Coarse Geometry of Groups and Spaces

群和空间的粗略几何

基本信息

批准号：
EP/V027360/1
负责人：
David Hume
金额：
$ 104.53万
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV027360%2F1
关键词：
Coarse Geometry Groups Spaces

项目摘要

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.

“组”是几何对象的对称性家族。 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群体，化学和材料科学的晶体学以及公钥密码学。小组理论享受与计算机科学的动态互动，尤其是在AI和机器学习的前进领域中；并塑造了我们对拓扑空间，几何学和数字理论的理解。许多天然的几何特性（例如生长，维度和曲率）已被深入研究其代数后果。给出两个例子，多项式增长的群体完全由格罗莫夫的显着多项式生长定理来完全解释，有许多具有指数增长的群体家庭。最近，发现了“中间增长”的群体的例子，这些群体非常神秘。格罗莫夫（Gromov）的另一项巨大贡献是表明，一个随机选择的群体几乎可以肯定是多功能的：也就是说，它是一个具有负弯曲的几何空间的对称性。这种方法的弱点是，这些几何特性中的许多弱点仅从高度对称的几何对象中传给了他们几乎完全含义的对称性的含义 - 众多的含义是Quasii quasii quasii quasii quasiimii''秤。为了研究更普遍的关系，我们需要以高度延伸的方式允许小组坐在几何对象内部的可能性。在几何群体理论所知的众多不变性中，在这个更一般的环境中，很少有能力得到富有成效的结果。 Poincaré概况基本上衡量了几何空间的牢固连接部分如何在各种不同的尺度上。已经确定双曲线群的庞加莱概况与与该组相关的分形的（共形）维度之间存在联系。分形是错综复杂的形状，在越来越小的尺度上暴露了自相似性，并且可以有许多不同但明智的方法来测量其尺寸 - 这不是必需的整数。分形的一个来源是双曲线群的边界：该组的可视化，如“来自无穷大”。我的提议的一个关键目标是精确揭示这种关系，以提高我们对双曲线组和分形的理解。通常，非常需要进一步的粗糙不变性。如果可以找到正确的不变性，几何群体理论中的许多结构结果可能具有天然的粗略类比。我有许多问题的想法，可以使用新的不变剂来处理，我将根据分析，代数拓扑，组合学，计算机科学和理论物理学来定义的启发。这项工作也有许多自然应用，因为找到和量化网络中良好连接的部分是广告算法，几何深度学习，蛋白质相互作用建模和图形中性网络的共同目标。这些技术的持续发展和改进具有工业和社会利益，从改善财务预测以及更好的3D面部和语音识别到更准确，更有效的药物设计以及复合材料的设计和测试。