Abstract Beauty

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Professor Jan Saxl of the University of Cambridge was fascinated by the abstraction of group theory when he first studied it at university, and has been working in the field ever since.

Groups are algebraic structures made of a set and an operation that combines any two of its elements to form a third element. The presence of groups in many areas of mathematics and other disciplines makes them a central organising principle. 
“By solving an abstract problem you can solve different questions in different subjects all at once. Abstraction a
llows you to do things you can’t do otherwise.”

“Octagonal coins have symmetry group D8. Theoretical chemists can explore what shapes molecules may have, using groups of symmetries. The graphs in which I am interested have a large group of symmetries, and are best studied using this group.” 

With others, Saxl has worked to classify the maximal subgroups of almost simple groups. “Simple groups are the building blocks of finite groups. They’re similar to primes in that all the finite groups decompose into chains of simple groups.”

He is studying maximal subgroups: “my groups invariably act on some set, whether the vertices of the coin or atoms of molecules. Subgroups are substructures; maximal are the largest possible substructures.”

He is also working on a big project about distance transitive graphs. “These are highly symmetric graphs, which have a very large symmetry group acting on them.” During his visit he was working with Professor Eamonn O’Brien at the University of Auckland “on an enormous group acting nicely on a very large graph. A graph for me is really a group acting on that graph. This graph has about 228 vertices, so it is impossible to draw. Eamonn can work with a group of size 430 on a computer.”

“The group is much larger, but more manageable. The Hoffman-Singleton graph (right) is complicated, but some aspects are much clearer within its group of symmetries, which is manageable despite its size.”

“Abstraction excites me; it is beautiful. Lots of people can enjoy the beauty of music, but not so many people can understand the beauty of abstract mathematics. It is a great pleasure seeing clever students learning to see its beauty.”

Saxl was in New Zealand in 2011 as a visiting Maclaurin Fellow.