The latest “This Week’s Finds” by John Baez discusses Felix Klein and his “Erlanger Programm”, which essentially was the idea that geometry should be understood as the study of Lie groups G, their subgroups H, and coset spaces G/H. This, supplemented with Cartan’s notion of a connection, allowing things that only locally look like G/H, is very much at the heart of our modern view of geometry. John gives links to quite a few things worth reading by and about Klein here. Another very interesting document is Klein’s own history of 19th century mathematics “Development of mathematics in the 19th century”.
I’m very much looking forward to the next installment of TWF, where John promises some insights into Hecke algebras. He also has a wonderful posting that generated an interesting discussion at the n-category cafe on the topic of mathematical exposition, entitled Why Mathematics Is Boring.
For some more mathematics blogging of the highest possible quality, see Terry Tao’s postings on his Simons lectures at MIT, here, here and here.
I wrote a bit about the LHC Theory Initiative here last year. They have just announced the award of two graduate fellowships and say that they will be awarding postdoctoral fellowships in the future. Unclear from this if they were successful in their efforts to get NSF funding, the solicitation of applications for the fellowship just mentions an older grant to Johns Hopkins.
NPR has run a two part series on the LHC (here and here). The first part features CERN theorist Alvaro de Rujula. I had the great pleasure of taking a particle theory course from him when I was a student at Harvard a very long time ago. He cut an impressive figure, and provided a survey of the subject that was both enlightening and entertaining.
Scott Aaronson provides quotes from someone else (Gian-Carlo Rota) whose lectures I attended around the same time, including one that ends “You and I know that mathematics, by definition, is not and never will be flaky”. I kind of agree with the sentiment in the full quote, but my experience with Rota back then was a rather weird one. For some misguided reason I had decided that since category theory was the most abstract kind of mathematics I had heard of, it would be a good idea to take a course on it. The only course on the subject was a graduate course down at MIT offered by Rota, so I started going down there to sit in on it. A few lectures into the course Rota all of a sudden announced that he had decided that only those students actually enrolled for credit should be taking the course, and that the several of us who were just auditing should leave. So we did, somewhat mystified (it’s not like the room was over-packed or anything). To this day, I still don’t know what that was about. Perhaps Rota knew that he was doing me a favor by stopping me from thinking about category theory at that point in my education, when in retrospect it seems likely that it really would have been somewhat of a waste.
There’s a lot more about Rota at this web-site. His capsule reviews in the back of the journal he edited, Advances in Mathematics, provided outrageous entertainment for many years (although some might at times think that they were, well, flaky…).
(Gian-Carlo Rota) whose lectures I attended around the same time, including one that ends “You and I know that mathematics, by definition, is not and never will be flaky”… although some might at times think that [Rota was] well, flaky.
Ahh, you must be careful always not to mistake mathematics for mathemeticians 🙂
I don’t understand physicists’ almost fetish like obsession with Lie groups. I know it’s nice and all, but I’m a bit confused as to what use all this mathematical theory is to the physics community? I thought physicists “just want the numbers”. I’m finding it hard to believe that you make use of all this when you compute electron shells, solve GR equations, etc, etc. Do you make use of it?
OMF,
Computing electron shells is a classic application of representation theory of the SU(2) group. Angular momentum operators correspond precisely to a basis of the Lie algebra of SU(2), all that stuff in QM books about how to use them to analyze atomic spectra is pure SU(2) group theory and representation theory.
For particle physics, you definitely need SU(3), not just SU(2), and this is a quite non-trivial example of a Lie group. Then there’s all that stuff about energy, momentum, special relativity: the Poincare group. Lie groups are extremely useful in physics.
Peter,
While I never liked Rota’s book reviews in Advances (I found the ones I read too superficial), I highly recommend his “Indiscrete Thoughts”, which is a collection of biographical sketches of matehmaticians he knew and many shorts essays on various topics.
If you have no time for anything else in the book, please read the 4 page biographical sketch on Alonso Church (who was a professor of logic at Princeton when Rota was a student there). And the one on Solomon Lefshetz. Pure gold.
Well, back in the days when I was doing nuclear structure shell model calculations, it very often was the case that diagonalizing the two-body matrices in a basis of irreducible representations of SU(3) — the “nuclear SU(3) model” [not to be confused with QCD SU(3)!] — gave very nice and elegant results that had a physical “meaning” (rotational bands). However, that’s probably not what you meant! 🙂
Except for 18.03 (Differential Equations–the largest class at M.I.T.), Rota disliked teaching large classes. I don’t know exactly why this was, but it was something that he was completely open about. He would frequently try to convince anyone who thought that they weren’t “getting it” to drop a class, as well as other tricks like asking those not enrolled to leave (but if they came back at the next lecture, he wouldn’t actually care). To his common refrain, “If you do not see this, I cannot explain it!” he would sometimes add, “And you should drop the course now!” Other professors certainly tried to winnow their classes down to what they felt were more manageable sizes too; only Rota was the least subtle. Perhaps this had something to do with his popularity, since most of his upper-level classes were indeed partially populated with people not up to grasping the material.
He didn’t paticularly like teaching that either. Well, at least not as much as he would have liked.
I always find it uplifting to encounter such diatribes from experts in the field, especially when one always found the subject opaque. It’s nice to see mathematics presented in a more human light instead of the rather dry and almost dogmatic presentations you normally get. It’s a bit like Feynman’s lectures, almost half apologising for the subject instead of handing down edicts from above. I think it’s better to learn a more fallable subject than a polished one.
He didn’t paticularly like teaching that [link goes to fascinating Rota talk on teaching undergraduate Ordinary Differential Equations courses] either.
Wow. I feel almost kind of ill reading this link. In a lot of ways it’s like reading a laundry list of the reasons why the DiffEQ classes I took as an undergraduate did not go well.
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Something that interested me in that link– in one brief section of that talk Rota appears to be advocating using differential algebra not as an advanced tool for those people who already have a firm grounding in working with differential equations, but actually as a teaching tool for giving introductory students a better understanding of what exactly is happening when differential equations are worked with. Is this tactic Rota proposes something which there are actually examples of out there in the world? I would be very interested to see an example of such a thing, since I personally find I have a much easier time with those mathematical subjects where I have been given a clear idea of how the math is to be viewed in the light of abstract algebra (or some other tool which similarly forces the foundational or axiomatic parts of the mathematical techniques at hand to be made explicit). Rota makes reference to “Cohen’s book of the twenties” as an example of an introductory treatment of differential algebra; does anyone have any idea what book exactly is being referred to here, or even which “Cohen” he refers to?
Maybe: Abraham Cohen, ‘An elementary treatise on differential equations’, 2nd ed., 1933?
Sounds like it, thanks.
Klein and Weyl are somewhat like Maxwell and Einstein as a pair. As time goes on, my respect for them just grows and grows – the luckiest thing that ever happened to me academically was to be shown the books “Elementary Mathematics from and Advanced Standpoint” at an early stage.
Thanks for that pointer.
-drl