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.]

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!

Blackboard cake

This birthday cake was made for Robin Knops, a Professor Emeritus at Heriot-Watt University, to celebrate his 80th birthday. The celebrations were part of a conference at ICMS in Robin’s honour, which was especially fitting as Robin was one of the founders of ICMS (the International Centre for Mathematical Sciences) in 1990. This cake was ordered from a Fife company called Sucre Coeur, with the mathematical equation suggested by Penny Davies.

The first equation is just a representation of the fact that the sum of Robin’s years is 80.

The second equation is the equation of conservation of linear momentum: rho = density, v = velocity, T = stress tensor, b = body force (e.g. something like gravity).  Penny says “The version on the cake refers to using an Eulerian (deformed) coordinate system, and this is more usual for fluids than for solid mechanics (which Robin has worked far more on and uses the reference or Lagrangian coordinate system).  This would normally have a suffix “R” on the rho and T terms, and so wouldn’t have been so easy to typeset on a cake.”

We wish Robin many happy returns (although his actual birthday is not until 30th December).

Surgery without tears

This photo is of the board of Rob Kirby, who recently visited the University of Edinburgh to give a series of talks on the topology of manifolds. It shows a version of the Thom-Pontrjagin construction for connecting two components of the pre-image (under the map g) of a point (0,0), but doing so in a neighbourhood of the blue arc connecting the two components.  The two components are in red, and as the homotopy progresses the red components are pushed towards each other until they meet at a point, like a saddle, and then become a cylinder joining the two components. This is basically surgery on a 0-sphere.

With thanks to Andrew Ranicki for taking the photo and providing the title for this post.

Fourier Piggy

I got sent this photo from Piggy who has a blog about exciting things he finds in science and maths. There are actually lots of fun whiteboards amongst his blog posts; for example this one about using mathematical analysis to design a strategy for hitting on the opposite sex.

In this photo, Piggy had to help a friend and explain to him the Fourier series representation of periodic and non periodic functions. He started easy, saying that Fourier had this great idea of resolving the problem in its various frequencies and then summing everything up. But then he got excited and dropped integrals here and there.