## 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!

## Doors Open Day

On Saturday 28 September 2013 many interesting buildings in Edinburgh opened their doors to the public for Doors Open Day. One of these was ICMS – the International Centre for Mathematical Sciences – on South College Street. ICMS are in a building that used to be a church and still has a beautiful stained glass window, but now the only worshipping which goes on is for mathematics! Visitors were invited to solve puzzles and play mathematical games, and to draw their favourite maths on this blackboard. What would you have drawn?

## Deformation theory and 3-folds

This photo is of the blackboard in the common room of the School of Mathematics at the University of Edinburgh. The workings on the left are by Michael Wemyss, drawn while he was talking about deformation theory (a generalization of differential calculus) with Will Donovan, as part of their work on the geometry of certain spaces, known as 3-folds. Will says, “Deformation theory lets us express the way in which a curve in a 3-fold can `move’ infinitesimally: we can then relate this to the (quantum) geometry of the 3-fold. I’ve tried to emulate Michael’s calligraphic F’s and G’s, but I haven’t had any success yet.” We don’t know who did the workings on the right.

## Knot concordance and slice movies

This board belongs to University of Glasgow researcher Brendan Owens. The pictures on the right show a ‘slice movie’ which is used to prove that two knots are concordant to each other. The pictures are essentially 3D slices through a 4D disc; the ‘singularity’ where two parts of the knot appear to intersect is an illusion caused by the particular projection.  The long exact sequences on the left are the technical algebraic topology used to explore the crazy world of 4-dimensional knot theory, where geometrical intuition so frequently fails.

## 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!

## Portable whiteboard

This photo sent in by Colin Beveridge is not your usual whiteboard! It’s a foldaway portable whiteboard called a Noteboard. Find out more details at http://thisisawesome.co.uk/. Colin was using this to work on a puzzle from MathsJam: “A chopstick (a thin uniform rod) lies at rest in a smooth, hemispherical bowl. The chopstick is longer than the diameter of the bowl.  What fraction of the chopstick is inside the bowl?”.

## 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!]

## Shmooing with the biologists

I couldn’t resist taking this photo in the office of Edinburgh mathematician Nikola Popovic. The central image is the result of a conversation between Nikola and visiting speaker Vahid Shahrezaei about the reproduction of yeast. Yeast cells normally reproduce asexually by budding, but cells of opposite mating types (a and α) can also reproduce sexually. The bulge which forms when a cell responds to pheromones of the opposite type is called a shmoo – this is a reference to a cartoon character of that shape invented in 1948. It’s a brilliant word which has found its way into other areas of science too, including a type of reproducing material good in economics, a larva found in sea urchins, and high energy cosmic ray survey instrument.

Other parts of the board refer to Nikola’s work on gene regulatory networks, where he is trying to model processes which have both ‘fast’ and ‘slow’ components, and to see how these components interact with each other.

## Fair coin puzzle

This board was sent to me via Twitter by Will Davies (aka @notonlyahatrack) who had listened to the podcast Wrong, But Useful (by Colin Beveridge (@icecolbeveridge) and Dave Gale (@reflectivemaths) ) and attempted the following puzzle:

I have three indistinguishable coins: one always comes up heads, one always comes up tails and the third is a fair coin. I pick one of the coins at random and toss it twice and get the same result both times. What is the probability I picked the fair coin?

You should have a go at the puzzle yourself before looking too closely at the solution 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-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.]