable action on a compact space, or C* exactness for a group. This work does have a functional analytic flavour, arising from the need to construct invariant means as limits. For example the integers are known to admit an invariant mean but no-one has ever seen one in the wild; they are observed to exist as weak limits of geometrically defined subsets known as Folner sets. There has been a lot of interest recently in the use of ideas from geometric group theory to construct such approximations to a mean and this is an ideal area for further exploration, allowing us to use geometry for constructions of objects discovered via analytic methods.
While much of geometric group theory arose from ideas in low dimensional topology (study of the mapping class group, the JSJ decomposition and Stallings' theorem all played a fundamental role) there is increasing interest in tackling problems in high dimensional topology. Given the remarkable recent proof of the Farrell-Jones conjecture for CAT(0) groups (by Bartels and Lueck), there is likely to be scope for a lot more interaction between these areas. In this direction my recent work with our Postdoc Aditi Kar shows that the classical sphere and torus theorems have a generalisation for even dimensional manifolds of real rank at least 2 and for quaternionic manifolds of dimension at least 8. Broadly speaking we show that any $\pi_1$-injective map of such a manifold to an aspherical manifold of dimension 1 higher is homotopic to a finite cover of an embedding. This may be viewed as a topological analogue of the celebrated geometric superrigidity theorem, at least in even dimensions, and we continue to explore the odd dimensional case.
I am also interested in applications of coarse geometry and graph theory to real world problems. My paper with Jim Anderson on construction of universal footprinting templates in DNA is currently being converted to a practical algorithm by an undergraduate Nuffield scholar with plans for the resulting protein assay templates to be built by our collaborator Keith Fox in the new Life Sciences Institute. More recently I have become involved in a project to model catastrophic failure in the the National Grid with the hope of developing new methods to prevent cascading blackouts. Tools from analytic graph theory are being used to detect optimal network decompositions during a crisis. this is joint work with Jacek Brodzki, Ruben Sanchez, Nick Wright and our student Max Fennelly at Southampton, together with engineers and mathematicians from Durham and Edinburgh.
1. Topological super-rigidity - a generalisation of the sphere and torus theorems and the JSJ decomposition to high dimensional topology.
2. Topologically amenable actions - generalisations of amenability to a wide variety of geometric and topological contexts. Funded by EPSRC grant
3. Applications of coarse geometry and analytic graph theory - protective islanding of energy networks, homological methods in data analysis. Funded by EPSRC grant
Professor Graham Niblo
Building 54 Mathematical Sciences University of Southampton Highfield Southampton SO17 1BJ
Telephone:(023) 8059 3674
Facsimile:(023) 8059 5147
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