Research on Enumerative and Probabilistic Combinatorics

本课题主要研究枚举组合学和概率组合学中一系列相互关联的问题。近年来，代数组合学与概率论，特别是随机过程之间出现了深刻而意想不到的联系。研究者的主要目标是探索这种联系，并基于代数组合原理开发新的方法和技术，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。首先是将分支过程理论和随机分析理论应用于组合对象的枚举，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漫步。二是研究具有Airy型极限分布的统计量枚举问题。三是研究多项式序列的代数性质及其在枚举组合中的应用，拓展经典的二项式枚举理论。最后，还将讨论几何随机图和各种组合对策的若干问题。这项研究属于组合学的一般领域。组合学的目标之一是找到排列、枚举和操作离散对象集合的有效方法。提出的研究解决了这些目标，结合了代数和概率的方法。纯数学的其他领域，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，都有深远的应用。组合学及其应用的持续研究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。

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.