EPSRC Reference: 
EP/J008508/1 
Title: 
From hyperbolic geometry to nonlinear PerronFrobenius theory 
Principal Investigator: 
Lemmens, Dr B 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Maths Statistics & Actuarial Sci 
Organisation: 
University of Kent 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 September 2012 
Ends: 
31 October 2013 
Value (£): 
98,832

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The classical PerronFrobenius theory concerns the spectral properties of nonnegative matrices, and is considered one of the most beautiful topics in matrix analysis with important applications in probability theory, dynamical systems theory, and discrete mathematics. Nonlinear PerronFrobenius theory extends this classical theory to nonlinear positive operators, and deals with questions like: When does a nonlinear positive operator have an eigenvector in the cone corresponding to the spectral radius? When does the eigenvector lie in the interior of the cone? How do the iterates of such operators behave? These questions arise naturally in a wide range of mathematical disciplines such as game theory, analysis on fractals, and tropical mathematics. Birkhoff showed that one can use Hilbert geometries to analyse these questions. Birkhoff's discovery of the synergy between nonlinear PerronFrobenius theory and metric geometry has only recently started to fully crystallise, and is the main theme of the project. We will focus on several central open problems concerning Hilbert geometries. Hilbert geometries are a natural nonRiemannian generalisation of hyperbolic geometry. Recent developments in metric geometry have triggered a renewed interest in Hilbert geometries, and opened up exciting opportunities to solve some of these problems.
Our first goal is to prove DenjoyWolff type theorems for Hilbert geometries, which provide detailed information about the dynamics of nonlinear positive operators without eigenvectors in the interior of the cone. The DenjoyWolff theorem is a classical result in complex analysis about the dynamics of fixed point free analytic selfmaps of the unit disc. Beardon discovered a striking generalisation of this result to fixed point free nonexpansive maps on metric spaces that possess mild hyperbolic properties. His work left open a number a fascinating problems some of which we hope to resolve in this project. Our second goal is to prove several conjectures by de la Harpe about the isometry group of Hilbert geometries. In a recent work we found a completely novel approach to these twentyyear old conjectures, which combines ideas from nonlinear PerronFrobenius theory with new concepts in metric geometry such as the Busemann points in the horofunction boundary and the detour metric. There appears to be an intriguing connection between the solution of de la Harpe's conjectures and the theory of symmetric cones, which we hope to unravel.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.kent.ac.uk 