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 V−E+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.