Extremal Graph Theory and Bootstrap Percolation

摘要奖的BENGINEERDMS-0302804建议的研究领域是极值图论和自举渗流。它们之间的距离并不遥远，因为第一种方法使用了许多概率工具，而第二种方法则需要许多组合思想。计算机科学的需要和离散模型在应用中扮演越来越重要的角色的需求，增加了极值图论的重要性，并提出了算法的观点。大约四十年来，渗流理论一直是概率论、组合学和物理学的一个活跃的研究领域。对标准渗流的各个方面的兴趣仍然很高，包括临界概率的估计。最近，人们研究了越来越多的标准渗流模型的变体，特别是， 这个过程家族被称为自举渗透。最近的应用来自不同的方面，例如时空动力系统。 物理学家的计算机实验已经揭示了一些有趣的非平凡的大尺度行为，并且已经证明了一些模型的许多深层次的数学结果。提出者的目标是研究临界概率下的渗流过程。提出者的工作是图兰定理在几个方向上的推广。一个方向是描述不包含某些导出子图的图族。二是研究超图，特别是三元系上的Turan型问题，并发展正则性和稳定性定理等通用工具。Bootstrap渗流是随机元胞自动机家族的一员，是图上的一个过程，其中每个点以一定的概率开或闭，并且这些状态随时间而变化。研究Bootstrap渗流，提出者的主要目的是描述相变、估计临界概率以及临界概率周围的窗口大小。我们的计划是证明转变是尖锐的，并研究不同的模型，其理解将有助于应用。

Abstract for award of Balog DMS-0302804The proposed research areas are extremal graph theory and bootstrap percolation. They are not far from each other, as many probabilistic tools are used in the first one, and many combinatorial ideas are needed in the second one. The need of computer science and demands from applications where discrete models play more and more important roles, increase the importance of extremal graph theory and suggests an algorithmic point of view. For about forty years now, percolation theory has been an active area of research at the interface of probability theory, combinatorics and physics. Interest in various aspects of standard percolation remains high, including estimates of critical probabilities. Lately more and more variants of the standard percolation models have been studied, in particular, the family of processes known as bootstrap percolation. Recent applications arise from different aspects, for example from spatio-temporal dynamical systems. Computer experiments performed by physicists have suggested interesting non-trivial large-scale behavior, and many deep mathematical results have been proved about a number of models.The proposer is aiming to study the percolation process at the critical probability.The work of the proposer is an extension of Turan's Theorem into several directions. One direction is to describe graph families which do not contain certain induced subgraphs. The other is to study Turan type of questions on hypergraphs, in particular on triple systems, and to develop general tools like regularity and stability theorems.Bootstrap percolation, a member of the family of random cellular automata, is a process on graphs, where each site is open or closed with a certain probability, and these states are changing with time.Studying bootstrap percolation, the main aim of the proposer is to describe the phase transition, estimate the critical probability, and the size of the window around the critical probability. The plan is to prove that the transitions are sharp, and to investigate different models, whose understanding would be helpful in the applications.