Scheme: Wolfson Research Merit Awards
Organisation: University of Warwick
Dates: May 2014-Apr 2019
Summary: Two systems with the same number of symmetries may nevertheless have qualitatively quite different symmetry. To more accurately measure symmetry mathematicians invented the algebraic concept of a group. Groups now play a central role in all areas of mathematics.
Groups themselves have symmetries, called automorphisms. A group is already an abstract mathematical object, and thinking about its group of automorphisms adds another level of abstraction. The main objective of my research is to bring the study of automorphism groups to a more concrete level by realizing them as
symmetry groups of some new geometric object. This enables one to visualize the automorphism group and to study it using geometric tools.
Paradoxically, the simplest groups often have the most complicated automorphism groups. One of the simplest of all (infinite) groups is called a free group. In early work with Marc Culler I built a geometric model for the automorphism group of a free group. This model has enabled mathematicians to settle many old questions about the group, though plenty of others remain. In addition to continuing to study these questions I am working with Ruth Charney to construct analogous spaces for a wider class of groups called right-angled Artin groups, which have recently played an important role in low-dimensional topology and geometry.
The geometric models we have constructed for automorphism groups turn out to be connected to a wide variety other areas of mathematics and science, from classical number theory to the study of phylogenetic trees.