Coarse Geometry of Groups and Spaces

“群”是几何对象的对称性族。本研究的主题-几何群论-旨在揭示对象的几何属性与其对称群的代数属性之间的联系。从一开始，群论就受到数学及其他领域应用的驱动，包括：伽罗瓦关于多项式根的原始理论，将物理系统的对称性与其守恒定律联系起来的诺特定理，物理学中的Lie和Kac-Moody群，化学和材料科学中的晶体学，以及公钥密码学。群论与计算机科学有着动态的相互作用，特别是在人工智能和机器学习的前沿领域;并塑造了我们对拓扑空间，几何和数论的理解。许多自然几何性质（如增长、维数和曲率）的代数结果已被深入研究。举两个例子，Gromov著名的多项式增长定理可以完全代数地解释多项式增长的群，有许多指数增长的群族。最近，人们发现了具有“中等增长”的群体的例子，这些群体非常神秘。另一个格罗莫夫的显着贡献是表明，一个随机选择的群体几乎肯定是双曲线：这种方法的一个弱点是，许多这些几何性质只能从高度对称的几何对象传递到它们的对称的几乎完全族-形式上，群和几何是拟等距的，这意味着它们在足够大的尺度上看起来“相同”。为了研究更一般的关系，我们需要允许组以高度扭曲的方式位于几何对象内部的可能性。在几何群论中已知的众多不变量中，很少有不变量在这种更一般的情况下表现得足够好，以给出富有成效的结果。我的建议涉及一个新的称为庞加莱轮廓的抗扭曲（粗糙）不变量家族，我最近介绍了它。庞加莱轮廓本质上衡量几何空间的连接部分在各种不同尺度上的鲁棒性。我已经建立了一个双曲群的庞加莱轮廓和与该群相关的分形的（共形）维数之间的联系。分形是复杂的形状，在越来越小的尺度上表现出自相似性，并且可以有许多不同但合理的方法来测量其维度-其中没有一个一定是整数。分形的一个来源是来自双曲群的边界：从“无限远”看到的群的可视化。我的建议的一个关键目标是准确地揭示这种关系，以提高我们对双曲群和分形的理解。几何群论中的许多结构结果很可能有自然粗糙的类似物，如果人们能找到正确的不变量。我有许多想法的问题，可以处理使用新的不变量，我将定义的启发工具，从分析，代数拓扑，组合，计算机科学和理论物理。这项工作也有许多自然的应用，因为寻找和量化网络中连接良好的部分是广告算法、几何深度学习、蛋白质相互作用建模和图神经网络的共同目标。这些技术的持续发展和改进具有工业和社会效益，从改进的财务预测和更好的3D面部和语音识别，到更准确和有效的药物设计和复合材料设计和测试。

A "group" is a family of symmetries of a geometric object. The subject of this research - geometric group theory - seeks to reveal the connections between the geometric properties of the object and the algebraic properties of its group of symmetries.From the very beginning, group theory has been driven by its applications in mathematics and beyond, including: Galois' original theory on the roots of polynomials, Noether's theorem relating symmetries of a physical system to its' conservation laws, Lie and Kac-Moody groups in physics, crystallography in chemistry and material science, and public-key cryptography. Group theory enjoys dynamic interactions with computer science, particularly in the advancing fields of AI and machine learning; and shapes our understanding of topological spaces, geometry and number theory. Many natural geometric properties (for example growth, dimension and curvature) have been intensively studied for their algebraic consequences. To give two examples, groups with polynomial growth are completely algebraically explained by Gromov's remarkable Polynomial Growth Theorem, there are many families of groups with exponential growth. Much more recently, examples of groups with "intermediate growth" were discovered, these groups are deeply mysterious. Another of Gromov's remarkable contributions is to show that a randomly chosen group is almost surely hyperbolic: that is, it is the group of symmetries of a geometric space with negative curvature.A weakness of this approach is that many of these geometric properties only pass from highly symmetric geometric objects to nearly complete families of their symmetries - formally the group and the geometry are quasi-isometric, meaning they appear "the same" at sufficiently large scales. To study the more general relationship we need to allow the possibility that the group sits inside the geometric object in a highly-distorted way. Of the multitude of invariants known to geometric group theory, very few behave sufficiently well in this more general setting to give productive results.My proposal concerns a new family of distortion-proof (coarse) invariants called Poincaré profiles, which I recently introduced. Poincaré profiles essentially measure how robustly connected parts of a geometric space can be on a variety of different scales. I have established that there is a connection between the Poincaré profiles of a hyperbolic group and the (conformal) dimension of a fractal associated to the group. Fractals are intricate shapes which exhibit self-similarity at increasingly small scales and there can be many different yet sensible ways to measure their dimension - none of which is necessarily an integer. One source of fractals is from boundaries of hyperbolic groups: visualisations of that group as seen "from infinity". A key goal of my proposal is to exactly reveal this relationship to improve our understanding of both hyperbolic groups and fractals.More generally, there is a great need for further coarse invariants. Many structural results in geometric group theory are likely to have natural coarse analogues if one can find the right invariants. I have many ideas of problems which can be dealt with using new invariants I will define inspired by tools from analysis, algebraic topology, combinatorics, computer science and theoretical physics. There are also many natural applications of this work, since finding and quantifying well-connected parts of a network is a common goal in advertising algorithms, geometric deep learning, protein interaction modelling and graph neural networks. The continued development and improvement of these techniques has industrial and societal benefits ranging from improved financial forecasting and better 3D facial and speech recognition, to more accurate and efficient drug design and composite material design and testing.