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Algebra

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!”

Inverse blackboard

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

David Cushing blackboardThis 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

Homological algebra blackboardThis 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

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.]


Borel subalgebras of F4

Jose blackboard July 2012This blackboard was drawn by Jos√© 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 F4, where F4 is one of the five exceptional simple Lie groups.