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的摘要该项目的重点是枚举和概率组合学中一系列相互关联的问题。近年来，代数组合学和概率论，特别是随机过程之间出现了深刻而意想不到的联系。这是调查的主要目标，探索这种联系，并开发新的方法和技术的基础上代数组合的原则，以及概率和随机过程的分析方法。该项目侧重于四个主要领域。第一个是应用分支过程和随机分析的理论来计数组合对象，并分析随机结构族的分布渐近性，包括随机树，随机森林，随机序列和各种随机游动。第二部分研究了具有Airy型极限分布的统计量的计数问题。 三是研究了多项式序列的代数性质及其在计数组合学中的应用，推广了经典的二项式计数理论。最后，我们也将探讨几何随机图与各种组合游戏的一些问题，这些研究属于组合学的一般范畴。组合数学的目标之一是找到有效的方法来安排，枚举和操纵离散的对象集合。拟议的研究与组合代数和概率的方法来解决这些目标。有深远的应用到其他领域的纯数学，包括代数，分析，数论，统计学和拓扑学，以及应用科学领域，如计算机科学，电气工程和管理科学。继续研究组合学及其应用将有助于生物信息学，互联网流量路由，网络通信和运筹学的进步，这将为工业和社会带来重大利益。