Random matrices, real trees, mortality models, and stepping-stone processes

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0405778Evans The PI extends the inter-disciplinary importance of random matrices by establishing further connections between random matrix theory, stochastic processes, analysis, combinatorics and mathematical physics. He brings recent ideas from metric geometry to bear on the analysis of algorithms for simulating uniform random trees and related questions about large random trees. He uses new constructions of exchangeable systems of interacting particles to create novel genetic models that extend previous ones of the stepping-stone type. He develops tractable and flexible probabilistic structures on the space of phylogenetic trees that will enhance exploratory data analysis and statistical inference in phylogeny. He provides new tools for mathematical demographers in their attempts to model mortality rates and understand the features of such models that give rise to Gompertz exponential increase for most of the lifespan and mortality plateaus in extreme senescence. Random matrices have recently been a focus of intense mathematical activity because of the connections they establish with seemingly disparate areas ranging from the Riemann Hypothesis (probably the outstanding unsolved problem in pure mathematics) to efficient design of cell phone networks. The PI provides new tools for understanding the structure of these objects. An understanding of the nature of the space of trees and ways of picking trees randomly is important in areas ranging from the design of data bases to constructing evolutionary family trees from DNA sequence data. The PI uses new ideas from metric geometry, a seemingly unrelated branch of mathematics, to advance the study of these objects. A current conundrum in demographic research is the fact that, for a vast array of species, mortality increases exponentially with age up to extreme old age, at which point it levels off. The PI shows that this behavior is a robust consequence of a whole general class of underlying models for the ageing process.