Extremal combinatorics and asymptotic enumeration

组合数学是研究有限结构的数学的一个分支。这些问题的普遍性表明了组合数学在纯数学的其他领域（最显著的是代数、数论、概率和拓扑学）以及在现实世界中的应用（离散优化、计算机科学）中的广泛适用性。组合数学最古老和最核心的部分之一是图论和枚举组合数学。图论为网络（如道路连接或互联网用户）建模，枚举组合学关注研究各种计数问题。极值图论是图论的一个广泛部分，研究各种图参数之间的相互作用。极值图论中的一个主要工具是所谓的Szemeredi正则性引理。这个工具（在70年代开发）已成为现代数学的基石之一。最近，利用从正则引理中获得的见解，Lovasz和Szegedy发起了对图极限的研究。拟议的研究项目解决了极值图论中的主要开放问题，旨在为正则引理、图极限的一般理论做出贡献，并通过开发将用于枚举组合学的新工具。

Combinatorics is a branch of mathematics studying finite structures. The generality of these questions suggests wide applicability of combinatorics in other areas of pure mathematics (most notably in algebra, number theory, probability, and topology), as well as in real-world applications (discrete optimization, computer science).One of the oldest and most central parts of combinatorics are graph theory and enumerative combinatorics. Graph theory models networks (such as road connections, or internet users), and enumerative combinatorics concerns studying counting questions of various kinds.Extremal graph theory is a broad part of graph theory which investigates interplay between various graph parameters. One of the main tools in Extremal graph theory is the so-called Szemeredi Regularity Lemma. This tool (developed in the 70's) has become one of the corner-stones of modern mathematics. Recently, using the insights gained from the Regularity Lemma, Lovasz and Szegedy initiated study of graph limits.The proposed research project addresses major open questions in extremal graph theory and aims contribute to general theories the Regularity Lemma, graph limits, and by developing novel tools which will be used in enumerative combinatorics.