Tomorrow morning I’ll head down to Princeton to attend the conference on Gauge Theory and Representation Theory at the IAS. Unfortunately I had to miss the first day of the conference (today), since I would have liked to have heard all the talks, most especially that of Dennis Gaitsgory on local geometric Langlands. Maybe someone who was there will explain to me what he talked about.
That might be even better than attending the lecture, since Gaitsgory’s pedagogical style seems to be rather daunting. Here is an article about his experience teaching linear algebra, and the Harvard Crimson last year ran this frightening account of what it was like to take Math 55 from him. Math 55 is a legendary honors math class for the most fanatical first-year students, and I have fairly vivid memories of my own experience with it (that year it was taught by Konrad Osterwalder and John Hubbard). From what I remember, the first row of the class was occupied by a sizable proportion of the winners of the previous year’s Math Olympiad, and being a rather average student in a math class was a new experience for me. The textbook for the course was a remarkable book by Loomis and Sternberg with the somewhat misleading title Advanced Calculus. It’s now available on-line. Osterwalder made a valiant effort to follow the text during the first semester, while Hubbard more-or-less winged it the second semester, entertaining us by going over in class research papers on dynamical systems and assigning us Spivak’s Calculus on Manifolds as something to work through during the reading period (about a week long) before final exams. Both Osterwalder and Hubbard seem to have been much mellower sorts than Gaitsgory though, since I remember working fairly hard on puzzling out problem sets, but also having a life with quite a lot of other things going on, nothing at all like the experience described in the Crimson article. Kids these days.
The first talk tomorrow morning is supposed to be Maldacena on integrability in N=4 SSYM. He really should be celebrating the day as the 10th anniversary of his amazing paper The Large N Limit of Superconformal Field Theories and Supergravity, which announced the AdS/CFT conjecture and was submitted to the arXiv on November 27, 1997. Work on this conjecture has dominated particle theory in a remarkable way over the last ten years. According to SPIRES, the paper has amassed 4897 citations, at a rate which has only accelerated in recent years, with 551 citations in 2006. It is now the third most heavily cited paper in particle physics, behind only those of Kobayashi-Maskawa and Weinberg. A simple extrapolation suggests that in another four years or so it should become the most heavily cited particle physics paper in the history of the multiverse. Several conferences are celebrating the anniversary, including one next month in Buenos Aires, and another in Fort Lauderdale. Davide Castelvecchi has a quite good popular article on the subject in Science News.
After it’s over, I’ll try and write something about the main topic of the conference, geometric Langlands. In the meantime, my ability to keep the comment section under control may be impaired. Behave.
Update: David Ben-Zvi is putting up his notes from the talks here.
He didn’t talk about Local Geometric Langlands at all, and focused on localization of Kac-Moody algebras (I’m basing this largely on the fact that he said this at the beginning of his talk, I didn’t follow very much of it).
*Not quite on-topic*
Just wondering. How many people finished math 55 in the year you took it?
I didn’t go to Harvard for undergrad, but in my freshman year I ended up doing sort of a mini-DIY version of “math 55” by taking the “honors” level real analysis and abstract algebra courses. (I finished most of the freshman + sophomore “non-proof heavy” math courses before I started university). The textbooks assigned were ones like Rudin and Royden for analysis, and Lang and Hungerford for algebra. The “honors” level courses also had a reputation for high drop rates.
This was the first time I ever had to put a lot of time into any math courses, where I ended up almost completely burning out. This was also on top of also simultaneously taking the “weedout” freshman physics courses.
Thanks. Here “localization” is the sort of geometric construction of representations I’ve always been interested in trying to connect to QFT, I’m sorry I didn’t get to hear the talk (although I might have also got lost when he went into the derived category…). By the way, I just got what looks like an excellent book in the mail when I got home: D-modules, perverse sheaves, and representation theory, by Hotta et. al. Lots about D-modules, and it shows explicitly how they are used in representation theory. No Kac-Moody groups, just the finite-dim theory, but it looks quite readable, unlike almost everything else in this subject. Something to read on the train tomorrow…
From what I remember, there were about 40 the first semester, 20 the second. The first semester I think I was an above average student in the class, the second semester, not so clear….
I just got that D-modules and perverse sheaves book too — looks good, I agree! I saw Gaitsgory give a talk at MIT a few weeks back and it was interesting, for the rather small percentage I understood . . . . He is a pretty good lecturer, although he moves rapidly (as one would expect).
His work (with Frenkel) on localization of modules for Kac-Moody algebras connects somehow to their formulation of local Langlands in characteristic 0. Frenkel’s book on the subject is not easy to read, although it seems as though if I could get through it I’d understand what they’re trying to do.
I must stereotypically respond, “Wow!” It’s not an easy feat to take Math 55. Although I am not in Harvard, I have heard of Math 55’s status among college-level math courses.
I’m somewhat curious to know if the course is more a graduate-type abstract math course or a high school Math Olympiad-type course, or somewhere in between?
I’m quite interested in finding out as, since you are a graduate of the course, you (of any persons I know) would be able to provide some feedback to the following query –
Does a course such as Math 55 help develop professional math skills from an early point onwards? i. e. would one be able to write one’s own research work (if albeit not completely professional) after completing the course?
Side-note: I have Sternberg’s and Rudin’s books, but somehow I found them too formalistic (and/or opaque) to learn from *and* understand the motivating reasons for the math simultaneously.
Can you suggest a more suitable option?
I’ve been spending a lot of time reading the Frenkel book, slowly understanding the details of what they’re doing. I find him relatively easy to follow, although it took me a while to get a feel for what he is trying to do and to see how some of the ideas fit together (there’s still a lot I haven’t understood). He’s a good expositor, especially compared to some other people in this field… I guess a very specific form of my question about Gaitsgory’s talk would be: what did he talk about that’s not explained in Frenkel’s book?
What exactly made the 2nd semester slightly harder for you?
The second semester wasn’t harder, actually I think Hubbard’s teaching style was such that you could get by with less work. But the twenty people who had dropped after the first semester were mostly not the best students in the class, so the twenty that remained were on the whole an impressive group. Luckily I’m not the competitive sort, otherwise I might have really not enjoyed that experience. But, in any case I was far more interested in my quantum mechanics class that semester, the beginning of a life-long love affair…
I didn’t know what my exact rank was in those honors level real analysis and abstract algebra courses I took in my freshman year. Through from I can recall anecdotally, I do remember there were at least two or three other folks who consistently performed better than me on assignments and exams, judging from a casual search of the piles of graded stuff. (The grader just left our graded assignments and exams on the front table for us to pick up ourselves, where it was easy to spot the ones with the better grades).
I was sort of a competitive type back in those days, which was one of the reasons I ended up not majoring in pure math. Another big reason was that I didn’t make it onto the Putnam exam team during my freshman year. It may sound silly now in hindsight, but it was a huge devastating blow to my ego at the time.
After my freshman year was over, I spent some idle time in the university library and came across several books on topics like engineering dynamics, particle physics, quantum mechanics, fluid dynamics, etc … which really grabbed my interest and attention. (I already had enough math background to be able to follow what these books were explaining at a basic rudimentary level). It took me awhile to mellow out from the hyper-competitive mentality, and eventually decided to change my major to physics. In hindsight, I’m glad that I found out early on as to what I was NOT interested in majoring in.
(This is getting off-topic, so I’ll stop here).
Math 55 is designed to prepare students to be professional mathematicians. It is not at all a math olimpiad problem solving course. It would normally cover the basics of real and complex analysis, some basic functional analysis (e.g. spectral theorem for compact operators), point set topology, introductory algebra, maybe some elementary Riemannian geometry (ala Spivak and do Carmo). The handful of students in it are usually exceptional; the year I was a freshman at Harvard the students in it included one who made full prof at Princeton before turning 30, one who is an associate prof at MIT, one who is at Stanford, etc. It’s full of the super fast thinkers who know a lot also.
In general it probably does not provide adequate background for doing research; no freshman course does. In normal circumstances even a very talented, hard-working kid would need several more years of courses before being able to do much of interest to researchers. Of course there are exceptional individuals who do research even in high school, someone like Drinfeld, but these people are anomalous, and their parents probably ought not to be emulated.
Gaitsgory’s talk (at least the part I kinda-sorta understood) started with the ind-scheme of opers on the formal punctured disc and its relation to representations of the corresponding Kac-Moody algebra (of the Langlands dual group) at the critical level. I think this stuff is in Frenkel’s book. I had a hard time following what he did after that though. I think the issue for me is that all of these topics — opers, representations of Kac-Moody algebras (and the completed enveloping algebra) at the critical level, localization of modules on finite or affine flag varieties, hecke eigensheaves, moduli stacks, etc — are not too awful to get an intuition for individually; but I get very confused as to how they’re all supposed to fit together to form a big-picture whole.
From the piece on teaching linear algebra:
“I found the job annoying for two reasons. First, the students were primarily non-math majors.”
This brings up one of my biggest bones to pick with mathematicians. Their insularity knows no bounds. While most fields have become more accepting of interdisciplinary work, mathematics has developed a significant population who are absolutely against any such work. Barbeckiism and abstraction are what is valued, not clarity, and certainly not application. For this reason, more and more engineering and science department are moving towards teaching their students calculus and other mathematics, just because mathematicians think it’s their way or the highway.
I hope odo’s comment helped answer your questions. The course is just an undergraduate course, although a fast-paced one, it’s still a long ways from research-level math. One thing it does for students is to give them enough background so that they can start taking some of the basic grad courses during their undergraduate years, which does get them closer to the point of being able to get into research early in their graduate careers.
I thought it was an ideal course for bright, intellectually ambitious students, throwing at them at much as they can handle. The Crimson article may have been exaggerated, but I don’t think it’s a great idea for first-year undergrads to be spending almost all their time on a math course, no matter how good. A great university like Harvard offers students so many wonderful opportunities to learn many different kinds of things, and they should take advantage of this. As well as taking advantage of being young and irresponsible…
After Math 55, as an undergraduate I took mostly physics courses, only a few math classes, including one graduate course. Most of my math education came later in life, not through taking courses (sometimes through teaching them…). As for books, Loomis/Sternberg is quite a document, but not so great pedagogically. Among undergrad analysis textbooks I’ve seen, the recent series from Princeton looks good, although I haven’t looked that closely at the books.
While one might aspire to be able a teach a top-rated course to people of all backgrounds, my (limited) experience has been that connecting with students who are interested in the topic that you’re teaching is much easier than fighting the very uphill (but worthwhile, of course) battle with students who are forced to take your course as a requirement… This is very much in line with the tone of the rest of Gaitsgory’s anecdote.
I also thought that spending 30-50 hours per week on problem sets for a single class is a bit excessive, and wondered if they’re learning anything else. On the other hand, I do like the idea of exposing the students to a lot of material early in their education so that they can see what’s coming and appreciate better what they’re learning in the context of the entire body of math. Perhaps this sort of thing is more suited to some kind of intensive summer program.
Hi — I’m posting notes to many of the talks
(in particular so far Gaitsgory, Beilinson, Ginzburg and Maldacena)
as we go along at
(available off my GRASP lecture notes page also) — many of them are transparency talks and I don’t even try with those, but
hopefully some of the transparencies will be posted later.
Gaitsgory’s talk discussed two realizations of representations
of Kac-Moody algebras: as D-modules on affine flag manifolds
(localization) and as coherent sheaves on spaces of Langlands parameters. The equivalence between the two is an important
case of the local geometric Langlands conjecture.
I haven’t checked carefully but I think the new results with Frenkel
(not covered in Frenkel’s book) involve the localization
on affine flags (I think their earlier work was on affine Grassmannians) and were presented with a more derived-algebraic-geometry viewpoint.
Thanks David. Congratulations on a fascinating talk today, I enjoyed it!
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Was Hubbard eccentric back then? He was pretty off his rocker when I took the freshman honors class with him at Cornell.
I wouldn’t say “off his rocker”, but definitely a bit eccentric, while highly enthusiastic. I found him rather entertaining, some of the more serious students in the class were a bit put off…