Last week I was in Edinburgh for a few days and managed to attend the last two days of the conference in honor of Sir Michael Atiyah’s 80th birthday. Atiyah is now retired, but he was one of the dominant figures in mathematics during the second half of the twentieth century, as well as perhaps the person most responsible for bringing together mathematicians and physicists around issues of common interest in geometry and physics. His interactions with Witten played an important role in several major developments, including the whole idea of “topological quantum field theory”. One major part of my mathematical education was spending quite a lot of time for a few years reading through Atiyah’s collected papers. He is at all times a very lucid writer, with his expository writings quite marvelous and uniformly worth reading.

The biggest news at the conference was the announcement by Mike Hopkins of his solution (with Mike Hill and Doug Ravenel) of most of an old problem in topology that goes back to the sixties, known as the Kervaire invariant problem. Hopkins in his talk labeled the new theorem a “Doomsday Theorem”, because it nearly finishes off the subject it deals with, by ruling out the existence of a certain class of possible interesting topological invariants in all the remaining open cases except one. I wasn’t looking forward to trying to explain this here on the blog, since what is involved are issues in stable homotopy far beyond my expertise, so I was pleased to find this morning that others have beat me to it with explanations. The slides from his talk are here, John Baez has a posting here (including a comment from Hopkins here), and the news was spread to the ALGTOP mailing list here.

Another report of impressive progress on a problem was Simon Donaldson’s talk on the problem of showing that a Fano manifold has a Kahler-Einstein metric if and only if it is stable. This is one of the big open problems in complex geometry, and Donaldson discussed the appropriate notion of stability and outlined a strategy for getting a proof. He is not claiming a proof, with significant work still to be done, but experts seem to believe that the goal is now within sight and the next few years will see a resolution of this problem.

Unfortunately I arrived too late at the conference to hear Witten’s talk, the slides of which are available here. He is continuing his work of the past few years on Geometric Langlands. Dijkgraaf gave a nice talk reviewing a wide range of topics connected in one way or another with topological strings. Perhaps his slides will soon become available, but the topics covered were similar to those of his talks at UCSB last spring (see here). Vafa gave a rather clear explanation of his program to try and get particle physics out of local F-theory models. I’m not convinced this does more than reinterpret GUT extensions of the standard model using quite complicated constructions, but you can see for yourself here. He didn’t talk about the crucial question of whether this approach makes distinctive predictions about supersymmetry breaking testable at the LHC.

One evening was devoted to a public program about the Higgs particle, with a panel discussion featuring Higgs himself. It was not clear to me how much got through to the public about the electroweak symmetry breaking issue and what we hope to learn at the LHC. As always, some of the public wanted to know about what string theory has to do with this question. Unfortunately they were not given the simple, accurate and easy to understand answer “nothing at all.”

**Update**: The web-site for the conference is here, and conference organizer Andrew Ranicki has set up another web-site here for various materials associated with the conference.

**Update**: Videos of the talks at Atiyah80 are now available at the web-site linked to above.

Hi Peter,

Very good post! Very useful too!

Can you suggest the most important / relevant papers to read from his collected works?

Regards.

In the spring of 1972 I was a post-doc at IAS. One day Mrs.

Atiyah dropped by my house. I enthused about how deep

and broad Michael’s work was. Then, getting reflective, I

said “But he hasn’t done any mathematical physics”. She

immediately piped up “He will, he will!”

Hopkins’s talk is already referenced in the Wikipedia article on the Kervaire invariant. I’m impressed!

John,

Here are some suggestions, taken from among the expository pieces. Most of the non-expository research articles are also quite readable, and include many classics. As an example, his paper with Bott on Yang-Mills in 2d is chock-full of remarkable ideas and results.

Algebraic topology and elliptic operators (Comm. Pure. Appl Math XX (1967))

Topology of elliptic operators (Proc. Symp. Pure. Math 16)

The index of elliptic operators (1973 AMS Colloquium Lectures)

Algebraic Topology of operators in Hilbert Space (LNM 103)

Classical Groups and classical differential operators on manifolds. (CIME Varenna 1975)

Geometry of Yang-Mills Fields (Pisa 1979)

The moment map in symplectic geometry (Durham Symposium 1984)

Anomalies and index theory (Lecture notes in physics 208)

Topological aspects of anomalies (Symposium on Anomalies, Geometry, Topology, 1984)

New invariants of 3 and 4 dimensional manifolds (Weyl symposium 1988)

The Geometry and Physics of Knots, CUP 1990.

Thanks for the post and the reading suggestions in the comments!

Thanks Peter!

Reading Atiyah is certainly no time spent unproductive! He’s written seminal works on a whole eclectic body of math and physics topics.

Admittedly, I also take a more-than-passing interest in what he has researched upon and written on over the years, along with E. Witten (my interest/s lie in tandem to such (gargantuan! and needless-to-say awe-inspiring) figures).

Please do provide more such posts. Recommended readings are even more welcome.

Regards.

(PS: Slightly off-topic, but if you have a Facebook account I would like to add you to my friends list.)

So, witten has been working on the Langlands program for 32 years. Nice to know that.

Peter, was the conference held at the James Clerk Maxwell building on the Kings Buildings campus or somewhere else? Just curious.

May I point out the recently published monograph: “Stable Homotopy around the Arf-Kervaire Invariant” by V. Snaith (Birkhauser Verlag, 2009) where we can find some history and background on the (non)-existence of framed manifolds of Arf-Kervaire invariant 1 in 2^n-2 dimensions, with the known exceptions?

Math goes quite fast, sometimes, as Mike Hopkins’ breakthrough shows!

Stevem,

The conference was held in a new building, the “Informatics Forum”.

Thanks Felipe,

For an interesting survey of the state of the Kervaire invariant problem before last week, see the introductory chapter of the Snaith book, which is available courtesy of Springer here

http://www.springerlink.com/content/t75067/front-matter.pdf

I am now maintaining an Atiyah80 home page http://www.maths.ed.ac.uk/~aar/atiyah80.htm

concerning the conference itself (including photos, cake, gift, wine label etc.) and subsequent mathematical developments.

Andrew,

Thanks, I’ll add that link to the main body of the posting. And thanks for all your work organizing the wonderful conference!

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For those interested: Witten has recently posted a preprint based on his Atiyah 80 talk.