Evolving Combinatorial Structures

该项目研究进化组合结构的概率模型，特别是划分、树和图值随机过程。要研究的具体课题包括组合马尔可夫过程的表示和刻画定理，连续统树和区间图的比例极限，一致的划分系统，树和图值过程，以及与随机矩阵和Levy过程的联系。这项研究的主要主题将是概率对称性，特别是交换性，对进化中的大型组合对象的结构属性的影响，因为这些结构属性影响这些过程的各个实际方面。作为这个项目的结果，我们应该进一步了解时变离散结构的模型，特别是划分、树和网络。这样的过程出现在不同学科的自然模型中，包括遗传学、物理学、生物学、计算机科学和统计学。特别是，理解图值过程在复杂网络的多样化和新兴领域具有潜在的深远应用。现实世界网络的有效模型与国家安全、公共卫生、社会学、计算机科学和物理科学等领域的问题相关。组合模型可能有用的其他领域包括系统发生学、机器学习、统计学和贝叶斯推理。

The project studies probability models for evolving combinatorial structures, particularly partition, tree and graph-valued stochastic processes. Specific topics to be studied include representation and characterization theorems of combinatorial Markov processes, continuum tree and interval graph scaling limits, consistent systems of partition, tree and graph-valued processes, and connections to random matrices and Levy processes. The dominant theme of the research will be the effect of probabilistic symmetries, especially exchangeability, on the structural properties of evolving large combinatorial objects, as these structural properties impact various practical aspects of these processes. As a result of this project, we should gain further understanding of models for time-varying discrete structures, especially partitions, trees and networks. Such processes arise as natural models in various disciplines, including genetics, physics, biology, computer science and statistics. In particular, understanding graph-valued processes has potentially far-reaching applications in the diverse and burgeoning field of complex networks. Effective models for real-world networks are relevant to problems in national security, public health, sociology, computer science and physical sciences. Other areas in which combinatorial models can be useful include phylogenetics, machine learning, statistics and Bayesian inference.