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# Dr Edward Crane

## Dr Edward Crane

B.A.(Cantab.), A.M. (Harvard), Ph.D.(Cantab.)

Senior Research Fellow
### Area of research

Probability and complex analysis
## Summary

## Biography

### Current projects

## Teaching

### Current projects

## Keywords

## Memberships

### Organisations

### Probability, Analysis and Dynamics

### Research themes

## Recent publications

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B.A.(Cantab.), A.M. (Harvard), Ph.D.(Cantab.)

Howard House,

Queen's Ave,
Bristol
BS8 1SD

(See a map)

+44 (0) 117 954 5661

edward.crane@bristol.ac.uk

My research is in probabilty and complex analysis.

My main project in probability concerns mathematical models of forest fires that display **self-organized criticality**. These are not intended to be realistic models of real-world forest fires, but instead they give us insight into complex stochastic systems that are driven into critical states by their own dynamics. The models that we study combining simple evolution rules and randomness to obtain this behaviour. The simplicity enables us to carry out exact mathematical analysis. In the critical state, the distribution of sizes of fires has a power law tail, and even the smallest possible change to the driving randomness has a positive probability of propagating into a macroscopic change in the state of the system. Mathematically, this research involves a mixture of probability theory and partial differential equations.

My recent work in complex analysis is about **circle packing**. The Koebe-Andreev-Thurston theorem says that given any triangulation of the sphere, we can find a packing of discs on the sphere with disjoint interiors, whose graph of tangencies is isomorphic to the given triangulation. There is a growing theory of discrete conformal mapping based on circle packing. My current project (with Ken Stephenson and James Ashe) involves extending this to a good theory of circle packing with movable branch points, in order to investigate the analogy with analytic functions that are not locally univalent. For example, we hope it will help to us to tackle the existence and uniqueness question for the circle packing analogue of rational functions.

My research is in probabilty and complex analysis.

My main project in probability concerns mathematical models of forest fires that display **self-organized criticality**. These are not intended to be realistic models of real-world forest fires, but instead they give us insight into complex stochastic systems that are driven into critical states by their own dynamics. The models that we study combining simple evolution rules and randomness to obtain this behaviour. The simplicity enables us to carry out exact mathematical analysis. In the critical state, the distribution of sizes of fires has a power law tail, and even the smallest possible change to the driving randomness has a positive probability of propagating into a macroscopic change in the state of the system. Mathematically, this research involves a mixture of probability theory and partial differential equations.

My recent work in complex analysis is about **circle packing**. The Koebe-Andreev-Thurston theorem says that given any triangulation of the sphere, we can find a packing of discs on the sphere with disjoint interiors, whose graph of tangencies is isomorphic to the given triangulation. There is a growing theory of discrete conformal mapping based on circle packing. My current project (with Ken Stephenson and James Ashe) involves extending this to a good theory of circle packing with movable branch points, in order to investigate the analogy with analytic functions that are not locally univalent. For example, we hope it will help to us to tackle the existence and uniqueness question for the circle packing analogue of rational functions.

- Probability
- self-organized criticality
- geometric function theory
- circle packing

- Crane, ET, Stephenson, K & Ashe, J, 2016, ‘Circle packing with generalized branching’.
*Journal of Analysis*, vol 24., pp. 251?276 - Crane, ET, Freeman, NP & Toth, BA, 2015, ‘Cluster growth in the dynamical Erdös-Rényi process with forest fires’.
*Electronic Journal of Probability*, vol 20. - Crane, ET, 2014, ‘Intrinsic circle domains’.
*Conformal Geometry and Dynamics*, vol 18., pp. 65-84 - Crane, E, 2012, ‘Relative Riemann mapping criteria and hyperbolic convexity’.
*Proceedings of the American Mathematical Society*, vol 140., pp. 2375-2382 - Crane, E, Georgiou, N, Volkov, S, Wade, A & Waters, R, 2011, ‘The simple harmonic urn’.
*Annals of Probability*, vol 39., pp. 2119 - 2177 - Crane, E & Short, I, 2011, ‘Rigidity of configurations of balls and points in the N-sphere’.
*Quarterly Journal of Mathematics*, vol 62., pp. 351 - 362 - Crane, E, 2008, ‘A note on the Hayman-Wu theorem’.
*Computational Methods and Function Theory*, vol 8., pp. 615 - 624 - Crane, E & Short, I, 2007, ‘Conical limit sets and continued fractions’.
*Conformal Geometry and Dynamics*, vol 11., pp. 224 - 249 - Crane, E, 2007, ‘A bound for Smale's mean value conjecture for complex polynomials’.
*Bulletin of the London Mathematical Society*, vol 39 (5)., pp. 781 - 791 - Crane, E, 2006, ‘Mean value conjectures for rational maps’.
*Complex Variables and Elliptic Equations*, vol 51 (1)., pp. 41 - 50

View complete publications list in the University of Bristol publications system

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