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The Floer Jungle
Posted on June 5, 2026 by woit
There’s a remarkable new book out about the life and work of Andreas Floer, entitled The Floer Jungle, co-written by writer Siobhan Roberts (author of some great biographies of mathematicians) and mathematician Helmut Hofer. Hofer has also given talks recently covering the material in the book, see for instance video here and slides here.
The story of Floer’s career and his work is a fascinating one. The book is written at a mixture of levels, starting out with some chapters explaining background at an easily accessible level, but then moving on to the details of the symplectic geometry and topology issues for which Floer’s work provided a breakthrough, some of which will be of most interest to experts.
Like many people, I first heard of Floer’s ideas from Michael Atiyah, in my case at the May 1987 Duke conference that I wrote about in detail here. I’ll refer to that posting for a description of the context for why the idea of "Floer homology" has wide significance beyond its origins. I can’t emphasize too much that if you’re at all interested in this area, you must read the write up of Atiyah’s talk, available here.
The new book ends with some discussion of the ongoing interest in Floer’s ideas, in particular describing a fall 2021 learning seminar at the IAS, organized by Akshay Venkatesh and Jacob Lurie. It ends with
Even as it stands, Floer theory is an enticing — perhaps irresistible, if intimidating — addition to the mathematical toolkit. "It seems absolutely terrifying, like something that’s not going to end well," Venkatesh said. But it also seems like a fundamental mathematical structure — "it occurs in so many places in topology of three and four manifolds, it makes you think it’s really something fundamental, like topology itself." Convening the Floer learning seminar, Venkatesh had no hidden motives in terms of his own research. Yet by the end, he had started to wonder… "I started to think, ‘Oh, it will be interesting to look for analogs of those things in number theory." It’s very pie-in-the-sky," he said. "But I would like to think about it: Some of the Floer-type structures in three- and four-dimensional topology, do they have shadows in number theory? That’s at least a question worth thinking about."
"It’s wonderful, this Floer idea," Venkatesh said. "I’m an outsider, but that much is clear, even to an outsider. It’s a wonderful thing."
One reason to suspect shadows of Floer theory in number theory is the long-standing analogy between 3-manifolds and number fields, which Peter Scholze has often emphasized as a guiding principle in some of the newer ideas about arithmetic geometry he has pioneered. From my own point of view, the striking thing about Floer homology is something Atiyah emphasized in his talk: Floer homology is the natural state space for a topological version of Yang-Mills theory. As described in my earlier posting, this was the beginning of the huge area of topological quantum field theory, with early work by Witten following up on Atiyah’s speculation. The structure of this TQFT looks very much like that of the Standard Model: it is a 4d theory with Yang-Mills gauge fields and fermions. Surely it’s not just a coincidence that this very deep and fundamental structure in mathematics is so close to the most fundamental and deep thing we know about physics.
I was going to add a reference to something by David Ben-Zvi, who has often written about these analogies between number theory, three and four manifolds, and quantum field theory. Looking for something to link to turned up notes to his recent Rademacher lectures, with the first of these especially relevant (see his website for the rest and for more).
Floer’s life ended early and tragically, with his suicide in 1991. He suffered from depression and mental health issues, likely aggravated by drug use. There isn’t a lot of material about this in the new book. I’ll add some recollections of the year I spent in Berkeley, during which I talked to Floer on a couple of occasions that I can remember.
During the academic year 1988-89 I was a postdoc in Berkeley at MSRI, and lived in the Ellsmere apartments on Dwight Way, with my bedroom window right across a narrow alleyway from the kitchen of the Barrington Hall student co-op. A couple days after moving in, I was awoken in the middle of the night by an unholy racket, which I finally discovered to be coming from the neighboring kitchen. A large group of students was banging on pots and pans, as loud as possible. This went on for an hour or so. I was worried that the apartment was a horrible mistake, but it turned out this wasn’t something they did regularly, it was some sort of special occasion. Various people explained to me that...