Daniel Quillen, one of the greatest mathematicians of the latter part of the twentieth century, passed away in 2011 after suffering from Alzheimer’s. For an appreciation of his work and an explanation of its significance, a good place to start is Graeme Segal’s obituary notice, and there’s also quite a bit of material in the AMS Notices.
It’s very exciting to see that the Clay Math Institute now has a project to make available Quillen’s research notebooks. Segal and Glenys Luke have been working on cataloging the set, producing lists of contents of the notebooks, so far from the earliest ones in 1970 up to 1977. Quillen’s work ranged very widely, and for much of the 1980s he was very much involved in what was going on at the boundary of mathematics and quantum field theory. His work on the Mathai-Quillen form provided a beautiful expression for the Thom class of a vector bundle using exactly ingredients that formally generalize to the infinite-dimensional case, where this provides a wonderful way of understanding certain topological quantum field theories. The Mathai-Quillen paper is here, see here for a long expository account of the uses of this in TQFT.
I’ve just started to take a look through the notebooks, and this is pretty mind-blowing. The Mathai-Quillen paper is not the most readable thing in the world; it’s dense with ideas, with motivation and details often getting little attention. Reading the Quillen notebooks is quite the opposite, with details and motivation at the forefront. I just started with Quillen’s notes from Oct. 15 – Nov. 13, 1984, in which he is working out some parts of what appeared later in Mathai-Quillen. This is just wonderful material.
Besides his own ideas, there are notes taken based on other people’s talks. See for instance these notes from a private talk by Witten in Raoul Bott’s office on Dec. 15, 1983.
I was already having trouble getting done a long list of things I am supposed to be doing. Having these notebooks available is going to make this a lot worse….