Geometric Group Theory, Random Graphs, and Isoperimetry

非正曲率是对快速增长的研究，例如人口爆炸，珊瑚礁的形状以及现实世界网络中的连接（例如，大脑的神经解剖学和互联网的结构）。该项目涉及几个不同的领域，每个领域都与几何群论有关，PI将开发和应用新的工具和技术来理解具有非正曲率的各个方面的空间。该项目的一个方面将涉及新的技术，用于研究广泛的空间是“层次双曲”，这些空间是由PI与M。哈根和A. Sisto和推广格罗莫夫的高度成功的概念，一个“三角形双曲空间”到一个更广泛的背景。PI打算进一步开发这一框架的一些应用，包括一个涉及一种新的方法来建造墙壁。该项目的另一部分将通过PI和他的合作者一直在开发的技术来研究“随机图”。随机图具有有趣的理论和应用兴趣;一个目标是解决渗流理论中的一个特定猜想。本项目研究几何群论及其与周围区域的相互作用。所研究的群体本身和建立更一般类别的群体（例如，CAT（0）群，外自同构群，相对双曲群）。 该项目的三个主要方面是：继续发展随机图中的Erdos-Renyi式“阈值定理”（与V. Falgas-Ravry和T. Susse）;研究群体中的高等等周函数（与C. Drutu）;和，研究代数和几何性质的分层双曲群和空间（与M。哈根和A. Sisto）。在该项目中，PI将致力于建立模型，用于研究广泛的群体和空间，同时加强几何群论与周围领域（包括组合学和几何测度理论）之间的相互作用。

Non-positive curvature is the study of rapid growth, such as that occurring in population explosions, the shape of coral reefs, and connections in real-world networks (e.g., neural-anatomy of the brain and the structure of the internet). This project involves several distinct areas, each with connections to Geometric Group Theory, in which the PI will develop and apply new tools and techniques for understanding spaces with various aspects of non-positive curvature. One aspect of the project will involve new techniques for studying a wide range of spaces that are "hierarchically hyperbolic"; these spaces were introduced by the PI together with M. Hagen and A. Sisto and generalize Gromov's highly successfully notion of a "delta-hyperbolic space" to a much broader context. The PI intends to further develop this framework with a number of applications, including one involving a new method for building walls. Another part of the project will study "random graphs" through techniques which the PI and his collaborators have been developing. Random graphs have both interesting theoretical and applied interest; one goal is to resolve a particular conjecture in percolation theory.This project studies Geometric Group Theory and its interactions with surrounding areas. The groups studied are interesting both by themselves and in establishing a paradigm for more general classes of groups (e.g., CAT(0) groups, outer automorphism groups, relatively hyperbolic groups). The three main aspects of this project are to: continue developing Erdos-Renyi-style "threshold theorems'' in random graphs (joint with V. Falgas-Ravry and T. Susse); study higher-isoperimetric functions in groups (joint with C. Drutu); and, study algebraic and geometric properties of hierarchically hyperbolic groups and spaces (joint with M. Hagen and A. Sisto). In the project the PI will work to establish models for studying broad classes of groups and spaces while also strengthening the interplay between geometric group theory and surrounding fields including combinatorics and geometric measure theory.