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The Newlander-Nirenberg theorem February 27, 2009

Posted by Joel Kamnitzer in Algebraic Geometry, differential geometry, Paper Advertisement.
3 comments

As you may have noticed, I haven’t written a post in a long time.  Fortunately my fellow bloggers have been doing a great job in my absence.  Though I haven’t been posting, I have been reading and enjoying the blog regularly.

If you want to know what I have been up to in the meantime, the short answer is “Bella”. For a longer, more mathematical answer, you could check out this post or look at the recent papers that I’ve posted on the arxiv with Sabin Cautis and Tony Licata.

In this post, however, I would like to discuss something completely different, namely the Newlander-Nirenberg theorem. Let me begin by recalling the setting. An almost complex structure on a smooth manifold M is an endomorphism J of the tangent bundle such that J^2 = -1 .

One way to get an almost complex structure is to start with a complex manifold. The definition of a complex manifold is just like that of a smooth manifold — ie in terms of an atlas — except that we require that the transition functions be holomorphic.

The underlying real manifold of a complex manifold has an almost complex structure. An almost complex structure which arises in this way is called integrable. The N-N theorem gives you a criterion for testing whether an almost complex structure is integrable.

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A warmup: GL_t for t not an integer February 25, 2009

Posted by Noah Snyder in Category Theory.
20 comments

This post is meant as a warm-up to my planned follow-up to David’s post.  You don’t have to have read his post to understand this one, but there are a few technical details at the end where I’ll refer you to the end of his post.  Most of what I learned here I learned from reading this expository paper by Ostrik which I read in preparation for some talks I gave my second year of grad school.

If you like to draw pictures, how do you think about the representation theory of groups?  Well, you use an oriented strand for some basic or fundamental representation V of a group, you orient the strand the other way for the dual representation, you use disjoint union for tensor product.  Now you can try to draw pictures for maps between tensor products V\otimes V \otimes V^* \rightarrow V^* \otimes V^* say.  Stacking these pictures is composition, and disjoint union is tensor product.  This should be pretty familiar to you if you’ve read the archives for “this week in mathematical physics.”

Since we’re looking at the category of representations of a group we have a bonus bit of information: this tensor category is symmetric.  There’s a canonical map V\otimes W \rightarrow W \otimes V which satisfies the relations of the symmetric group.  In pictures this can be drawn using a crossing.  (Warning: this is not a crossing in 3-dimensional space, you need to either think of your pictures as being in 4-dimensions or not embedded at all.)

Ok, so what is \mathrm{GL}_t?  It should be a linear symmetric tensor category with a representation V that has no properties other than having dimension t.  What does it mean to have dimension t in picture language?  It means that a closed loop should have the value t.  So here’s our proposed category \mathrm{GL}_t:

  • Objects are collections of oriented points on a line.
  • Morphisms are linear combinations of oriented strands (unembedded or in dimension greater than 4 so that the crossings satisfy the relations of the symmetric group) whose boundaries match the objects that they’re mapping between.
  • Composition is stacking of diagrams with the relation that a closed loop can be removed for a multiplicative factor of t.
  • Tensor product is disjoint union.

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Deligne’s “La Categorie des Representations du Groupe Symetrique S_t, lorsque t n’est pas un Entier Naturel.” February 25, 2009

Posted by David Speyer in Uncategorized.
25 comments

In the requests section, Noah asks for help understanding Deligne’s paper “La Categorie des Representations du Groupe Symetrique S_t, lorsque t n’est pas un Entier Naturel.” Peter Arndt refers us to two papers of Knop extending the construction. I’ve been reading them, and I think I understand what’s going on. Moreover, this material is making me think of combinatorial question which seem interesting to me, although others may already have thought about them. I should warn everyone that I had never come across this material until Noah’s question, so there is a risk that I will say something completely ignorant. I also have made no attempt to make my notation match anyone else’s, because I don’t like their notation; if someone wants to compile a dictionary, that would be great.

Nonetheless, here is my attempt at explaining Deligne’s construction in English, with only low level category theory, and with pictures!

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Threading! February 21, 2009

Posted by Ben Webster in Uncategorized.
18 comments

EDIT: Threading was turned off, due to popular demand.

Well, ask, and ye shall (eventually) receive.  WordPress.com has finally added support for threaded comments, and we’ve added it to the blog.  Everyone enjoy.

More application materials blogging February 20, 2009

Posted by Ben Webster in jobs.
18 comments

So, maybe it’s a bit late in the season to be worrying about CV grooming, but I’m curious: what are people’s opinions on the talk list in one’s CV?

Obviously there’s some point where one wants to include every public talk one has ever given, and then clearly some point where one stops. Once one stops, then one has to decide which are worthy of inclusion, and it’s completely unclear to me how one decides this. How many is too many?  Does one slant toward recent talks? Toward a diversity of different talks? Toward particularly prestigious fora? This is the sort of point for which there seems to be no guidance online, so let’s create some.

There is no p-adic 2 pi i February 18, 2009

Posted by David Speyer in Uncategorized.
10 comments

You may not know this, but we can see the search terms people use to find our blog. Yesterday, four people came to our blog using the search string “p-adic 2 pi i”. Presumably, people want to know what the p-adic analogue of 2 \pi i is.

There isn’t one, and there is a good reason why. I assume that we can all agree that the most important property of 2 \pi i is that it is the period of the complex exponential function. Unfortunately, there are no continuous periodic functions on \mathbb{Q}_p except for the locally constant functions. The reason is very simple. Suppose that f : \mathbb{Q}_p \to \mathbb{Q}_p was periodic with some nonzero period \lambda. Then we would have f(k \lambda) = f(0) for every integer k. But, in the p-adic toplogy, the integers are dense in any neighbourhood of the identity. So f would take the value f(0) infinnitely often near 0, and would thus be constant in a neighbourhood of 0. The same argument gives that f is constant in a neighborhood of any a. (And the same arguement applies if you take f to be complex valued.)

There is an interesting \mathbb{F}_p[[t]]-analgoue of 2 \pi i, related to the Carlitz exponential. But that is a more complicated, and more interesting, subject.

Judging from our searches, what people want to know today is when Ed Witten’s seminar meets. I’m afraid I don’t know that!

Update: This post has drawn some comments which are far smarter than what I wrote. There is a way to make p-adic sense of 2 \pi i, although I don’t understand what it is yet. Come and see our very smart commenters try to explain it to me!

Update: The conversation seems to have stopped for the moment, but I am still trying to understand these period rings, with help from Jay. If I get it, I’ll be sure to post an explanation here.

Toric Varieties and Fans February 18, 2009

Posted by David Speyer in Uncategorized.
4 comments

In my previous post, I told you how to create a toric variety from a collection of cones. I also told you that the cones had to satisfy certain, unspecified compatibility conditions. (And that, starting from any lattice polytope, you could build a legitimate collection of cones.)

In this post, I want to tell you what these conditions are. I also want to introduce the way that most experts think about these matters, in terms of fans. For most people, fans are less intuitive than the collections of cones I used in the previous post. But, if you are going to want to think about toric varieties, you should eventually learn how to use the fan language.

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About that field isomorphism… February 15, 2009

Posted by Ben Webster in Galois theory.
10 comments

There was a question about the isomorphism of fields between \mathbb{C} and \bar{\mathbb{Q}}_\ell. I just stuck that in this previous post as a comment, so let me elaborate a tiny bit.

The important thing about this kind of field isomorphism is that it tells you almost nothing interesting. What makes \mathbb{C} interesting is its topology, and when you lose that, \mathbb{C} becomes a rather floppy, uninteresting object.

So, I guess, my point is, this result is helpful as a security blanket, when you have to face the \ell-adics the first time, but it actually has few really important consequences.

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The Grothendieck trace formula as categorification, I: the category and the comparison theorem February 12, 2009

Posted by Ben Webster in Algebraic Geometry.
17 comments

The proof of the Weil conjectures is one of these great achievements of modern mathematics which has more trouble than it should diffusing out of its original field. It’s understandable why it would be scary to people. It involves things over characteristic p, which is a pretty scary place, and a lot of hard algebraic geometry.

On the other hand, I think it’s a shame that so many people who like to think about categorification don’t realize arithmetic geometers have been doing it for decades; thus, I’m going to try to write some posts which translate.

The important fact here is the function-sheaf correspondence. In this post, I’ll just try to tell you what kind of sheaves we’ll have, and then in a forthcoming one, we’ll talk about the functions (surprisingly, yes, it makes sense to talk about the sheaves first).

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Well, that was much ado about nothing February 12, 2009

Posted by Ben Webster in math life, selling out.
3 comments

Hmm, it seems after all that, the original NSF appropriation made it throught the House/Senate conference, so assuming the Senate actually votes on the bill, we should be set.

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