## 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.

## 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.

## 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.

## Instanton geometry

This week’s whiteboard is a joint effort between José Figueroa-O’Farrill and PhD student Paul Reynolds, who study objects in mathematical physics called instantons. These are special solutions to the Yang-Mills field equations which play important roles both in Physics and Mathematics. For example, in quantum physics instantons describe the process of tunnelling, where particles can get through barriers that classically they would be unable to surmount. In mathematics, they were instrumental in Donaldson’s work on 4-dimensional manifolds, for which he was awarded the Fields Medal.

Instantons appear in all sorts of dimensions, and José and Paul are specifically interested in 8 dimensions. Four-dimensional instantons are intimately related to the quaternions, whereas the 8-dimensional instantons are intimately related to the octonions.

The first bit of the blackboard (to the left) explains that the 8-dimensional instanton equations (which consist of 7 independent conditions) can be viewed as the components of a momentum mapping. This generalises a similar fact about 4-dimensional instantons. In the 4-dimensional case the geometry is that of a hyperkähler momentum mapping, but the geometry of the 8-dimensional equations remains unknown.

Hyperkähler manifolds can be constructed by taking the quaternions and quotienting out by certain group actions. Paul is trying to do something similar with the octonians: trying to find the right quotients that will give interesting manifolds which fit the geometry of the 8-dimensional instantons. The part of the board to the right describes his attempts at finding the geometry of a quotient in a special case.