From hyperbolic geometry to nonlinear Perron-Frobenius theory

The classical Perron-Frobenius 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 Perron-Frobenius 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 Perron-Frobenius 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 non-Riemannian 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 Denjoy-Wolff 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 Denjoy-Wolff theorem is a classical result in complex analysis about the dynamics of fixed point free analytic self-maps of the unit disc. Beardon discovered a striking generalisation of this result to fixed point free non-expansive 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 twenty-year old conjectures, which combines ideas from nonlinear Perron-Frobenius 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.

The classical Perron-Frobenius theory is one of the most beautiful topics in matrix analysis with many important practical applications. It forms the basis for the study of Markov chains, which is one of the pillars of modern probability theory and is widely used in finance, risk analysis, and statistics. Perron-Frobenius theory is also used by Google in their page-rank algorithm, which is an essential part in their search engine. Nonlinear Perron-Frobenius theory extends the classical theory to include nonlinear positive operators. It has a definite scope for real-world applications. So far the most striking applications have been within academia, but certain pathways to real-world applications are emerging. For example, nonlinear Perron-Frobenius is used in engineering as a framework to analyse the performance and the dynamic behaviour of so-called discrete event systems, which are used as models for transportation networks, telecommunication networks, and manufacturing systems. It is also used in computer systems engineering to analyse synchronisation and concurrency (resource sharing) phenomena. Although the focus of this project is on fundamental issues, it will certainly advance the state-of-the-art in nonlinear Perron-Frobenius theory and thus enhance its potential for applications.

I have a close working relation with the Maxplus group at INRIA (Saclay), which has a strong mandate to develop applications of nonlinear Perron-Frobenius theory. They are involved in various joint research projects with industry. The work with Walsh (INRIA) on the project will strengthen our collaborative partnership and increase the probability of non-academic exploitations of my work on nonlinear Perron-Frobenius theory. Besides the collaboration with INRIA, I am also setting up links with computer scientists and control engineers to explore applications in computer systems engineering. I have organised with Sarah Spurgeon from the Engineering the department at Kent a Networks@Kent day, to discuss common interests and facilitate collaborations between control engineers, computer scientists and mathematicians. I am also investigating with Elizabeth Ogston, who is a computer scientist at the TU-Delft, possible applications in the analysis and design of distributed algorithms to aggregate information, or synchronise the nodes, in large scale computer networks. Over the course of the project I will develop these links further.

To increase the user-base of nonlinear Perron-Frobenius theory it is essential to train people. In 2010 the University of Kent started an MSc in Mathematics and Its Applications. As part of this MSc programme I will offer in 2012 or 2013 a module called "Topics in nonlinear analysis and optimisation", which contains an introduction to nonlinear Perron-Frobenius theory and discusses applications to Markov decision processes, game theory, and optimisation.

Our department is actively involved in various projects with secondary school students from Kent. Each year the school runs a several masterclasses, which are very well attended by pupils from local secondary schools. As part of the project I will develop four masterclasses related to (nonlinear) Perron-Frobenius theory and its applications. They will take place at the University of Kent and are supported by the Royal Institution of Great Britain. In addition the PDRA and I will set up, in 2012, a summer project for a secondary school student for which funding will be sought from the Nuffield Foundation. To enhance my public communication skills, I have requested funds to attend a training day organised by the Royal Society.