## Combinatorics of non-crossing partitions

## Inverse blackboard

Christian Perfect and David Cushing spent the morning computing inverses using their special inverse blackboard.

Going left-to-right and top-to-bottom, we have:

- The inverse of a 2×2 matrix;
- The commutative diagram for the universal property of the inverse limit;
- A variable which is inversely proportional to another;
- The inverse of a complex number;
- The derivative of the inverse of a function;
- The inverse tangent of 1;
- The axiom for group inverses;
- Part of the definition of the logical inverse.

Thanks Christian and David for this wonderful idea!

## Cubic derivation

This photo is from PhD student David Cushing from Newcastle University. He was deriving the formula for solving a cubic equation, and says it looks messy because it was done ‘live’! The history of the solution to the cubic is a very interesting one, with all sorts of mathematical duels and underhanded dealings by 16th century mathematicians: Tartaglia, Cardano and Ferrari. You can read one account of it here: http://www.storyofmathematics.com/16th_tartaglia.html. The solution to the cubic was actually what started the development of complex numbers!

## Kernels and cokernels

This board belongs to Newcastle University PhD student Tom Fisher, who is doing research in homological algebra. He was proving that the category of chain complexes over a ring is abelian; this bit is concerned with showing that it has kernels and cokernels.

[Thank you to Christian Perfect for sending in the photo!]

## Maximal oriented Wicks forms

This 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 ^{-1}a^{-1}b^{-1}c*, 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.]

## Borel subalgebras of F4

This blackboard was drawn by José Figueroa-O’Farrill and Paul de Medeiros at the Isaac Newton Institute for Mathematical Sciences in Cambridge, as part of their research in mathematical physics. It is a diagram of abelian ideals of a Borel subalgebra of F_{4}, where F_{4} is one of the five exceptional simple Lie groups.