Edmund Harriss sent me these blackboard photos, taken on his trip to China to visit Luo Jun, a mathematician at Sun-Yat-Sen University. They produced several of these boards as they argued and discussed the topology of tilings and of self-similar fractals. Edmund points out that some of the statements on the boards are false!
This photo was taken in the former office of Sandy Davie, a recently retired professor at the University of Edinburgh. The writing on the board is the result of conversations with another professor, Istvan Gyongy, about partial differential equations in Sobolev spaces. It’s a great example of a typical mathematician’s blackboard, with chalk being drawn over old chalk and some evidence of half-hearted rubbing out with hands. Mathematicians usually prefer to make this kind of a ‘mess’ rather than interrupt the flow of ideas to clean the board properly.
Wishing everybody a merry Christmas and a mathematical new year! I can’t remember who sent me this lovely photo, but if it was you then please write in to give your name and the background to these great doodles!
Today’s blackboard belongs to Nathan Barker at the University of Newcastle. Nathan works with Thompson groups, which were first studied in 1965 and provide examples of interesting phenomena in group theory.
Top-left is a realisation of the automorphism group of one of Thompson’s groups as a tree. On the right is a subgroup that may or may not have the property Nathan wants, and then Poincare’s disc model of the hyperbolic plane, and an annular strand diagram. The jumpy line drawing at the bottom is the Cayley graph of PSL(2,Z) – the group of all 2×2 integer-valued matrices with determinant 1, where matrices A and -A are considered to be the same.
The bottom-left section is some quantum computation stuff by somebody called Ben that nobody but him understands.
Thanks to Christian Perfect for taking and sending me this photo, and for providing details of what’s on the board.
Readers of this blog may also be interested in an article detailing the relationship between mathematicians and blackboards: http://www.sps.ed.ac.uk/__data/assets/file/0020/60518/Chalk.pdf
Another picture from our wonderful blackboard artist Ryan Budney in Victoria, Canada. He’s taken a 4-dimensional sphere and removed a knotted 2-dimensional sphere (i.e. the surface of a beach ball), and tried to triangulate the resulting space. Triangulation means breaking up a complicated space into simpler pieces – triangles! – and then explaining how these pieces fit together. The triangulation here has precisely one vertex (a 0-dimensional triangle), one edge (a 1-dimensional triangle), 4 triangles (i.e. your usual garden-variety triangles), 5 tetrahedra (3D triangles) and the two pentachora (4-dimensional triangles). The bottom picture shows how the pieces are glued together. The top picture is completely unrelated.
The knotted 2-dimensional sphere is called a Cappell-Shaneson knot because it is sitting inside a ‘homotopy 4-sphere’. That is, the space the knot is sitting in looks like a normal sphere from a distance but might turn out not to be upon closer inspection. If it isn’t, then it would provide a counterexample to the Poincaré Conjecture, which is still open in 4 dimensions (but was famously solved for 3 dimensions by Grigori Perelman in 2003).
Will Donovan temporarily transformed his dining room table in London into a whiteboard using a plastic tablecloth. Here he’s trying to understand “Young symmetrizers“, which correspond to the grids of boxes that he’s sketched. These can be used in the representation theory of symmetric groups, to help us understand the possible types of symmetry that can be associated to objects in space. And they also have interesting algebraic properties all of their own…
To see more of Will’s mathematics, check out his gallery where he has posted many many photos of his changing black/whiteboards over the years.
A South African teacher, known on Twitter as @Pyfagoras, sent me this photo of his board. It shows how to turn a cube into an Eulerian graph by adding in the diagonals across each face of the cube. In an Eulerian graph, every vertex has even degree (that is, an even number of edges coming out of it) and it is so called because Euler proved that for such graphs it is possible to start anywhere and traverse every edge in the graph exactly once, returning to the start. Furthermore, such a circuit is impossible if the graph contains a vertex of odd degree. For the cube on the board, there are 6 edges coming out of each vertex, meaning it is not only Eulerian, but also a regular graph. You can read more about it in its incarnation as the 16-cell.
This is Julian Hall‘s whiteboard. Julian works in the area of optimization, and he is trying to figure out better ways to implement the simplex method to solve very large linear programming problems. Network flow problems are special cases of linear programming problems; for example, how a company should organise deliveries from various warehouses to various shops, or how a company should organise its workforce to complete a set of activities most efficiently. On the board here, these network problems are being analysed to understand why the simplex method sometimes runs much faster when the objective costs of a linear programming problem are perturbed randomly.
If you have access to a black or white board of interesting mathematics, please take a photo and send it to me (along with a short description) at Julia.Collins@ed.ac.uk!
Jane Walker from ICMS sent me this photo that she took at the Theory of Infinity Conference (also known as the Third European Set Theory Conference). The concept of infinity raises many difficult questions in Set Theory, many of which were written about by Georg Cantor and Bertrand Russell, and they are still being discussed today. The little dog in the picture is called Marley, and his photo is being posted around the world in order to raise money for the Dog’s Trust. He has a blog and can be followed on Twitter as #flatmarley.
Simon Willerton drew these knots on a board at the University of Edinburgh when he was a graduate student in 1995, and today they are still there! You can see in the bottom left corner there is a message to new students that these knots must never be rubbed off, so there is every hope that they will persist for many years to come. Simon’s thesis was about Vassiliev invariants, which are a way of constructing knot invariants using the extended space of singular knots (i.e. not just the usual knots, like those drawn on the board, but knots which may intersect themselves or have crossings where more than two strands are present). It is still an open question in mathematics whether Vassiliev invariants can be used to distinguish any pair of knots.
