Research Fellows Directory
Professor Martin Bridson
University of Oxford
My interests span geometry, group theory and topology. I am interested in the nature of symmetry. The symmetries (automorphisms) of any structure form a group, so groups are everywhere in mathematics. Geometry is equally ubiquitous: identifying an appropriate geometry is often the key to unravelling a mass of complicated structure. I am also interested in measures of complexity that lend precision to the idea that problems are hard, and in rigidity phenomena.
A “presentation” is a concise way of describing a group; precisely how concise one can make it is a question I have studied intensely in the last few years. It is provably impossible to extract all of the information a presentation contains, in general, but for groups arising in geometry, the complexity of information-extraction is intimately connected with large scale features of spaces. For example, in the presence of favourable (non-positive) curvature conditions, most algorithmic problems about groups can be solved rapidly. But Henry Wilton and I proved that even in the presence of non-positive curvature, there is no algorithm that can determine which groups can act as symmetries of finite objects. Adaptations of this theorem show that natural decision problems in other areas of mathematics are also unsolvable.
All finite groups are built from a finite number of families of atoms (simple groups). Evans, Liebeck and I, drawing on deep results from other branches of mathematics, have recently identified which of these natural families harbour the Bridson-Wilton undecidability phenomenon and which are more tractable.
With other coauthors, I proved that the groups associated to many 3-dimensional geometries, such as knot complements, are uniquely determined by their actions on finite sets – an instance of “profinite rigidity”, a topic that has occupied a lot of my thought in recent years. In all these endeavours, there is a constant, beautiful interplay of geometry and algebra.