Non-positive curvature, geometry, and geometric group theory

从1990年起，几何群论迅速发展，成为现代数学的一个重要领域，与许多其他分支有着深刻的联系和丰富的接口。它提供了大量的新工具，解决了这些领域长期存在的中心问题，一个值得注意的例子是，非正曲立方体复形（或高维）的强大理论被用来解决三维流形的几何和拓扑学的其余主要问题。另一个重要的应用程序集围绕着深刻的见解，几何群论已提供到结构的映射类组的表面和自同构群的自由团体，对象是重要的，在各种各样的数学背景下，和对象的丰富的结构保留许多奥秘，决议将产生广泛的影响几何，数论和数学物理。非正曲率的表现（重铸远远超出其原来的微分几何领域）发挥了核心作用，在许多这些主题。这个研究项目的目的是攻击一系列的问题，在主流的几何群论，涉及方面的非正曲率，利用一些重大突破，近年来，并有潜力在数学的相邻领域中开启重要的应用。成功不仅需要严格理解代数，拓扑和几何等一些具有挑战性的主题：还需要在解释和扩展有关表面及其模量，外层空间和cultural技术的最新进展方面具有很大的新奇。除了推进数学的核心，学生在现代数学的这个中心领域的培训将提高英国在该地区的人力资本，以及其在数学科学基础研究的前沿地位-其经济和社会的重要性已经在政府的产业战略和规划的背景下得到了很好的证明后英国脱欧时代。这属于EPSRC数学科学几何和拓扑研究领域的福尔斯，但也有可能为数字经济议程和网络安全议程做出贡献。

Geometric group theory grew enormously from 1990 onwards to become a major field of modern mathematics with deep interconnections and rich interfaces with many other branches. It has provided a remarkable number of new tools that have led to the resolution of longstanding central problems in those fields, a notable example being the way in which the powerful theory of non-positively curved cube complexes (or high dimensions) was used to solve the remaining major questions about the geometry and topology of 3-dimensional manifolds. A further important set of applications revolves around the deep insights that Geometric Group Theory has provided into the structure of mapping class groups of surfaces and automorphism groups of free groups, objects that are of importance in a wide variety of mathematical contexts, and objects whose rich structures retain many mysteries, the resolution of which would have wide impact in geometry, number theory, and mathematical physics. Manifestations of non-positive curvature (recast far beyond its original realm of differential geometry) play a central role in many of these topics.The aim of this research project is to attack a range of problems in the mainstream of geometric group theory that involve aspects of non-positive curvature, harness some of the great breakthroughs of recent years, and have the potential to unlock significant applications in adjoining fields of mathematics. Success will require not only a rigorous understanding of a number of challenging topics spanning algebra, topology and geometry: it will also require great novelty in interpreting and extending the latest advances concerning surfaces and their moduli, Outer space, and cubulation techniques. Besides advancing the core of mathematics, the training of the student in this central area of modern mathematics will enhance the UK's human capital in the area, and its position at the forefront of fundamental research in the mathematical sciences - a strength whose economic and societal importance has been well documented in the context of the government's industrial strategy and planning for the post-Brexit era. This falls within EPSRC Mathematical Sciences Geometry and Topology research area , but also has the potential to contribute to both the digital economy agenda, and the cybersecurity agenda.