Discrete Potential Theory and Applications

这个项目致力于纯数学的基础研究。它的基础是对电子网络理论（被视为抽象的数学工具）和图上随机游走理论之间的相互作用进行了深入研究。该项目解决了该框架内的四个特定主题：—图和群上随机游走的泊松边界；我们遵循由Kesten引发的群论的积极传统，通过考虑在群体上随机行走并将其行为或与之相关的边界结构与群体的代数性质联系起来，间接获得关于群体的结果。该项目受益于申请人最近提供泊松边界标准的强有力结果，以及将随机有限图而不是随机漫步与群体联系起来的新想法，利用Lovasz等人最近的图论——本杰明和施拉姆意义上的离散保形均匀化；我们寻求加强申请人的新结果，与上述相关，回答了本杰明和施拉姆的一个问题。这种强化将提供泊松边界上的新结果。——极值有限图与随机有限图中覆盖时间与覆盖成本的关系。图的覆盖时间是数学和计算机科学中的一个重要概念，甚至被物理学家研究过，但它很难计算，甚至很难近似。使用申请人介绍的覆盖成本的概念，我们试图通过将其分解为两个步骤来简化许多类图的覆盖时间的近似：显示它接近覆盖成本，并计算（可证明更易于处理的）覆盖成本。这些主题位于数学的不同领域，近年来都有大量的研究活动。它们通过电子网络、随机漫步及其相互作用的总体主题相互联系，并共享更精细的相互联系。该项目旨在通过为每个子主题单独产生新的结果以及建立或加强它们之间的联系来作出贡献。该项目的结果将引起几个研究团体的兴趣，包括图论、概率论、（离散）势论和群论。

The project is devoted to basic research in pure mathematics. It is based on the well-studied interplay between the theory of electrical networks -seen as abstract mathematical tools- and the theory of random walks on graphs. Four specific topics within this framework are addressed by the project:-- The Poisson boundary for random walk on graphs and groups;We follow an active tradition in group theory, triggered by Kesten, where results about groups are obtained indirectly by considering a random walk on the group and relating its behaviour, or the structure of a boundary associated to it, to the algebraic properties of the group. The project benefits from a recent strong result of the applicant providing a criterion for the Poisson boundary, as well as a novel idea of associating a random finite graph rather than a random walk with a group, exploiting the recent theory of graphons by Lovasz et. al. -- Discrete conformal uniformization in the sense of Benjamini & Schramm;We seek to strengthen a new result of the applicant, related to the above, that answered a question of Benjamini & Schramm. Such a strengthening will provide new results on the Poisson boundary.-- The relationship between the cover time and the cover cost in extremal and random finite graphs.The cover time of a graph is an important concept in mathematics and computer science, and is even studied by physicists, but it is very hard to compute or even approximate. Using the concept of cover cost that the applicant introduced, we seek to simplify the approximation of the cover time for many classes of graphs by breaking it down into two steps: showing that it is close to the cover cost, and computing the (provably more tractable) cover cost.These topics lie in different areas of mathematics, all of which have seen a lot of research activity in recent years. They are interlinked by the general theme of electrical networks, random walks, and their interplay, and share further finer interconnections. The project aims to contribute by producing new results individually for each sub-topic as well as by establishing or strengthening connections between them. The project's results will be of interest to several research communities, including Graph Theory, Probability, (discrete) Potential Theory and Group Theory.