Not Even Wrong

The International Congress of Mathematicians will be taking place in Madrid relatively soon, in late August. One tradition at this conference is the announcement of the Fields Medals, and I’m getting embarassed that I’m not hearing any authoritative rumors about this (other than about Tao and Perelman); if you have any, please send them my way. One other tradition is to have speakers write up their talks in advance, with the proceedings available at the time of the conference, so already some write-ups of the talks to be given there have started appearing on the arXiv.

Last night, Michele Vergne’s contribution to the proceedings appeared, with the title Applications of Equivariant Cohomology. On her web-site she has a document she calls an exegesis of her scientific work, this gives some context for the equivariant cohomology paper. She also is co-author of a book called Heat Kernels and Dirac Operators, which has a lot more detail on some aspects of this subject. Finally, there has been a lot of nice recent work in this area by Paul-Emile Paradan.

Equivariant cohomology comes into play when one has a space with a group acting on it, and it mixes aspects of group (or Lie algebra) cohomology and the cohomology of topological spaces. There are various ways of defining it, the definition that Vergne works with is a bit more general than the one more commonly used. It involves both differential forms on the space, and generalized functions on the Lie algebra of the group.

The beauty of equivariant cohomology is that it often computes something more interesting than standard cohomology, and you can often do computations simply, since the results just depend on what is happening at the fixed points of the group action. There’s a similar story in K-theory: when you have a group action on a space, equivariant K-groups can be defined, with representatives given by equivariant vector bundles. Integration in K-theory corresponds to taking the index of the Dirac operator, and in the equivariant case this index is not just an integer, but a representation of the group. The index formula relates cohomology and K-theory, and one of Vergne’s main techniques is to work with the equivariant version of this formula.

In the case of a compact space with action of a compact group, there’s a localization formula that tells you how to integrate representatives of equivariant cohomology classes in terms of fixed point data. In many cases, this leads to a simple calculation, one famous example is the Weyl character formula, which can be gotten this way. New phenomena occur when the group action is free, and thus without fixed points. This was first investigated by Atiyah (see Lecture Notes in Math, volume 401), who found that he had to generalize the index theorem to deal with not just elliptic operators, but “transversally elliptic” ones. Such operators are not elliptic in the directions of orbits of the group action, but behavior of the index is governed by representation theory in those directions.

Vergne has been studying examples of this kind of situation, and it is here that generalized functions on the Lie algebra come into play. Integrating the kind of interesting equivariant cohomology classes that occur in the transversally elliptic index theory case over a space gives not functions but generalized functions on the Lie algebra. There’s a localization formula in this case due to Witten, who found it and applied it to 2d gauge theory in his wonderful 1992 paper Two Dimensional Gauge Theories Revisited.

This kind of mathematics, growing out of the equivariant index theorem, is strikingly deep and beautiful. It has found many applications in physics, from the ones in 2d gauge theory pioneered by Witten, to more recent calculations of Gromov-Witten invariants. It leads to a mathematically rigorous derivation of some of the implications of mirror symmetry in special cases, and a wide variety of other results related to topological strings. My suspicion is that it ultimately will be used to get new insight into the path integrals of gauge theory, not just in 2 dimensions but in 3 or 4.

Update: Vergne has another nice new paper on the arXiv. It’s some informal notes on the Langlands program which she describes as follows:

These notes are very informal notes on the Langlands program. I had some pleasure in daring to ask colleagues to explain to me the importance of some of the recent results on Langlands program, so I thought I will record (to the best of my understanding) these conversations, and then share them with other mathematicians. These notes are intended for non specialists. Myself, I am not a specialist on this particular theme. I tried to give motivations and a few simple examples.

It would be great if more good mathematicians wrote up informal notes like this about subjects they have learned something about, even if they are not experts. The notes are entitled All What I Wanted to Know About Langlands Program and Was Afraid to Ask.

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).

Taking off tomorrow for a long weekend, internet access may be spotty. Here are some things that may be of interest:

HEPAP is meeting today and tomorrow, the presentations given at the meeting are available here.  JoAnne Hewett is there and has a posting on Cosmic Variance.
The Seed article with various physicist’s views about what to expect at the LHC that was discussed here earlier is now available online.

There’s an article about Jim Simons in Newsday (via Angry Physics).

Maybe a cosmologist can comment on the significance of this, but over at CosmoCoffee there’s a discussion of a new paper reanalyzing the latest WMAP data and coming up with a scalar spectral index n_s= .969 +/- .016. This is now 2.0 sigma away from 1, instead of the 2.7 sigma of the earlier analysis. This deviation from 1 was widely sold as evidence for inflation (since the simplest inflationary models give values slightly less than one), the fact that it is now only a 2 sigma effect seems to make this case a bit weaker.

The Institut Henri Poincare in Paris will be having a three-month-long program on Groupoids and Stacks in Physics and Geometry. The web-siter there contains a good associated overview of the subject.

Bruno Kahn has an excellent expository article on motives.

Over at the Edge web-site Lawrence Krauss has a piece called The Energy of Empty Space That Isn’t Zero. It’s partly about the cosmological constant, and discusses a workshop on Confronting Gravity that he organized back in March, which brought many prominent theorists together at a Caribbean resort to discuss physics, travel in a submarine, and hang out at the “private island retreat” of the funder of the event, science philanthropist Jeffrey Epstein.

Krauss has many provocative things to say about the current state of theoretical physics, including perhaps the most concise and vivid description I’ve read in a while:

It’s been very frustrating for particle physicists, and some people might say it’s led to sensory deprivation, which has resulted in hallucination otherwise known as string theory.

He also has a somewhat longer skeptical take on extra dimensions, together with an attempt at positive spin:

Many of the papers in particle physics over the last five to seven years have been involved with the idea of extra dimensions of one sort or another. And while it’s a fascinating idea, but I have to say, it’s looking to me like it’s not yet leading anywhere. The experimental evidence against it is combining with what I see as a theoretical diffusion — a breaking off into lots of parts. That’s happened with string theory. I can see it happening with extra-dimensional arguments. We’re seeing that the developments from this idea which has captured the imaginations of many physicists, hasn’t been compelling.

Right now it’s clear that what we really need is some good new ideas. Fundamental physics is really at kind of a crossroads. The observations have just told us that the universe is crazy, but hasn’t told us what direction the universe is crazy in. The theories have been incredibly complex and elaborate, but haven’t yet made any compelling inroads. That can either be viewed as depressing or exciting. For young physicists it’s exciting in the sense that it means that the field is ripe for something new.

I hadn’t realized how many of the physics departments at the top universities in the US have instituted undergraduate string theory courses.  The only one I was aware of was MIT’s 8.251, String Theory for Undergraduates, taught by Barton Zwiebach, who developed a textbook for the course, A First Course in String Theory. 

Maybe now that there’s a textbook, that is what has caused other institutions to follow suit.  Caltech has Physics 134, String Theory, and Carnegie-Mellon has Physics 33-652, An Introduction to String Theory.  Stanford goes its competitors one better by having two undergraduate courses in string theory: Physics 153A, Introduction to String Theory I, and Physics 153B, Introduction to String Theory II. This last course even promises to explain to students how string theory is connected to particle physics. 

Alex Vilenkin has a new popular book out about cosmology, entitled Many Worlds In One. It’s mainly about the extremely speculative end of cosmology, and much of it is devoted to explaining the author’s ideas on eternal inflation, creating the universe by tunneling out of nothing, and the anthropic landscape, together with stories about how he came to these ideas. It contains various amusing anecdotes, especially about Alan Guth. Sean Carroll is credited with the following story:

One of the leading superstring theorists, Joseph Polchinski, once said that he would quit physics if a nonzero cosmological constant were discovered. Polchinski realized that the only explanation for a small cosmological constant would be the anthropic one, and he just could not stand the thought.

He also describes the reaction to his anthropic arguments back during the years when these were not all the rage like they are now:

After one of my seminars, a prominent Princeton cosmologist rose from his seat and said, “Anyone who wants to work on the anthropic principle – should.” The tone of his remark left little doubt that he believed all such people would be wasting their time.

Vilenkin’s book covers much the same ground as Susskind’s, although from the point of view of a cosmologist, not a particle physicist. A huge amount is made of the supposed anthropic “prediction” of the value of the cosmological constant (any news of the rumor from Sean Carroll of new work by prominent Princeton cosmologist Paul Steinhardt showing this is bunk?). Unlike Susskind, Vilenkin at least doesn’t seem to be on a campaign to attack the “Popperazi” and convert everyone to anthropics, but he demonstrates a similar lack of concern for the fact that the ideas he is discussing don’t lead to much if anything in the way of a testable experimental prediction.

Here’s his scientifc program for 21st century physics, which he hopes will be spent working on the anthropic landscape:

First, we will need to map the landscape. What kinds of vacua are there, and how many of each kind? We cannot realistically hope to obtain a detailed characterization of all 10⁵⁰⁰ vacua, so some kind of statistical description will be necessary. We will also need to estimate the probabilities for bubbles of one vacuum to form amidst another vacuum. The we will have all the ingredients to develop a model of an eternally inflating universe with bubbles inside bubbles inside bubbles… Once we have this model, the principle of mediocrity can be used to determine the probablility for us to live in one vacuum or other.

Unfortunately for this research program, it has yet to even begin to get off the ground, and there are very good arguments that it can never succeed. There are an infinite number of possible vacua, and trying to make this finite so one can do statistics requires putting in cutoffs, with results then strongly depending on the cutoff. The large numbers of these vacua make any attempts to identify ones that agree with the real world computationally completely intractable. Even if one could do this, all evidence is that one would end up with broad statistical distributions for many of the parameters of the standard model, providing no useful prediction of what new experiments will see, or any insight into why these parameters have the values that they do.

The Cao-Zhu paper at the Asian Journal of Mathematics that is supposed to have a complete proof of the Poincare and geometrization conjectures is still not available, but the introduction to the paper has been posted there.

If you’re not getting enough string theory bashing today, head over to John Horgan’s Scientific Curmudgeon blog, where he has a posting entitled Pulling the Plug on Strings. It contains a wide selection of string-puns (or whatever you call such things), and he has decided to refer to string theory advocates as “yarn-heads” and braniacs. For his trouble, his comment section is under assault by the usual suspects. There’s also this site, containing a graphic mentioned here before which I refuse to admit to finding funny. The proprietors have an interesting way of dealing with the comment section.

Sabine Hossenfelder has an excellent posting on Science and Democracy.

This past week I’ve spent some time at the 26th International Colloquium on Group Theoretical Methods in Physics, being held here in New York at the CUNY Graduate Center. It was ably organized by Sultan Catto, who somehow convinced me to give a short talk on the blog and the book, one where I think I disappointed people by keeping string-bashing to a minimum. I enjoyed seeing people at the conference, and there were some good talks, including one by Greg Moore on his recent work with Dan Freed and Graeme Segal (see here and here).

Urs Schreiber has been putting his notes on-line about elliptic cohomology. Lots of interesting material, but his comment that he expects the landscape of superstring theories to be equal to the spectrum of elliptic cohomology sounds frightening. Maybe he means a different landscape…

I was quite sorry to hear of the recent death of Irving Kaplansky. Kaplansky was an algebraist, and director of MSRI when I was there in 1988-89. At the time I wasn’t much interested in algebra, so didn’t talk to him about math, but he was responsible for making MSRI a really wonderful place to work.

Update: The slides from Yau’s talk at Strings 2006 are now available here.