Research on Enumerative and Probabilistic Combinatorics

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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.