Is there an integral condition for zero torsion? January 27, 2011
Posted by David Speyer in Uncategorized.10 comments
Sooner or later, I’m going to want to mention the Levi-Cevita connection in my course, so I want to understand the meaning of the fact that it has no torsion. I think I have enough understanding for teaching purposes. See these Mathoverflow threads 1 2 and these John Baez essays 1 2 3 (scroll down) for some of the ideas I might present.
However, I am left with a question. I haven’t found any integral characterizations of zero-torsion, only infinitesimal ones. Let me explain what I mean by some examples:
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A public service announcement January 26, 2011
Posted by Ben Webster in jobs.31 comments
At the request of Jim Humphreys, I’m making a little PSA about postdoctoral positions at UMass: there seems to have been some confusion over the labels used for the job searches on MathJobs. The label “POSTDOC” is a position is statistics. If you are not a statistician, and wish to have a job at UMass next year, make sure you applied for the Visiting Assistant Professor position, labeled “VAP.”
Just so there’s something for people to discuss: why is the mathematics community so bad at coming up with a unifying terminology for postdoctoral positions? The academic ranks on the tenure track as named very consistently throughout the US in both public and private institutions, but postdoctoral ones are a very confusing mishmash. You can be a “Fellow,” a “Instructor”, a “Professor” or even a “Member” with all kinds of possible adjectives without any kind of clear terminology behind them. I’m sure this is mostly just for historical reasons, but is there any hope of rectifying the situation?
Representation theory course January 24, 2011
Posted by Joel Kamnitzer in representation theory, teaching.9 comments
Well, like David, I am teaching a course this semester and writing up notes.
My course is on representation theory. More specifically, I hope to cover the basics of the representation theory of complex reductive groups, including the Borel-Weil theorem. In my class, I have started from the theory of compact groups, for two reasons. First, that is the way, I learned the subject from my advisor Allen during a couple of great courses. Second, I am following up on a course last semester taught by Eckhard Meinrenken on compact groups.
Feel free to take a look at the notes on the course webpage and give me any feedback.
Very soon, I will reach the difficult task of explaining complexification of compact groups. As I complained about in my previous post, I don’t feel that this topic is covered properly in any source, so I am bit struggling with it. Anyway, the answers to that post did help me out, so we will see what happens.
Cluster algebras and canonical bases in Oregon January 22, 2011
Posted by David Speyer in Uncategorized.10 comments
Nick Proudfoot and I are organizing a week long workshop in June 2011 on connections between Cluster Algebras and Canonical Bases. Our target audience is graduate students and postdocs who would like to get up to speed on the many fields which overlap here, including cluster algebras, representation theory of Lie groups and quantum groups, and perverse sheaves. Roughly speaking, Nick is handling logistics, I am handling math, and we are going to try to rope Ben in one way or another.
To quote from Nick’s description of the workshop:
The workshop will be aimed at graduate students and postdocs, with most of the talks given by the participants. We do not expect any of the participants to be experts in all of the subjects that are represented in this workshop. Rather, we hope to bring together participants with diverse backgrounds, and to weave these backgrounds together into a coherent picture through a combination of lectures and informal discussion sessions.
This is a really fun family of subjects, and one which is leading to a lot of exciting research right now, so it should make an excellent workshop.
Some financial support is available, see our webpage for details.
Crowdsourced department ranking January 13, 2011
Posted by Ben Webster in math life.5 comments
So, I was reading The Monkey Cage this morning, and happened upon a post about crowdsourcing rankings of political science and sociology departments. Basically, there’s a website that lets you put in an arbitrary list of options (which it somewhat unfortunately insists on calling “ideas”) and gives the internet as a whole to vote on them pairwise. Of course, the next step was obvious (and while, yes, it was procrastination, in my defense it was actually incredibly easy), so I set up such a listing for math departments. If you have nothing better to do with your time, you can go and vote a bit. You can also see the results, though of course, at the moment they are pretty meaningless (not that they won’t be meaningless after lots of people vote, but I think at the moment, some schools have received no votes either way).
Before anybody complains about the schools listed: I just took the listing of graduate programs in mathematics in the United States and Canada from the Notable Math Wiki. Obviously, it was a little unfortunate to have to be so nationalistic (continentalistic?) but otherwise, I think the overwhelming number of pairs for anyone would have been two schools they had never heard of. If somebody else wants to set up an option for schools in different parts of the world, of course, they are free.
EDIT: I decided the full list was just too unwieldy; I eliminated all the schools whose “score” (roughly their probability of being liked better than a random school) was below 40 (though, of course, the remaining ones are going to spread out now). Interestingly, the results are not nearly as “conventional wisdom” as I expected; Northwestern is a lovely school, but I don’t think many would rank it above Harvard, MIT, and Princeton as it is at the time of writing. If that’s a statistical fluke, presumably it will go away a lot faster now, as the remaining schools will get voted on more often.
My course begins January 7, 2011
Posted by David Speyer in Algebraic Geometry, teaching.12 comments
I just put up lecture notes for the first lecture from my course Algebraic Geometry II, a course on the complex approach to algebraic geometry, loosely taught out of Claire Voisin’s book.
The mathematical content of my opening lecture is something I have often considered as a blog topic: Seven ways of computing the cohomology of 
I am going to have my students take turns preparing electronic notes, which I will edit and post on the course website. Come read along!
Notes to date:
Differential forms
Sheaves
Sheaf cohomology
Sheaf cohomology II, de Rham cohomology
Complex differential forms
Dolbeault’s Lemma
Cohomology Vanishes on Polydiscs
deRham computation of topological cohomology of affine varieties — preview
Cartan’s Lemma
Gluing resolutions
