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Group theory

Combinatorics of non-crossing partitions

Iain Gordon whiteboard
The board is a result of of Bin Shu from East China Normal University visiting Iain Gordon at the University of Edinburgh. In Iain’s words: “We were discussing the combinatorics of non-crossing partitions, and its generalisations to a bunch of different finite groups. The two diagrams are Cayley graphs of symmetric groups where one calculates this combinatorics. Most of that text is in black, but we were lazybones, so there is also a bit of green writing from my PhD student, about braid group actions on tensor categories and their asymptotic limits. Amazingly, the two topics are linked, through moduli spaces in algebraic geometry!”

Maximal oriented Wicks forms

Stacey_Aston_blackboardThis board was created by Stacey Aston, a PhD student at Newcastle University. She is doing research in the area of geometric group theory and was trying to work through one of her supervisor’s papers on maximal oriented Wicks forms.

A Wicks form is a word on the fundamental group of a graph on a surface (and it seems that nobody has written a Wikipedia article about it yet!). For genus 1 surfaces, the maximal oriented Wicks form is abc-1a-1b-1c, a word of length 6 (that can be written as a single commutator, which is a fun challenge). The maximal oriented Wicks form of a surface with 2 holes is 18 letters long, and then the word on a genus 4 surface has over 13000 letters!

[Many thanks to Christian Perfect for sending me in this photo and explanation.]

Thompson groups

Nathan Barker - Thompson GroupsToday’s blackboard belongs to Nathan Barker at the University of Newcastle. Nathan works with Thompson groups, which were first studied in 1965 and provide examples of interesting phenomena in group theory.

Top-left is a realisation of the automorphism group of one of Thompson’s groups as a tree. On the right is a subgroup that may or may not have the property Nathan wants, and then Poincare’s disc model of the hyperbolic plane, and an annular strand diagram. The jumpy line drawing at the bottom is the Cayley graph of PSL(2,Z) – the group of all 2×2 integer-valued matrices with determinant 1, where matrices A and -A are considered to be the same.

The bottom-left section is some quantum computation stuff by somebody called Ben that nobody but him understands.

Thanks to Christian Perfect for taking and sending me this photo, and for providing details of what’s on the board.

Readers of this blog may also be interested in an article detailing the relationship between mathematicians and blackboards:

Young symmetrizers…on a tablecloth

Will Donovan - tablecloth mathsWill Donovan temporarily transformed his dining room table in London into a whiteboard using a plastic tablecloth. Here he’s trying to understand “Young symmetrizers“, which correspond to the grids of boxes that he’s sketched. These can be used in the representation theory of symmetric groups, to help us understand the possible types of symmetry that can be associated to objects in space. And they also have interesting algebraic properties all of their own…

To see more of Will’s mathematics, check out his gallery where he has posted many many photos of his changing black/whiteboards over the years.

Cayley graph

Cayley group of F_2Our photo this week is by Christian Perfect, a PhD student at the University of Newcastle. He was drawing a Cayley graph, which is a picture encoding the structure of a group. This particular picture is a drawing of the structure of F2, the free group on two generators. This group is generated by the elements {x,y,x-1, y-1} with x and x-1 being right and left arrows, and y and y-1 being up and down arrows respectively. Each element of the group is then a vertex of the graph. For example, to find the vertex corresponding to xyx2y-1 you start in the middle and go right, up, right, right, down.

This picture can also be interpreted (if you are a topologist!) as the universal cover of the ‘8’, or infinity, symbol.

Christian actually took a series of photos as he was drawing the graph. I’ve put these together into an animated gif for your viewing pleasure (but beware – it’s 30Mb!).