Dr. Michael Bishop, one of my high school teachers, passed away last Wednesday, December 9.

I don’t know the circumstances — last I saw him, in July, he was in great health and full of his usual enthusiasm. I realise that sbseminar might not be the best place for an obituary, but I hope you’ll bear with me for a moment.

I owe a lot to Michael, and several of my fellow students who I’ve talked to in the last few days feel the same way. My decision to study physics and mathematics at university — well, maybe that was a foregone conclusion, but it was certainly strongly influenced by him! Perhaps the quickest way I can say what I need to say is that even as a high school student, he treated me, and my fellow students, as intellectual peers, and we enjoyed an experience with him much more like university (perhaps even grad school).

His “day job” was teaching chemistry, but he had little patience for the exam curriculum (indeed, he advocated that the school withdraw from the state examination system, something that was actually considered for a while during a particularly depressing curriculum revision), and where he really got going was as “Master in charge of academic extension”. For years he very successfully trained students for the International Chemistry Olympiad. He also put together “Kaleidoscope Eyes”, a magazine published in the school for all the various extension projects going on. At some point I got interested in the tautochrone and brachistochrone problems, and Mike suggested I go read Volume 2, Chapter 19 of the Feynman lectures in order to learn the calculus of variations, and I wrote a long and rambling piece about my experience solving these problems. The highpoint of “Kaleidoscope Eyes” was, for me, a piece written by two students about Hittite grammar — illustrated with a beautifully chosen extract of the Hittite legal code, and titled “Bestiality in the Ancient World“.

He also taught some extra classes — he decided that the best response to the state physics curriculum being lame was to finish teaching it a term early, and then do something fun. Thus, we got courses on Lagrangian dynamics, on special relativity, and on quantum mechanics. The quantum mechanics course was a triumph — teaching to students who’d never met a matrix or solved a differential equation, he managed to get us to the point we could successfully estimate the first ionisation energy of H_2^+ (the molecule H_2^+ is just a hydrogen molecule with only one electron: the ionisation energy is the gap between the lowest and second lowest eigenvalues)! I’m still impressed by that one. (Hints: you can estimate eigenvalues by optimising parameters in a test function, you can use bezier splines to translate “qualitative” knowledge about eigenstates into test functions, and you can guess that the second eigenstate must be antisymmetric, by orthogonality.)

I was in Australia in July and I stopped by the school one afternoon and found Michael, and we went down to Bill and Tony’s to have a coffee. He told me about his latest adventures — he’d been playing “viking chess” with some students. The rules are not well attested, so they’d decided to try to reconstruct the rules by playing variations and seeing what worked best. Experimental paleoludology!

Thank you Michael, for all you did! I suspect there’s a whole crowd of young Australian scientists out there who’ll miss you as much as I will.

A few results

1 (Bjorner, Eidelman and Ziegler) Suppose we have a finite collection of great circles on a sphere, none of them through the north or nouth pole. Let be the set of regions in the complement of these circles, and suppose that every region is a triangle. Put a partial order on by if is south of every circle that is south of. Show that, for and , there is some such that if and only if and .

2 (Mozes, see also IMO 1986.3) Let be a finite graph, and let be a real valued function on the vertices of . Consider the following (solitaire) game: find a vertex for which is negative. Replace by and, for every vertex that neighbors , decrease by . The game ends if all of the are nonnegative. You and I start playing with the same graph and the same . Show that, if my game ends in moves at position , then your game will end in the same position, in the same number of moves.

3 (Poincare, Birkhoff and Witt) Define to be the ring generated by , and , subject to the relations , and . Show that any element of can be expressed uniquely as a sum of elements of the form . (Uniqueness is up to rearranging the sum and combining like terms.)

4 (Jordan and Holder) Let be a finite group. Let

by two sequences of subgroups such that is normal in , with simple, and the same is true for the ’s. Then and the quotients are a permutation of the quotients .

What do all of these have in common? You can remember all of their solutions by drawing the same figure — the diamond!

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This post is based on a conversation I had with Allan Adams at Mathcamp a few summers ago, and I was reminded of it by an aside in Mike Freedman’s talk in Scott’s backyard on Friday. As usual with blog posts based on other people’s talks, all good ideas in this post should be attributed to Allan and Mike and all mistakes to me. Furthermore I think everything I say here is obvious to people who actually know physics.

My basic confusion was how physical intuition (in particular in quantum field theory) could be applied to so many mathematical settings when there’s only one physical world so there’s no reason to think any intuition built up within that single example would apply any more generally than that one example. What Allan pointed out to me is that it’s not true that physicists are only studying one example. Although there may only be one fundamental theory of physics, by looking at various particular physical systems the limiting behavior becomes its own theory. The physics at the surface of a black hole can be thought of as its own example; the physics of superconductors is its own example; etc. Because all of these examples are physical (they involve minimizing actions, they’re quantum, etc.) they have a lot of attributes in common, so intuition and general techniques can be developed by understanding their commonalities.

Mike made two comments in his talk (on K-theory and superconductors) that flesh out this idea further. He was discussing the BCS superconductor and explained that when physicists refer to a theory by initials they’re not just being polite, what they mean is that you’re studying the mathematical model rather than any particularly instantation of it. In particular, the model doesn’t care if there are exactly 10^9 electron pairs or the exact composition of the material, it is studying the abstract setting that appears in the limit. By calling it the “BCS superconductor” they mean that in some sense they’re studying the physics of a different world. In particular, in the BCS setting since you’re assuming that there’s a huge sea of electron pairs the “vacuum” consists of this huge sea. This explains how physicists can develop intuition for more general notions of vacuum: they’re not always studying the absolute vacuum, they’re also studying other systems with states that have the properties of being a “vacuum.” This particular vacuum has a delightfully strange property. Since a new electron pair doesn’t change the underlying vacuum, in this “world” electric charge isn’t preserved!

I am coming to the point in my career where I will be expected to take graduate students, and I’d like some advice about finding problems for them. How responsible am I for making sure that a problem is solvable and not already under attack elsewhere? I have a (private) list of problems that might be suitable for attack with tropical methods, or using cluster algebras. In most cases, the reason that I have not worked on these problems myself is that I would have to do a fair bit of research to find out the current state of the field and make sure that I wasn’t missing something stupid. Is it fair to pass this sort of thing off to a graduate student?

News is circulating on the internet that Israel Gelfand has died. My ultimate source for this is LiveJournal, so take it with a grain of salt, but it’s not hard to believe, given that the guy was 96. Anyways, seminars will never be the same again.

This seems pretty cool: this website allows you to draw a picture of the LaTeX symbol you would like, and then searches the popular “Comprehensive LaTeX Symbol List” for them. It’s far from perfect, but still seems worth bookmarking.

Also, it gets smarter as people use it, so be sure to tell the website when it finds the right symbol, so it gets smarter for other people.

So, in another thread Scott C. questioned why I would say that TQFT’s produce all numerical invariants of n-manifolds. There is a good reason for this, though maybe not good enough.

The reason is that there is a construction for starting with a numerical invariant, and extending it to a TQFT:

• Start with a numerical invariant of n-manifolds.
• Now consider your favorite n-1-manifold X.
• Consider the vector space spanned by n-manifolds with isomorphisms of their boundary to X.
• The space has an inner product, given by gluing two n-manifolds along X, and summing their invariants. Quotient by the kernel of this and call this .

Congratulations. You’ve got your TQFT. This construction is a little weird; in particular, it doesn’t seem to have to be monoidal, which is bad. Still, it tells you TQFT’s are in some sense easier to construct than you probably thought.

Presumably, one can extend this further by giving an n-2-manifold the category whose objects are formal sums of manifolds bounding it, with morphism spaces given by gluing, etc. Anyways, this is what I had in mind when writing the infamous sentence in question.

I think this past year, we were all assuming that there were more NSF postdocs, due to the stimulus, but I hadn’t heard numbers, until I decided to look them up on the NSF website. I thought you guys might be interested to see as well. These may not be perfectly accurate (I just searched the database for awards in the calendar year 200*), but they’re close enough.

2009: 56
2008: 41
2007: 29
2006: 30
2005: 32

On a lighter note, I’m informed that there’s recently been a graphic novel published on the life of Bertrand Russell. I haven’t read it myself, but LogBlog likes it. I’m of a mixed opinion about Russell’s achievements as a mathematician, but no one can deny he had an interesting life. Of particular interest to some readers is the on-going tour, coming (maybe) to a town near yours soon.