Enumerative Combinatorics and Probabilistic Method

项目简介项目负责人（PI）将研究枚举组合学和概率方法中一系列相互关联的问题。近年来，人们在代数枚举和随机过程之间，特别是在经验过程和布朗运动中，发现了深刻而意想不到的联系。本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，如有根树、停车函数和多色结构。另一方面，“双跳”中的随机图演化可以用期望族数接近1的分支过程来描述。从代数的角度来看，PI计划解决分支过程中的组合自然应用和组合学中的概率结果。她将通过在代数枚举中应用成熟的技术来研究具有近临界后代分布的分支过程的行为。她也将从分支过程的概率论研究某些随机结构的渐近性质。从随机的角度来看，PI计划开发随机图理论和通过分支过程随机生成树的随机模型，并将这些模型与布朗运动理论联系起来。重点是对随机过程和微积分的技术和结果的组合理解。PI期望将该模型应用于图枚举，并研究随机图和其他随机结构的分布渐近性。除了上述核心研究计划外，PI还打算研究由概率方法和极值组合引起的三个代数问题。第一个是与图兰问题相关的递归；第二个是关于差异理论，第三个是关于平衡向量。这里的目标是扩展我们对离散系统代数结构的知识，并开发组合学的新方法。

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.