A correspondent wrote in to tell me about a wonderful web-site, called People’s Archive. Their idea is to do in-depth interviews at a peer-to-peer level with the great thinkers and creators of our time. They’ve been doing this for a few years, only recently providing open access to much of the content on their site.
The two interviews of people closest to my interests are ones of Sir Michael Atiyah and Murray Gell-Mann. The interviews are very long, several hours. So far I’ve made my way through the Atiyah interview (which is in 93 pieces), mostly just reading the transcript, and have poked around a bit in the Gell-Mann interview (which is in 200 pieces).
Atiyah is on just about every mathematician’s list as one of the very few greatest figures in the second half of the twentieth century. He’s also had a major impact on the relation of mathematics and physics. The interview essentially provides a long memoir of his life, concentrating on his mathematical research work, explaining in detail how it came about and how it evolved. It’s truly wonderful, with all sorts of interesting stories, together with insights into mathematics and how it is done at the highest level.
The interview begins with his childhood in Khartoum, then discussing his later education in England, ending up at Cambridge where he was a student of Hodge. One story he tells (segment 21) is about Andre Weil’s reaction when Atiyah showed him his work at the time he was a student. The segment is called “how not to encourage somebody.” Atiyah also later on talks about his mathematical heroes, especially Hermann Weyl. Physicists often confuse Weil and Weyl, who were two rather different characters. They both did important work on representation theory with Weyl responsible for, among many other things, the representation theory of compact Lie groups, and the exponentiated form of the Heisenberg commutation relations (what mathematicians call the Heisenberg group). Weil was responsible for the geometric construction of representations of compact Lie groups (Borel-Weil theory), and a general theory of representations of Heisenberg-like groups (known as the Segal-Shale-Weil, or metaplectic representation).
Atiyah tells about the importance of his years spent at the IAS in the fifties and the people that he met there. It was one of the great meccas of mathematics at the time. He tells in detail the story of how the index theorem came about (segment 43), and the crucial role provided by the Dirac operator in linking together the analysis and the topology. The Dirac operator was rediscovered by him and Singer during their work. He also explains the important role from the beginning of equivariant versions of the theorem, in providing motivating examples and requiring the most general and deepest sort of proof.
During the 1970s Atiyah started to get deeply involved in interactions with physicists, and he recalls going to MIT to discuss instantons with them, meeting a young Edward Witten in Roman Jackiw’s office there (segment 67). He describes in detail his interactions with Witten, especially his prodding of Witten that led to the discovery of the TQFT for Donaldson theory (segment 71), something that took Witten quite a lot of effort before he came up with the necessary twisting of supersymmetry to make this work. He also tells the story of the famous dinner at Annie’s in Swansea where, in discussions with Atiyah and Segal, Witten came up with his Chern-Simons theory. The idea was so compellingly correct that Witten decided the next day to not give the talk he had planned, but to talk about this new theory born only the night before.
In his comments on the future (segment 74), Atiyah refers to the new ideas brought into mathematics from QFT as “high energy mathematics”, and predicts that mathematics in the future will make crucial use of the sort of “infinities of infinities” that occur in QFT structures, but that mathematicians until recently have had no real idea how to approach. He also makes some interesting comments about what sort of problems it is best for graduate students to work on, and gives (segment 90) a wonderful description of the importance of beauty in mathematics and his own definition of it.
All in all, it makes fantastic reading, I hope the company that put this together will clean up the transcripts and put them out in book form.
I haven’t had the time to go through all of the Gell-Mann interview, but it also contains all sorts of valuable history. One little-known fact that Gell-Mann mentions is that the SU(3) eight-fold way that he got the Nobel prize for came about because, after he had spent a long time trying to generalize SU(2)xU(1) unsuccessfully, a mathematics assistant professor (Richard Block) finally explained to him that what he was doing was trying to find a certain kind of Lie algebra, and the one he was looking for was the Lie algebra of SU(3).