Atiyah memorial day

On Saturday, we went to the memorial event for Michael Atiyah, held in the magnificent Playfair Library in the University of Edinburgh’s Old College buildings. (Not named after John Playfair, Professor of Mathematics in Edinburgh and responsible for Playfair’s Axiom, a form of the Parallel Postulate in Euclidean geometry.)

The first speaker was Nigel Hitchin, who made an attempt to explain the Atiyah–Singer Index Theorem to us. He began with Euler’s polyhedral formula VE+F = 2, on which the left-hand side is an alternating sum of combinatorial data and the right-hand side a topological invariant of the sphere. Using cohomology, the numbers on the left can be replaced by dimensions of vector spaces (the spaces of functions on vertices, edges and faces); then, using K-theory (another subject in which Atiyah played an important part), these can be replaced by vector spaces of differential forms. Applications include the 28 bitangents to the plane quartic (where Atiyah was proud of the theorem that a real quartic with no real points has exactly four real bitangents), and the structure of topological insulators.

Jean-Pierre Bourgignon, speaking by video link, told us something of Atiyah’s presence in mathematical physics, involving spinors and the Dirac operator, and something of his role in the setting up of the European Mathematical Society, of which he was individual member number one.

The other two speakers, Nick Manton and José Figueroa-O’Farrell, talked about physics rather than maths. Manton told us about work he and his students and colleagues had done on skyrmions, inspired by Michael Atiyah (but not directly Atiyah’s work). Figueroa-O’Farrell told us that Atiyah’s influence on Ed Witten had healed the divorce between maths and physics pointed to by Freeman Dyson in the 1970s, and claimed that now physics had an enormous influence on maths (an overstatement in my opinion, backed up by the statement that the Jones polynomial at a certain root of unity appears in the work of Witten).

After lunch, there were more personal recollections from a variety of people including Lord Mackay of Clashfern, the eminent lawyer and former Lord Chancellor, who had been a student of mathematics with Atiyah and had remained a close friend all his life; and William Duncan, former CEO of the Royal Society of Edinburgh, who had worked closely with Atiyah during his presidency.

About Peter Cameron

I count all the things that need to be counted.
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2 Responses to Atiyah memorial day

  1. Maxis Jaisi says:

    Any source on the application to 28 bitangents of plane quartics?

  2. A bitangent to a curve is a line which is tangent at two different points.

    The 28 bitangents to a general quartic form a famous configuration, with an interesting Galois group Sp(6,2). (This means that you obtain the coordinates of the bitangents by solving an equation over the field of coordinates of the quartic, and the Galois group is Sp(6,2), a doubly transitive group of order 28.)

    You can construct a real model as follows. Take two concentric overlapping ellipses, for example x2+2y2 = 1 and 2x2+y2 = 1. Their union is a (special) quartic curve with equation

    (x2+2y2−1)(2x2+y2−1) = 0.

    Now replace the right-hand side by a small negative constant. This shrinks the four lobes of the figure slightly, giving four closed curves. Each pair of these curves has four bitangents, giving 24; in addition, each of the curves is non-convex and has a bitangent meeting it at the tips of the two “horns”, giving four more, or 28 in all. This is the maximum number for any quartic, I believe.

    I believe I learned this construction from Graham Higman. You can see a picture, the Trott curve, on Wikipedia. I have no idea where to find Atiyah’s work on this problem.

    Apologies: superscripts don’t seem to work in comment boxes: it should be, e.g., x squared + 2 y squared.

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