Random trees and tree-valued stochastic processes are of particular importance in combinatorics, computer science, phylogenetics, and mathematical population genetics. Using the framework of abstract "tree-like" metric spaces (so-called real trees) and ideas from metric geometry such as the Gromov-Hausdorff distance, Evans and his collaborators have recently pioneered an approach to studying the asymptotic behaviour of such objects when the number of vertices goes to infinity. These notes survey the relevant mathematical background and present some selected applications of the theory.

Autorentext

Steve Evans received his undergraduate degree from Sydney University in Australia and his PhD from Cambridge University in the United Kingdom. His first position was in research and analysis at the Commonwealth Bank of Australia. This was followed by a post-doc at the University of Virginia. He has been a faculty member at the University of California at Berkeley since 1989. Evans has numerous publications in the general area of stochastic processes, with a particular emphasis on probability on algebraic and topological structures. He also works on aging and mortality, population genetics, and phylogenetics -- particularly the application of computational methods to historical linguistics. Evans is a recipient of the Rollo Davidson Prize, a Presidential Young Investigator Award, an Alfred P. Sloan Foundation Fellowship, and the G. de B. Robinson Prize of the Canadian Mathematical Society. He is a Fellow of the Institute of Mathematical Statistics and has held a research professorship from the Miller Institute for Basic Research in Science.

Inhalt

Around the Continuum Random Tree.- R-Trees and 0-Hyperbolic Spaces.- Hausdorff and Gromov-Hausdorff Distance.- Root Growth with Re-Grafting.- The Wild Chain and other Bipartite Chains.- Diffusions on a R-Tree without Leaves: Snakes and Spiders.- R-Trees from Coalescing Particle Systems.- Subtree Prune and Re-Graft.