Last month I spoke at the workshop Subfactors, higher geometry, higher twists and almost Calabi-Yau algebras. The workshop was part of the six-month program Operator algebras: subfactors and their applications at the Newton Institute. I talked about the paper The classification of 3-Calabi-Yau algebras with 3 generators and 3 quadratic relations written with Izuru Mori. The slides for my talk are here (Talk-3-CY_algebras).
I was rather surprised to be asked to speak at the workshop because I don’t know much about subfactors. I’ve been an interested spectator ever since Vaughan Jones proved his remarkable Index Theorem in 1984. But my own work has not interacted with subfactors at all. Still, I was keen to go because of my general interest in subfactors and because it was a chance to return to Cambridge, one of the most beautiful cities in the world.
I lived in Cambridge for about 4 months in late 1975 and early 1976. I had long hair and a beard then. (I had just finished my undergraduate degree at the University of Canterbury in New Zealand.) I had a job at the Pye Electronics factory. Pretty low-level stuff but my appearance made potential employers wary. After a couple of months packing things in cardboard boxes with styrofoam bubbles I shaved off my beard, cut my hair, borrowed 50 pounds from Barclays Bank, bought a suit and briefcase, and found a job in London at Datastream working as something like a quant, before the word “quant” was invented. While living in London I visited Cambridge several times with my girlfriend, Gillian, who was doing her Ph.D. in pharmacology at University College. We punted on the Cam, picnicked, and drank champagne under the willows, on sunny days. Good memories. I spent another 3 months in Cambridge in late 2006, again at the Newton Institute. So, it was nice to be invited to return even though I felt I didn’t have much to add to the workshop. I was tempted to decline but decided to go once the slate of speakers was drawn up: at least half the speakers were not experts on subfactors, and I was quite interested in the topics they planned to speak on. So, I went. And I was glad I did. I found the workshop very stimulating. Lots of interesting talks. I was delighted to hear and meet Akhil Matthews whose blog I have been reading since he was in high school!
What is a subfactor? First, what is a factor? A von Neumann algebra, of the -flavor not of the von Neumann regular algebra flavor, is called a factor if its center is . A subfactor is a von Neumann subalgebra of a factor that is itself a factor. It’s best to read about these things on Wikipedia. If is a subfactor of a factor Jones Index Theorem concerns the number , the index of in . Roughly speaking, measures the size of relative to . The Index Theorem says the index can be any real number or for an integer , and all these values occur. If you are interested, I suggest you read the survey article On the origin and development of subfactors and quantum topology by Vaughan Jones. That article was written in 1995. A more up-to-date account can be found in the very beautiful and stimulating article The classification of sub factors of index at most 5. If nothing else, I suggest you read the introduction; the parallel with Galois theory provides a helpful framework.
At the workshop, Iyama gave a very beautiful talk about pre-projective algebras aimed at non-experts. It was a “perfect” talk: the “perfect” examples and the most efficient and elegant perspectives on the main themes. Gus Lehrer spoke about some of his old work on Temperley-Lieb algebras and categories. For 30 years, people have asked me whether I had me Gus. I suppose they thought that all antipodean algebraists knew each other. Heck, I had only met Vaughan Jones once before even though we lived in the same town for several years when we were in high school, him as a student at Auckland Grammar, me at King’s College. (I should boast that Vaughan’s Fields Medal means that New Zealand has more Fields Medallists per capita than any other country.) Auckland Grammar and King’s College have been keen rivals for the best part of a century, especially in sporting events, the most important of which is the annual rugby match between their First XV’s. Probably, sometime in the early 1970’s Vaughan and I stood in the rain on opposite sidelines supporting our respective schools.
A couple of months before the workshop I talked about my work with Izuru in our weekly Algebra & Algebraic Geometry seminar. It wasn’t a good talk. I had also spoken about our work in Shanghai in late 2015. That wasn’t a good talk either. I think I am finally beginning to understand what we proved, and I think the talk in Cambridge reflected this.
The organizers scheduled me to speak on my birthday. They had also scheduled the conference dinner for that day too 🙂 The dinner was at Christ’s College. The size and magnificence of the college is breathtaking. Founded in 1437. Alumni include Darwin and John Milton…