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

PI通过建立随机矩阵理论、随机过程、分析、组合学和数学物理之间的进一步联系，扩展了随机矩阵的跨学科重要性。他将度量几何的最新思想应用于模拟均匀随机树的算法分析和大型随机树的相关问题。他使用相互作用粒子的可交换系统的新结构来创建新的遗传模型，扩展了先前的踏脚石类型的遗传模型。他在系统发育树的空间上开发了易于处理和灵活的概率结构，这将增强系统发育中的探索性数据分析和统计推断。他为试图建立死亡率模型的数学人口统计学家提供了新的工具，并了解这些模型的特征，这些特征导致了大多数寿命的贡珀兹指数增长和极度衰老时的死亡率停滞。随机矩阵最近成为激烈的数学活动的焦点，因为它们与看似不同的领域建立了联系，从黎曼假设（可能是纯数学中未解决的突出问题）到手机网络的有效设计。PI为理解这些对象的结构提供了新的工具。从数据库的设计到从DNA序列数据构建进化家谱，了解树木空间的性质和随机选择树木的方法都是很重要的。PI使用了米制几何（一个看似无关的数学分支）的新思想来推进对这些物体的研究。目前人口统计学研究中的一个难题是，对于大量物种来说，死亡率随着年龄的增长呈指数增长，直到极度衰老，然后趋于平稳。PI表明，这种行为是整个老化过程的一般基础模型的可靠结果。

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.