Coarse Geometry of Groups and Spaces

群和空间的粗略几何

• 批准号: EP/V027360/2
• 负责人: David Hume
• 金额: $ 58.52万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV027360%2F2
• 关键词: Coarse; Geometry; Groups; Spaces

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.

“群”是几何对象的对称族，本研究的主题 - 几何群论 - 旨在揭示对象的几何性质与其对称群的代数性质之间的联系。 ，群论受到其在数学及其他领域的应用的推动，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、李和卡克-穆迪物理学、化学和材料科学中的晶体学以及公钥密码学中的群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先进领域；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（例如增长、维数和曲率）的代数后果都得到了深入研究，举两个例子，具有多项式增长的群可以通过格罗莫夫卓越的多项式增长定理完全代数地解释。最近发现了许多具有指数增长的群的例子，这些群非常神秘，它表明随机选择的群可能几乎是双曲线的。 ，它是具有负曲率的几何空间的对称群。这种方法的一个弱点是，许多几何属性只能从高度对称的几何对象传递到其对称性的近乎完整的族 - 形式上，群和几何是准等距，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，我们需要考虑到该群以高度扭曲的方式存在于几何对象内部的可能性。几何群理论中，很少有人在这种更一般的环境中表现得足够好以产生有效的结果。我的建议涉及一个新的防扭曲（粗略）不变量系列，称为庞加莱轮廓，我最近引入了庞加莱轮廓，它本质上衡量了连接部件的鲁棒程度。我已经确定，双曲群的庞加莱轮廓和与该群相关的分形的（共形）维数之间存在联系。分形是表现出自身的复杂形状。 -在越来越小的尺度上的相似性，并且可以有许多不同但合理的方法来测量它们的维度 - 分形的一个来源不一定是整数：双曲群的边界：该群的可视化“从”看到的。无穷大”。我的提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。更一般地说，非常需要进一步的粗不变量。几何群论中的许多结构结果可能具有如果能够找到正确的不变量，我有很多可以使用新的不变量来处理的问题的想法，我将在分析、代数拓扑、组合学、计算机科学和理论物理学的工具的启发下定义这些问题。这项工作还有许多自然应用，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已经在工业界和社会中得到广泛应用。社会效益包括改进的财务预测和更好的 3D 面部和语音识别，以及更准确和高效的药物设计以及复合材料设计和测试。