Extremal and probabilistic combinatorics

Lu博士和Szekely博士在他们以前在组合学和图论方面的工作的基础上，继续他们对图形可视化和图形绘制、随机图形模型、与化学相关的树指数、生物分子序列进化模型以及物种进化的系统发育重建等问题的研究。他们还继续他们在极值集合论、极值图论和离散几何的基本问题上的长期项目。他们致力于的具体问题包括*进一步发展图形的交叉数理论和更好的图形绘制算法；*改进超图上Turan型极值问题的当前界，这是极值组合学中最困难的问题之一；*提高我们对性质B的认识，其中众所周知的困难问题为新技术提供了试验场；*扩展组合数学中现有的概率工具，使其适用于新的或更一般的情况；*致力于统一随机图模型；*寻找关于反链和子集族相交的新见解；*更好地理解兰迪克指数，这是一个已被引入并在化学中广泛使用的极值图论概念；*继续研究生物分子序列进化模型和系统发育树重建算法，特别是最大似然法，并在倒置随机函数的背景下研究系统发育问题。人们越来越需要理解离散数学中的“最优”极端结构和“典型”随机结构，因为这种理解通常会导致新的算法。这个项目将研究关于结构的基本组合问题，并将寻找离散数学在计算机科学、生物学和工程学中的各种应用。此外，提出者计划继续与来自工程、生物、统计学和计算机科学的同事进行跨学科合作，并期望解决他们的一些新问题。

Drs. Lu and Szekely build on their previous work in combinatorics and graph theory and continue their research on problems related to graph visualization and graph drawing; random graph models; tree indices relevant for chemistry; models for biomolecular sequence evolution; and phylogeny reconstruction for species evolution. They also continue their long-running projects on fundamental questions in extremal set theory, extremal graph theory, and discrete geometry. The particular problems on which they work include* the further development of the theory of crossing numbers of graphs and of better graph drawing algorithms;* improving current bounds on Turan type extremal problem on hypergraphs, one of the toughest problems in extremal combinatorics;* improving our knowledge on Property B, where notoriously hard problems provide testing ground for new techniques;* extending current probabilistic tools in combinatorics to be applicable in a new or more general scenario;* work towards the unification of random graph models;* searching for new insights on antichains and on intersecting families of subsets;* reach a much better understanding of the Randic index, a concept of extremal graph theory that has been introduced and widely used in chemistry;* continuing the study of models of biomolecular sequence evolution and phylogenetic trees, of phylogenetic tree reconstruction algorithms, in particular the maximum likelihood method, and investigating the phylogeny problem in the setting of inverting random functions.There is increasing demand to understand ``optimal''extreme structures and ``typical'' random structures in discrete mathematics as this understanding often leads to new algorithms This project will investigate basic combinatorial questions about structures and will look for various applications of discrete mathematics in computer science, biology, and engineering.In addition, the proposers plan to continue their interdisciplinary collaborations with colleagues from engineering, biology, statistics, and computer science, and expect to solve some of their new problems.