Non-positive curvature, geometry, and geometric group theory

几何群论从1990年开始迅速发展，成为现代数学的一个主要领域，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，这些工具解决了这些领域长期存在的核心问题，一个值得注意的例子是，非正弯曲立方体复合体（或高维）的强大理论被用来解决关于三维流形的几何和拓扑的剩余主要问题。另一组重要的应用围绕着几何群论对映射类曲面群和自由群的自同构群的结构提供的深刻见解，这些对象在各种数学环境中都很重要，其丰富的结构保留了许多奥秘，其解决将对几何，数论和数学物理产生广泛的影响。非正曲率的表现（重塑远远超出其原始领域的微分几何）在许多这些主题中起着核心作用。本研究项目的目的是研究几何群论中涉及非正曲率的主流问题，利用近年来的一些重大突破，并有可能在相邻的数学领域开启重大应用。成功不仅需要对代数、拓扑和几何等一系列具有挑战性的主题有严格的理解，还需要在解释和扩展有关曲面及其模、外空间和计算技术的最新进展方面有很大的新颖性。除了推进数学的核心之外，在现代数学的核心领域对学生的培训将提高英国在该领域的人力资本，并提高其在数学科学基础研究的前沿地位——在政府的工业战略和后英国脱欧时代的规划背景下，这种力量的经济和社会重要性已经得到了很好的证明。这属于EPSRC数学科学几何和拓扑研究领域，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.