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Local systems — what do you know about connections? May 28, 2009

Posted by David Speyer in Uncategorized.
23 comments

I’m working on the next local systems post, which will be about connections on vector bundles. It’s getting really long though. So, a question for my audience: Should I

(1) Assume you have already seen the definition of a connection, and of integrability (also known as flatness, also known as zero curvature) and skip directly to relating this concept to the other ideas I have discussed?

(2) Build connections from scratch, with motivation and definitions?

Don’t you wish every election featured such great choices? Ralph Nader would just go home and take up knitting!

Writing a math paper on a wiki May 27, 2009

Posted by Ben Webster in blegs, crazy ideas, math life.
38 comments

Combining a couple of previous topics, I was wondering: is there a good platform for writing a math paper on a wiki? This seems like a desirable goal, both for small groups of collaborators and for any MMORPG’s (massively multiplayer online research project groups), and I’ve never seen such a thing, but I’ll hold off on crankily complaining about its absence until the blog readership has had a chance to tell me whether it’s out there.

Here’s what such a thing would have to include:

  • The ability to take in proper TeX code, including packages, bibtex, anything else people use in arXiv papers, and produce some kind of reasonable preview. Obviously, it wouldn’t have to be precisely what LaTeX would produce, but it would have to be readable. Clearly this is somewhat possible, since WordPress and Wikipedia do a decent job with it.
  • The ability to sync with a local copy quickly and easily (hopefully with something roughly approximating svn).
  • All the usual wiki features (user control, full history, etc.)

I feel like this is not a lot to ask, since all aspects of it seem to be in wide use in different programs, but I’ve never seen the whole package brought together. Am I just missing out?

Bleg: Independence of points in Picard group May 27, 2009

Posted by David Speyer in Algebraic Geometry.
2 comments

For reasons discussed in my post below, I would like to construct a curve \overline{S} (over \mathbb{C}) and a finite map \overline{F}: S \to \mathbb{C}) with the following properties:

(1) The map \overline{F} is ramified somewhere. (This will be automatic if \overline{F} is not an isomorphism.) Let R \subset \overline{S} be the ramification locus and let S be \overline{S} \setminus R.

(2) The coordinate ring of S has no nontrivial units. This can be thought of as saying that R is not too large.

I think I have such a construction. But my argument that S has no units is basically that I need a bunch of elements of a Picard group to be linearly independent, and I can’t see any relations between them. Obviously, this needs some help! Details follow:

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How not to prove the Jacobian conjecture May 27, 2009

Posted by David Speyer in Algebraic Geometry.
9 comments

A few years ago, I got curious to see how people attack the Jacobian conjecture. The Jacobian conjecture says the following:

Let k be a field of characteristic zero, and let F: k^n \to k^n be an unramified map. Then F is an isomorphism.

The conjecture is usually stated algebraically, not geometrically. That statement goes as follows: Let the map F be given by \displaystyle{(x_1, \ldots, x_n) \mapsto \left( f_1(x_1, \ldots, x_n),\ \ldots,\ f_n(x_1, \ldots, x_n) \right)}.
Define J(F), the Jacobian of F, to be \det(\partial f_i/\partial x_j). Then the algebraic statement is

Let f_1,f_2, …, f_n are n polynomials in n variables such that J(f_1, \ldots, f_n) is in k^*. Then the f_i generate the ring k[x_1, \ldots, x_n].

The geometric meaning of J(F) is that J(F) vanishes at the points where F is ramified1.

I have noticed a pattern in many of the false proofs. Today, I’m going to tell you how to spot these proofs. The warning sign is the phrase “but k[x_1, \ldots, x_n] has no nontrivial units.”

Disclaimer: I am not an expert on the Jacobian conjecture. I will not referee manuscripts on the subject unless they are relevant to my published work in some specific way. Please do not send me preprints on the subject (unless you are a personal friend).

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Is massively collaborative mathematics scalable? May 25, 2009

Posted by Ben Webster in blog triumphalism, crazy ideas, fun problems.
13 comments

I’ve been watching, though not particularly intently, Tim Gowers’s attempt massively collaborative mathematics. I’m not sure if I’ve looked hard enough to judge, but it certainly looks as though it were quite successful. This of course, answers Tim’s original question “is massively collaborative mathematics possible?” positively, but I still have to wonder if it’s sustainable in the long term. Of course, it never seems smart to bet against the possibilities of the Internet combining disperate contributions into valuable knowledge. Certainly, I would say people have tended to underestimate the possibilities of real advances coming from the technology of wikis and blogs. At the same time, it seems hard to imagine that people will really have the energy and time, not to mention mental organization, to follow several such projects at all closely. One of Tim’s take away lessons from the project seemed to be that it shrank in number of participants faster than he expected. And this was in a collaboration prominently featuring two Fields medalists and promoted on what is probably the world’s most prestigious math blog! It seems more likely that as the number of such projects expands the average number of participants will shrink until most are functionally equivalent of the collaborations we are used to today, just with more efficient coauthor location. By which I mean, the important advance will not be the number of people involved, but rather the identity of them.

Not that the value of efficient coauthor location should be minimized! The broader array of people we can stay in contact with due to the Internet is a huge boon to mathematics. It’s just that I suspect any concern over how we will deal with the allocating credit in a 20 person collaboration is a bit premature, at least outside of exceptional cases.

On the other hand, I’m kind of excited about the possibility of proving myself wrong, but haven’t been able to come up with any good projects. Does anyone wanna do that massively collaboratively?

Working equivariantly for the action of a monoidal category May 22, 2009

Posted by Ben Webster in Category Theory.
15 comments

I recently got an email question from Sergey Arkhipov with a question, which I couldn’t answer to my own satisfaction, so I thought I would throw it open to the peanut gallery.

One construction I’ve used a lot in my recent work is the equivariant derived category for the action of a group G on a space X (in basically whatever category you like). This is basically the poor man’s way of understanding sheaves on the quotient stack of that space by the group.

But, of course, one could forget that there was ever a space there, and just remember that you have a category of sheaves on X, which the group G acts on. So, questions:

  1. Is there a construction of the equivariant derived category which makes no reference to the space and just uses the category of sheaves?
  2. If there a generalization of this construction where the action of G can be replaced by one of an arbitrary monoidal category?

The first question is in that class of things I’m sure I could do myself if I forced myself to sit down and do it: the answer is something like replacing the category with the category of locally constant sheaves on BG valued in your category. The second, I’m less sure about.

The Grothendieck trace formula as categorification, II: the function-sheaf correspondence May 21, 2009

Posted by Ben Webster in Algebraic Geometry.
16 comments

So, I started this series a while back with this post, by talking about some sheaves on an algebraic variety.  This post will probably not make a lick of sense to those who haven’t read the first part yet, though you’re welcome to try.  The upshot of that was that one has a machine into which one can can feed in

  1. a variety X defined over \mathbb Z (i.e. defined by polynomials with integer coefficients)
  2. a choice of stratification of X (a good example might be the orbits of an algebraic group with finitely many orbits, so for example, the Schubert cells on a Grassmannian)
  3. a set of acceptable local systems valued in your favorite field k (though, if want to do things properly, your favorite field should probably be \mathbb Q_\ell).

and receive out the other end a category.  The cool fact about this category is that you can think of it as sheaves on the complex variety X_{\mathbb{C}} or its analogue X_{\bar{\mathbb{F}}_p} over characteristic p (let p be your favorite prime) and you will get the same answer.  The former category has the advantage of involving geometry that you probably care about, like the cohomology of smooth varieties, and the latter has the advantage that there is a Frobenius acting.

What I’d like to explain in this post how to analyze the structure of this category, and what that has to do with categorification.   This will require a bit of machinery, but believe me, the result will justify it.

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Thoughts on graduate school: an addendum May 18, 2009

Posted by Ben Webster in math life.
6 comments

Of course, this process could go on endlessly, but I think there was an important point that Noah didn’t emphasize enough: talk to people.

There are a few categories of talking that deserve special attention.

  • You should make a point of going to conferences whenever possible (it can be extremely easy to get travel money for conferences as a grad student), even if they’re not exactly your field.  If you have something to speak about, and can get a speaking spot, even better.  If you’re wondering how one goes to conferences, there’s a simple algorithm. 
    1. read the AMS math calendar
    2. request funding for any ones that sound interesting
    3. rinse and repeat.
  • You should do whatever you can, non-annoyingly, to cultivate relationships with mathematicians, especially ones who are older.  They can give you valuable advice, serve as good references, and can be good collaborators.

I feel like it can’t be emphasized enough: mathematics is a social activity.  You’ll never learn it properly from books and papers, and you can’t rely on your advisor to tell you all the things you need to know.  Rather, you have to talk to the people around you, and make sure you have people around you to talk to.

Of course, different levels of talking are good for different people.  I’m a pretty sociable guy, and that shows in my mathematical work (it’s been almost 3 years since I’ve written a solo paper and don’t have any on the horizon), but even if you don’t want to collaborate with people, you really do need to talk to them about math.

Mike on Topological Quantum Computing, at Georgia May 18, 2009

Posted by Scott Morrison in conferences, low-dimensional topology, quantum computing, talks.
12 comments

I’m here at the 2009 Georgia Topology Conference and Mike Freedman is about to start talking about the current proposal for building a topological quantum computer. I’ll try liveblogging his talk; there’s a copy of the slides at http://stationq.ucsb.edu/docs/Georgia-20090518.pptx (PowerPoint only, sorry!) if you want to see the real thing. I think he recently gave a version of this talk in Berkeley recently, so some of you may have already heard it. I’ll fail miserably at explaining everything he talked about, but ask questions in the comments!

Mike says that the point of the talk will be to explain how it is that there’s a “topological” approach to building a computer, and try to give an idea of the mathematics, physics and engineering problems involved.

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Extended TFTs May 13, 2009

Posted by Chris Schommer-Pries in differential geometry, low-dimensional topology, mathematical physics, Paper Advertisement, QFT, Shamelss Self Promotion, tqft, websites.
10 comments

So I’ve finally managed to bang my dissertation into something more or less ready for public consumption. It is basically finished (except for some typos and spell checking).

It is available on my new website.

The title is “The Classification of Two-Dimensional Extended Topological Field Theories”.

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