Not Even Wrong

This week the Simons Center is hosting a workshop on Differential Cohomology and its applications in physics. I won’t try and give an explanation of what differential cohomology is here, with a little luck the videos of the talks will soon be on-line. Very briefly, this subject is about an extension of the usual sort of cohomology theory that provides finer information. It was discovered independently by Deligne in an algebraic geometry context (his construction is often called “Deligne Cohomology”) and by Jim Simons and Jeff Cheeger in a differential geometry context. The subject made its appearance in physics first through Wess-Zumino-Witten terms in non-linear sigma models and the Chern-Simons term in gauge theories.

Dan Freed’s first lecture included an extensive discussion of one recent example that uses a generalized cohomology theory, and thus generalized differential cohomology, see here for details. Mike Hopkins discussed his work with Singer which led to this paper, and some ongoing work from a more generalized perspective. He started with some history, explaining that things began with a specific example he noticed in work on topological modular forms that Witten had found around the same time in work on the partition function of the fivebrane. He described this initial impetus as like discovering that they both were looking at the same intriguing specific tropical fish, with attempts to understand it leading to a huge ferocious formalism he characterizes as a shark that lept out of the tank.

In the afternoon, Jim Simons gave a wonderful description of the early history of his work on Chern-Simons invariants and Cheeger-Simons differential characters, leading up to recent work trying to prove that certain properties uniquely characterize this kind of theory. He began his talk by noting that only one small piece of chalk was available and complaining “I paid all this money for this place and all I get to use is one broken piece of chalk?”. The story started when he tried to work out a combinatorial formula for the signature in 4 dimensions, by analogy with what one does starting with the Chern-Weil formula for the Euler characteristic. In the signature case, the evaluation of a 4d Pontryagin class leads to the study of a 3-form on the boundary, which he investigated with Chern, leading to Chern-Simons theory. This is much the same problem as the one that (more than a decade later) I started working on as a graduate student in physics, trying to figure out how to calculate the second Chern number of a lattice gauge field configuration.

Finally Krzysztof Gawedzi gave an interesting talk reviewing the by-now-extensive history of the use of this kind of mathematics in physics, including various incarnations of the notion of a “gerbe”. Unfortunately I’m back in the city now, hope to follow the rest of the workshop via video at some point.

There’s an explanation at the latest Abstruse Goose.

To recycle some of my own writing, from page 107 of NEW, the book:

When I was a graduate student at Princeton, one day I was leaving the library perhaps thirty feet or so behind Witten. The library was underneath a large plaza separating the mathematics and physics buildings, and he went up the stairs to the plaza ahead of me, disappearing from view. When I reached the plaza he was nowhere to be seen, and it is quite a bit more than thirty feet to the nearest building entrance. While presumably he was just moving a lot faster than me, it crossed my mind at the time that a consistent explanation for everything was that Witten was an extra-terrestrial being from a superior race who, since he thought no one was watching, had teleported back to his office.

And, before anyone takes this seriously, I certainly don’t believe this is the explanation for the “M” or that any actual teleportation occurred. To quote the next paragraph of the book:

More seriously, Witten’s accomplishments are very much a product of the combination of a huge talent and a lot of hard work. His papers are uniformly models of clarity and of deep thinking about a problem, of a sort that very few people can match. Anyone who has taken the time to try and understand even a fraction of his work finds it a humbling experience to see just how much he has been able to achieve.

Update: Clifford Johnson at Asymptotia points out a recent talk by Witten to a non-specialist audience about knots. It there is a Martian plot going on here, at least it has led to some wonderful insights about mathematics and quantum field theory that human beings might never have otherwise been able to figure out…

Update: The America COMPETES Reauthorization has just passed the House and will go to the president to be signed, something no one expected to happen a week or so ago, more details about the legislation here. I gather that it authorizes 5 to 7% increases for science agencies. Problem is that these are not the actual appropriations, which are still up in the air, awaiting action next year by the next Congress. But this does indicate that there is bipartisan willingness to at least pay lip service to protecting the research and development part of the budget.

I learned from a colleague last night about recent events bringing together the topics of the title of this posting, something that one wouldn’t have thought was possible. Last Wednesday there was a showing in Berkeley of Edward Frenkel’s short film Rites of Love and Math, together with the Yukio Mishima film Rites of Love and Death that inspired it. Frenkel is a math professor at Berkeley, and one of the leading figures in geometric Langlands research (which he describes as a “grand unified theory of mathematics”). He’s also a wonderful expositor, almost single-handedly making the beauty of a subject initially renowned for its obscurity accessible to a much wider audience. Recently he has worked with Witten on relations of geometric Langlands to quantum field theory, and with Langlands and Ngo on relations to number theory. At the same time, while a visiting professor in Paris, he co-directed (with Reine Graves) and acted in this new film.

MSRI was one of the two sponsors of the showing of the film, but pulled out of this role recently, for reasons explained here by MSRI director Robert Bryant. He had found that some people in the math community were upset by the film and MSRI’s involvement with it, feeling that it glamorized an objectionable view of the relationship of women to mathematics. There’s a plan to organize some sort of event at MSRI to discuss the issues brought up by the film and the decision to withdraw sponsorship.

I still haven’t seen the film, although I gather that a DVD will soon be available. Congratulations to all involved in this for finding a unique way to make mathematics and mathematicians look interesting and worthy of media coverage. I had no idea it was still possible to stir up controversy in the Bay area with art involving transgressive sex, and would never have thought that using research mathematics was the way to do it.

Update: Andrew Ranicki has written a review of the film for the London Math Society newsletter, available here. He identifies the notorious equation in question (5.7 of http://arxiv.org/abs/hep-th/0610149), and makes the comment that, sartorially, this film is a breakthrough, since, in other films:

By and large, male mathematicians are portrayed as crazies who are smart and lovable, but badly dressed. Likewise for female mathematicians, although they tend to be better dressed. This said, in the film under review, the actors are either very well dressed, or not dressed at all.