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Topology

Self-intersection numbers for a fence

whiteboard Seifert pairingThis 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).

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Backdrop for Sir Michael Atiyah

(c) James GlossopToday 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!


Knot concordance and slice movies

Brendan Owens blackboardThis 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.


Maximal oriented Wicks forms

Stacey_Aston_blackboardThis 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.]


Intersecting 4k-manifolds

Patrick Orson Feb 2013This 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.


Surgery without tears

Kirby topology 2012This 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.


Magic squares

Magic square blackboardThis blackboard comes from the website Blackboard of the Day or BBOTD. If you click on ‘Mathematics’ at the top of the page you get taken to loads of wonderful mathematics blackboard photos, and from time to time I will post my favourite pictures on this blog too.

This board was the result of a conversation between the Fields medallist Cédric Villani and an unnamed audience at the University of Lyon. Cédric started by showing them the magic square in the bottom right corner, where every row, column and diagonal adds to 65. This can be better understood by joining the top edge to the bottom edge and the left edge to the right edge to form a torus, as drawn in the top left corner. The audience weren’t so impressed because this sort of magic square is well known and understood. Thereupon Cédric drew a magic hexagon (top right) in which the sum of every vertical and diagonal line added up to 38. Moreover, he claimed that this was the only such magic hexagon. Attempts to draw this hexagon on a torus did not provide further understanding and the audience was left frustrated.

Finally, the discussion became topological and the group realised that they could draw the Borromean rings by folding sides of a cube in half, as shown in the middle drawing.


Bundles from the ’60s

Zeeman, Sanderson, RourkeThis photo has been the first to break my unwritten rule of only showing boards and not people. It was sent to me by Brian Sanderson at the University of Warwick, who is also the right-hand person in the photograph. He says that the picture dates from 1965 and shows Christopher Zeeman pointing to a block bundle. In topology, block bundles in the PL category are analogous to normal bundles in the smooth category. The block bundle idea was Brian’s and was subsequently fully developed with Colin Rourke, in the centre. Christopher, on the left, came up with the name on a flight to Oberwolfach, a mathematical retreat in the Black Forest of Germany.


Scary index theory

Index TheoryThis board was found in Michael Singer‘s office in Edinburgh.  He tells me that it was started by a student coming to him saying that they were scared of Index Theory – an understandable emotion in my opinion. The Atiyah-Singer index theorem is a powerful result which relates topological properties of a manifold with its analytical and geometric properties. [For more explanation, you could try reading my other blog post on this: http://haggisthesheep.wordpress.com/2010/09/20/heroes-of-topology/.%5D  Using the theorem, Michael Singer was then able to explain a result to the student using analysis that had previously only been understood via geometry and algebra.


Browder 2012 conference

McDuff May 2012What better place to collect great blackboard photos than at a topology conference! This conference was in honour of Bill Browder, one of the great figures of 20th century topology, and the photo was taken by student Alexi Hoeft during a talk by the eminent mathematician Dusa McDuff. You can see more photos from the conference on this Facebook page (no sign-in required) and I may well be posting more of them on this blog!