Research on Enumerative and Probabilistic Combinatorics

枚举与概率组合学研究

• 批准号: 0245526
• 负责人: Huafei Yan
• 金额: $ 12.49万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245526&HistoricalAwards=false
• 关键词: Research; Enumerative; Probabilistic; Combinatorics

Abstract for Yan DMS-0245526The proposed project focuses on a series of interrelated problems in enumerative and probabilistic combinatorics. In recent years deep and unexpected connections have been seen between algebraic combinatorics and probability theory, in particular, stochastic processes. It is the primary objective of the investigator to explore such connections and to develop new methods and techniques based on algebraic-combinatorial principles, as well as analytic methods of probability and random processes. The project focuses on four main areas. The first is to apply the theory of branching processes and stochastic analysis to the enumeration of combinatorial objects, and to analyze the distributive asymptotics for families of random structures, including random trees, random forests, random sequences, and various random walks. The second is to investigate the enumeration of statistics with an Airy type oflimit distribution. The third is to study the algebraic properties and the applications of sequences of polynomials in enumerative combinatorics, which would extend the classical theory of binomial enumeration. Finally, several problems on geometric random graphs and various combinatorial games will also be investigated.This research is in the general area of combinatorics. One of the goals of combinatorics is to find efficient methods for arranging, enumerating, and manipulating discrete collections of objects.The proposed research addresses these goals with a combined algebraic and probabilistic approach. There are far-reaching applicationsto other areas of pure mathematics, including algebra, analysis, number theory, statistics, and topology, as well as to areas of applied sciences such as computer science, electrical engineering, and management science. Continuing research in combinatorics and its applications will contribute advances in bioinfomatics, internet traffic routing, network communications, and operations research, which would bring significant benefits to industry and society.

YAN DMS-0245526的摘要拟议项目着重于一系列相互关联的列举和概率组合问题。近年来，在代数组合学和概率理论（尤其是随机过程）之间已经看到了深层和意外的联系。研究人员的主要目的是探索基于代数组合原理的新方法和技术，以及概率和随机过程的分析方法。该项目着重于四个主要领域。首先是将分支过程的理论和随机分析的理论应用于组合对象的列举，并分析随机结构家族的分布渐近学，包括随机树，随机森林，随机序列，随机序列和各种随机步行。第二个是通过通风类型的Limit分配调查统计数据的列举。 第三个是研究多项式组合中多项式序列的代数特性和序列的应用，这将扩大二项式枚举的经典理论。最后，还将研究有关几何随机图和各种组合游戏的几个问题。这项研究在组合学的一般领域。组合主义者的目标之一是找到有效的方法来安排，枚举和操纵对象的离散集合。拟议的研究以组合代数和概率的方法来解决这些目标。纯数学的其他领域有深远的应用，包括代数，分析，数量理论，统计和拓扑，以及应用科学，计算机科学，电气工程和管理科学等应用科学领域。对组合技术及其应用的继续研究将在生物信息学，互联网流量路由，网络通信和运营研究方面取得进步，这将为工业和社会带来巨大的好处。