Enumerative Combinatorics and Probabilistic Method

枚举组合学和概率方法凯瑟琳·华飞·严普罗杰首席研究员(PI)将研究枚举组合学和概率方法中一系列相关的问题。近年来，在代数计数和随机过程之间发现了深层次和意想不到的联系，特别是在经验过程和布朗运动中。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，分枝过程编码了各种组合结构，如有根树、停车功能和多种颜色的结构。另一方面，“双跳”中的随机图演化可以用期望家族大小在1附近的分支过程来描述。从代数的观点来看，PI计划解决组合数学中分支过程和概率结果的组合应用。她将通过在代数计数中应用成熟的技术来研究具有接近临界子代分布的分支过程的行为。她还将从分支过程的概率理论研究某些随机结构的渐近性质。从随机的观点来看，PI计划通过分支过程为随机图和随机生成树的理论建立随机模型，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希望将该模型应用于图的计数，并研究随机图和其他随机结构的分布渐近性。除了上面概述的核心研究计划外，PI还打算研究由概率方法和极值组合产生的三个代数问题。第一个是与图兰问题相关的递推；第二个是关于偏差理论，第三个是关于平衡向量。这里的目的是扩展我们对离散系统上的代数结构的知识，并在组合学中发展新的方法。

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.