Non-positive curvature, geometry, and geometric group theory

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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.