Enumerative Combinatorics and Probabilistic Method

枚举组合学和概率方法

• 批准号: 0070574
• 负责人: Huafei Yan
• 金额: $ 7.77万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070574&HistoricalAwards=false
• 关键词: Enumerative; Combinatorics; Probabilistic; Method

Enumerative Combinatorics and Probabilistic MethodCatherine Huafei YanPROJECT SUMMARYThe Principal Investigator (PI) will study a series of interrelated problems in enumerative combinatorics and probabilistic method. In recent years deep and unexpected connections have been found between algebraic enumeration and stochastic processes, in particular in the empirical process and in Brownian motion. The primary objective of this research is to explore this connection. The starting point is the probabilistic model of branching processes. On one hand, branching processes encode various combinatorial structures such as rooted trees, parking functions, and multi-colored structures. On the other hand, random graph evolution in the \double jump" can be described by branching processes with the expected family size near one. From the algebraic standpoint, the PI plans to address the combi- natorial applications in branching processes and probabilistic results in combinatorics. She will investigate the behavior of branching processes with near critical offspring distributions by applying well-developed techniques in algebraic enumeration. She will also study the asymptotic properties of certain random structures from the probability theory of branching processes. From the stochastic standpoint, the PI plans to develop stochastic models for the theory of random graphs and random generated trees via branching processes, and to relate such models to the theory of Brownian motion. The emphasis is a combinatorial under- standing of the techniques and results of stochastic processes and calculus. The PI expects to apply the model to graph enumeration, and to investigate the distributive asymptotics of random graphs and other random structures.In addition to the core research program outlined above, the PI intends to pursue three algebraic problems arising from the probabilistic method and extremal combinatorics. The first is a recurrence associated with Turan problems; the second is on discrepancy theory, and the third is on balancing vectors. The objective here is to extend our knowledge of algebraic structures on discrete systems and to develop new approaches in combinatorics.

列举组合和概率方法Catherine Huafei Yanproject摘要首席研究员（PI）将研究一系列列举组合和概率方法中的相互关联问题。 近年来，在代数枚举和随机过程中，尤其是在经验过程和布朗运动中，发现了深厚的意外联系。 这项研究的主要目的是探索这种联系。 起点是分支过程的概率模型。 一方面，分支过程编码各种组合结构，例如根树，停车功能和多色结构。 另一方面，可以通过分支过程来描述\双跳的随机图形进化。从随机的角度出发的某些随机结构的渐进性。为了研究随机图和其他随机结构的分布渐近学。在上面概述的核心研究计划外，PI打算追求由概率方法和极端组合学引起的三个代数问题。 首先是与Turan问题相关的复发。第二个是关于差异理论的，第三个是关于平衡向量的。 这里的目的是扩展我们对离散系统代数结构的了解，并开发组合学中的新方法。