The Research Works Act January 20, 2012
Posted by Ben Webster in Uncategorized.add a comment
Sigh. Congress is trying (again) to undermine the NIH’s open access policy. As usual, you should write your congress-critters. You can do that easily from OpenCongress here. My letter is below the fold. (more…)
Why do symplectic manifolds need to be closed? January 14, 2012
Posted by Ben Webster in symplectic geometry.8 comments
In a comment on my last post, plm suggests that my condition about the rules of turning energy functions into flows be itself time invariant is the only way to justify requiring that symplectic forms be closed.
While I agree that this is a good way of thinking about what closeness is supposed to mean, and maybe the best, I would dispute that it’s the only one. It’s a very reasonable condition from the pure math side as a kind of “flatness.” (more…)
What is a symplectic manifold, really? January 9, 2012
Posted by Ben Webster in mathematical physics, symplectic geometry.7 comments
I’m teaching a graduate course in symplectic geometry and GIT this semester, and am going to try to produce some posts related to lectures I’m giving there. Hopefully, this will help me think things through and put some new exposition out there on the internet.
So, obviously, the first question is “what is a symplectic manifold?” Now, wikipedia will tell you it’s a manifold equipped with a non-degenerate closed 2-form. Certainly that’s right, but it doesn’t tell a novice in symplectic geometry much. Why think about such a structure?
So let me try to put a different spin on this. This isn’t all that new of a spin (in fact, Henry Cohn wrote almost exactly the same thing here), but I don’t know of anywhere symplectic manifolds are really presented like this: I want to think of a symplectic manifold as a space where one can do a particular flavor of classical mechanics. (more…)
Rationality of the zeta function mod p December 12, 2011
Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.5 comments
Here’s a neat argument about counting points that you could present at the end of a second course in number theory. I’m sure it’s not original, but, hey, that’s what blogs are for!
Let 
and that the eigenvalues of 
Here I am using the Lefschetz hyperplane theorem to know what 
This is, of course, a famously hard theorem. The claim about the eigenvalues is the hardest part, but simply the existence of a matrix for which this formula holds is already quite hard; the first proof was due to Dwork.
What I am going to show you is that there is a much easier proof of the above formula modulo 
Drink with me to days gone by November 22, 2011
Posted by Noah Snyder in Off Topic.4 comments
This may not be of interest to most of our readers, but I have sad news that’s relevant to many of the bloggers. Last weekend Raleigh’s burned down. It was the traditional place for beers after the seminar for which this blog is named, and the first draft of my qual syllabus was originally written on a Raleigh’s napkin (back when they had napkins that were perfect for writing math on). It’s always sad to lose a place that felt like home. Have a drink outside in memory.
Farey fractions, Ford circles, and SL_2. October 18, 2011
Posted by Scott Carnahan in group theory, Number theory.12 comments
The topic of this post came up during a conversation with some physicists about the fractional quantum Hall effect (which is quite fascinating, but I don’t feel particularly qualified to discuss). I have decided to set it down here in the hope that, as long as I have an internet-capable device with me, I won’t have to rederive it in front of people again. Some of this material appears in Apostol’s Modular functions and Dirichlet series in number theory and Conway’s The sensual form. I’d be happy to hear about other good treatments.
The many disguises of rhombus tilings October 17, 2011
Posted by David Speyer in Uncategorized.2 comments
For a while now, I thought I should write up a blog post on the many different combinatorial objects which are in bijection with rhombus tilings of centrally symmetric polygons: various constructions with reduced words, oriented matroids, projections of hypercubes, strongly separated sets and so forth. But I kept putting it off because I knew it would take a long time to write correctly, with all the motivation and lots of figures it deserved.
Yesterday I had a very nice conversation about rhombus tilings with Lionel Levine, and I decided it was time to consolidate my knowledge and fill in the gaps. So I sat down and dumped everything I could think of into a question on MO. Note that this is a question and even a community wiki one — if you know of more results to add, please head over there and add them!
Reflecting on the sociology of mathematics, it seems to me that we are seeing a growth in ways to do a quick and sloppy job publishing something. Fifteen years ago, this would have been a survey article that would have taken weeks for me to research and edit. Five years ago, this would have been a blog post written over several days. Now I’ve written something much less polished, but I was able to do it in an evening in between taking care of my baby. I’m not sure whether it’s good or bad, but it seems to have been the only way I could get this written at all.
The NSF and career-life balance October 9, 2011
Posted by Noah Snyder in math life, NSF madness, WANT.3 comments
The NSF recently announced some new policies concerning work-life balance. There seems to have been a publicity push about it on the part of the White House, as it made the regular news. The main changes seem to be adding flexibility to grant rules for new parents. Mostly pretty obvious stuff like letting people delay the use of their grant if they go on parental leave. Good ideas to be sure, but mostly just catching up to what they already should have been doing.
This reminded me of one of my favorite ideas I’ve heard for an NSF policy change which would help career-life balance. Currently the MSPRF postdoc policy reads:
Changes in the host institution will be approved only under extremely unusual and compelling circumstances… Securing a position at an institution other than the proposed host institution is not considered an “extremely unusual and compelling circumstance.”
The suggestion is to change this by adding the line:
Nonetheless, if the fellow has a partner who is unable to procure a job near the sponsoring institution, and both the fellow and their partner have job offers in other city, that will be considered compelling circumstances.
Things learned today in calculus class October 5, 2011
Posted by David Speyer in Uncategorized.2 comments
Usain Bolt can accelerate at 
For anyone who is going to be teaching about computing derivatives numerically, my students really enjoyed looking at the data in this paper. (I give them just a scan of table 1, and have them do the analysis themselves.)
Any other great data sources?
Subfactors of index less than 5 October 4, 2011
Posted by Noah Snyder in Uncategorized.9 comments
Masaki Izumi, Vaughan Jones, Scott Morrison and I recently uploaded to the arXiv the 3rd and final part of the four part series “Subfactors of index less than 5.” This is a project we’ve been working on for a long time (since Emily, Scott and I started running Planar Algebra Programming Camps in spring of ’08), and after three years and a lot of work from many people it’s very exciting to finally have made it there.
In this post I’ll state the main theorem, say a few words about the history, and then explain the main takeaway lesson we learned in this project.
