## Accidental pies

This is the blackboard from the University of Edinburgh School of Mathematics common room from September 2015. I particularly noticed it because of the inadvertent appearance of P(ies) in the centre! I’m also enjoying the idea of pulsational energy.

## Don Zagier on modular forms

Théophile Cantelobre, a student at the Lycée Louis-le-Grand, sent me this photo from a summer school he’s been attending in Bremen. He writes:

“Yesterday night, we had an improvised lecture by Don Zagier introducing us to modular forms. Ben Heuer, a PhD student in number theory in London, continued on to finish an explanation started earlier about the connections between elliptic curves, Galois representation and modular forms and notably the implications towards Fermat’s Last Theorem.

A friend of mine (Richard) took a picture of the board… it’s a special one: a lecture given by Don Zagier with John Conway in the audience!”

It certainly looks like an exciting and dynamic lecture judging by the board!

## Shape optimization

I found this board at the International Centre for Mathematical Sciences (ICMS) during a conference about shape optimisation and shape geometry. Talking to some of the delegates there I found out that the field is all about finding the best shapes to suit a purpose, and these purposes might come from physics, engineering, architecture, or simply pure mathematics. For example, the Reuleaux triangle is a shape of constant width (it has the same diameter wherever you measure it) but has an area 12% less than a circle of the same diameter. This makes it more efficient for making coins (our 50p piece is a similar shape but with 7 sides), manhole covers and even buildings.

I asked conference delegate Jimmy Lamboley what his favourite shape was, and he laughed and said “Anything but a sphere!”. He explained: ” The sphere is so often the answer to minimisation problems that I love to find the problems where it isn’t the case.”

The Reuleaux triangle is a great example of a shape more efficient than a circle. What’s amazing is that the corresponding question for 3D shapes (which shape of constant width has minimum volume?) is a problem still waiting for a solution.

## Fractal decorations

I found this blackboard in the maths common room of Lafayette College, a beautiful old campus university near the small town of Easton (just north of Philadelphia in the US). While the board contains some nice mathematics, I was particularly taken by the psychedelic fractal border on top of the board. I believe this was created by Professor Cliff Reiter, who has done a lot of research into visualisation and fractals, and who has an interesting textbook on the subject.

## Combinatorics of non-crossing partitions

## Board of pi

This board by David Cushing at Newcastle University attempts to calculate pi very crudely by counting how many squares a circle covers and using the formula

Area = pi x r^{2}

to get a value of 2.98. Can you do better?

This exercise was part of a series of activities carried out to celebrate “Ultimate pi day” on 3/14/15 (in US date format). There were also many other attempts by people to estimate the value of pi using nonstandard methods. Did you take part in the festivities and, if so, how?

## Backward differential equations

This board was created in the common room of the School of Mathematics at the University of Edinburgh by David Siska and Arnaud Lionnet. Arnaud is visiting David in Edinburgh, and they are working on backward stochastic differential equations and stochastic partial differential equations, which are on the interface between probability and analysis.

## Self-intersection numbers for a fence

This is the board of Chris Palmer, a postgraduate at the University of Edinburgh. He has recently been working on finding a simple chain level Seifert pairing for the Seifert surface of a link. This is related to his supervisor Andrew Ranicki’s recent talk in which he used surgery theory to find a chain level pairing. The figures show how to compute the self intersection numbers for a fence (a one-dimensional simplicial complex that is a deformation retraction of the Seifert surface).

## Backdrop for Sir Michael Atiyah

Today the University of Edinburgh was privileged to welcome award-winning photographer James Glossop to the School of Mathematics. His task was to photograph Sir Michael Atiyah for an article in The Times (to appear next week) and he asked for a blackboard to be decorated with mathematical equations to form the backdrop to this photo. It was a joint effort between Andrew Ranicki, Julia Collins, Patrick Orson, and (of course) Sir Michael, and contains all their favourite formulae, numbers and ideas in mathematics. What would *you* have drawn on the board?

James kindly took this photo of the board after he was finished getting his shots of Sir Michael. I like the heavenly light shining in from above!

## Niels Bohr Institute

This is the blackboard in the common room at the Niels Bohr Institute in Copenhagen, taken by Andrew Jackson. It is advertising a colloquium given by Julia Collins about Peter Guthrie Tait, with a helpful diagram of a vortex cannon to show people what to expect in the talk. Unrelatedly, there is a lot of matrix algebra on the left and what seems to be a half-rubbed-off torus on the right. Proof that even in a physics institute, much of the work is really mathematics! We have no idea what the cartoon at the top signifies.

## 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 ^{-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.]

## Multi-type branching processes

This board comes from the office of Peter Keller, a new postdoc working at the University of Edinburgh. He is currently thinking about multi-type branching processes in biology and wanted to draw some pictures on the board to help him figure out what was going on. In a branching process you have a population of individuals who each produce a random number of offspring, and you want to ask questions like “How big will the population be at a certain time in the future?”, or “What is the chance of the population ever becoming extinct?”. In a multi-type branching process, there are different types of individuals. For example, when bacteria reproduce, they may either replicate themselves, or they may produce a mutant form. Mutants can then only replicate into mutants. We can then ask what the proportion of mutants to normal individuals is likely to be after a certain amount of time.

I have no idea what the reference to martingales and larks is. Please leave a comment if you think you know!

## Momentum

This blackboard photo was taken as part of an art exhibition entitled “Momentum” in which photographer Alejandro Guijarro spent 3 years travelling to the great Quantum Mechanics institutions of the world and photographing the blackboards as he found them. In the exhibition, the photographs were then displayed life-sized. More information on the ideas behind the project can be found on the website of the gallery curators.

## Intersecting 4k-manifolds

This board was created in the office of University of Edinburgh PhD topologists Carmen Rovi and Chris Palmer during a discussion with Patrick Orson about Wall’s theorem on the non-additivity of signatures. That is, what happens to the signatures when you glue together three or more 4k-manifolds in such a way that the three boundaries intersect in a (4k-2)-manifold? The answer to this question, and the pictures in the bottom right of the board, help to explain in a geometric way what is going on with an algebraic construction to do with L-theory in the top right of the board.

## Christmas party song requests

On coming back to the School of Mathematics in Edinburgh in the New Year, I was amused by the blackboard in the common room. Christmas party songs had been written on the board during our end-of-year party, but in January, rather than rubbing them off to make space for mathematics, people had just decided to write their maths in the gaps between words. One of our postgraduates, Hari Sriskantha, noted that mathematics is like a weed that grows in unwanted parts of blackboards.

Do you have examples of this happening on your public blackboards too? If so, send them in!