Here is a very basic question that has come up in some work I’m doing with Diane Maclagan. There is lots of algebraic geometry in our intended application, but I think that what I really need is a better understanding of the underlying linear algebra. First, let me review some even more basic ideas. Let and be two finite dimensional vector spaces over a field and let be a bilinear pairing. For a subspace of , define the subspace to be the space of those vectors such that for all . We can also define for any subspace of .

Then the “Fundamental Theorem of Bilinear Pairings” is the following: for any subspace of , we have . In particular, if and only if

OK, that was pretty simple. My situation is that, instead of having a pairing down to the ground field, we have a bilinear pairing into another finite dimensional vector space. 

What is the new fundamental theorem characterizing ? In particular, under what conditions do we have ?

Many thanks!

Mohib Ali asks:

Can you talk little bit about the ISI Journals which reply faster than others, in other words they take minimal time for their decision about acceptance or rejection?

Unfortunately, the answer is “No. Sorry. I wish I knew something to tell you about that.” (more…)

So, Scott’s comment about being dictator of a math department got me thinking (very idly), and I came up with a hypothetical for you guys: imagine you earn a preposterous amount of money one way or another, and for whatever reason you decide that the best use of it is a new institution dedicated to research in mathematics (I don’t actually agree with this premise, but one can always assume that one has a truly preposterous amount of money, and have already spent a bunch of it on family planning NGOs and smart growth advocacy, or whatever other causes you might think are a higher priority than mathematics). What model would you go with? (more…)

Algebraic stacks are essential to my research.  This is more acceptable now than it was twenty years ago, but it still presents a bit of a language barrier.  Most mathematicians, I think, don’t know what a stack is in the way that they know what a manifold or a scheme is.  So I want to use this post to explain what stacks are, with an eye towards their appearance in mathematical physics.  I won’t quite define them (see Vistoli’s notes for that), but I’ll get you a lot closer than Harris & Morrison do (see p. 139), hopefully close enough to be comfortable that you know what’s going on when someone says “stack”.

Let’s start by saying that a space is what you get when you start with a set and then add some geometry. Maybe make the set into a manifold, maybe make it into a scheme; you can choose your favorite category.   The elements of the set become the points of your space.

A stack is what you get when you start with a groupoid instead of a set, and then add geometry. A groupoid, remember, is a category whose morphisms are all isomorphisms.  This means that the points of a stack aren’t just elements of some set; they also come equipped with a bunch of relations, telling you which points are isomorphic to each other.

So why would anyone try to make a groupoid into a geometry?

(more…)

Subversion, often abbreviated as SVN, is a “version control system”. Prompted by Nathan’s request to hear about collaborative software for mathematicians, and the comments on Ben’s post on the subject, I’m going to briefly describe how you might use Subversion to collaborate on a maths paper. Even better, I’m offering to set up a subversion repository for any mathematician who’d like to try it. Jump to the bottom if you already have your subversion-fu, and just want the goodies.

(more…)

So, looking at Google Reader stats today, I discovered that our blog has a bit over 400 subscribers (for reference, that’s a bit over half of the n-category cafe’s subscriber base, and between a third and fourth of Terry Tao. I guess it pays to update regularly).

On the other hand, our comments feed has exactly two subscribers. My question is: who’s the other one?

Nathan Dunfield (a new commenter!) supplies our first request:

How about a discussion of long-distance collaboration tools and methods, beyond just using email and talking on the phone? It seems like there are a lot things that might work, e.g. pointing a cheap webcam at piece of paper, using collaborative text editors (e.g. SubEthaEdit), IM’ing (some clients have LaTeX support, I tnink), virtual whiteboards (e.g. Scriblink.com), but which might also turn out to be useless in practice for all sorts of annoying technical reasons. So it would be interesting to hear from people who have had success or failure with various methods.

Unfortunately, I have nothing insightful to say on this topic (I would be really excited to hear if any one else has exciting ideas, for reasons which will be clear below). This is a little sad, since I’m a perfect candidate for having done something interesting in this area. I’m pretty technophilic, even for a mathematician, and am currently writing two different papers with two different people in Germany, and working on another paper in a group of 4 where I don’t think more than 2 of have been in the same state simultaneously in over a year. Almost all this work has been done over email, or face-to-face, with occasional phone conversations and one video chat on Skype (but with no attempt to write anything on a board or paper, just gesticulation). In particular, the last paper I mentioned has been written entirely while we were all permanently in different locations (me in Princeton and Boston, one in Oregon, one in California, and one in Amherst), and generated an enormous number of emails, I think around 500 (thank Gbus for Gmail).

So, why haven’t I done anything more exciting? (more…)

So, maybe I’m a little late to this party, but I wanted to comment on a recent XCKD (yes, Isabel blogged it first, though I added it to our del.icio.us before she blogged it). This arranged several scientific fields, by order of purity. Of course, the mathematician is standing at the far end of the scale (saying “Oh hey, I didn’t see you guys all the way over there”). The same point has circulated in saying form as

“Biologists defer to chemists. Chemists defer to physicists. Physicists defer to mathematicians. Mathematicians defer only to God”

(does anyone have an attribution for this quote? My Google-fu was insufficient).

Now, I think it’s clear that I love XKCD (one of these days, I’ll get my shit together and drive out to the Boston meet-up. Who knows, maybe next Saturday), but I can’t help but disagree a bit with the premise of this comic. I think it misses something pretty important about mathematics. (more…)

You may have noticed that there’s a new tab up at the top of the page. That’s because I’ve added a new page called Requests where our blog’s readers are encouraged to suggest blogging topics. Of course, we’re not qualified to discourse on everything (and we do try to stay roughly on topic) so we may not be able to do everything, but I’m really curious to see what good suggestions there are out there. Other bloggers should feel free to poach ideas as well, and link in comments there.

Dear MathJobs.org,

Let me just start by saying I’m a big fan. Really, you’re a most excellent website, who filled a definite need in the mathematical community. You’ve improved the life of pretty much everybody who’s applied for academic jobs in math in the past couple of years, and I’m sure you’ll continue to do a good job of this.

But, true to my nature, I’m not writing to discuss your manifold virtues, but the one painfully obvious one which is missing. I’m referring, of course, to implementing RSS feeds. I mean, lists of job openings which are sporadically updated are precisely the sort of information RSS was designed to deal with. In your present set-up, it’s very easy for people to miss job openings if they go a little too long between checking the website, a problem which could be rectified in a…well…really simple way by RSS. I mean, Craigslist has RSS feeds working. Do you really see yourself as the sort of website who cannot keep up with Craigslist? I think you can do better.

Best wishes,
Ben Webster

EDIT: Well, MathJobs came through as promised.

After such prompt and thoughtful response from the MathJobs crew (see the comments below), I feel like a bit of a heel for the (fairly moderate) level of snark displayed above. I will, however, defend the choice to voice my request in blog form. It wasn’t just because our blog has been hurting for material lately; it also provides a forum for people to discuss what else they would like to see out of MathJobs. Maybe someone else has had some tiny aspect of the website that’s been nagging them. I mean, like I said, MathJobs is great, but no website is perfect. So…anybody else have bright ideas?