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Best practices: Hispanic serving institutions and the production of future Latino doctorates in science and engineering

This study expands research begun as a baccalaureate origins study of Latino (US citizens) who obtained their doctorates in science and engineering between 1993 and 1997. Data was disaggregated according to gender, Hispanic subgroup, and broad field of science. Institutions were ranked according to their rate of productivity in these categories, regardless of their size. Hispanic Serving Institutions (HSIs) figured prominently in these rankings. Recruitment and retention strategies, as well as cultural characteristics, at eleven of the most productive HSIs have been studied in an effort to develop a research-based set of best practices in the production of future Latino doctorates in S&E at Hispanic Serving Institutions.## The Newlander-Nirenberg theorem February 27, 2009

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.

I am teaching a course in symplectic geometry this semester and I’m trying to do a fair bit of complex geometry, so I thought that I should cover this theorem. However, before the past week, I didn’t know much about the proof of this theorem. Now, that I’ve understood a few things, I would like to share them with you.

Recall from David’s post, the notion of a $(0,1)$ vector field. Actually, David discussed $(0,1)$ forms, but $(0,1)$ vector fields are just the dual notion. Namely they are those complex valued vector fields $M$ which are eigenvectors for $J$ with eigenvalue $-i$.

The N-N theorem can be stated as follows:

Theorem
An almost complex structure $(M, J)$ is integrable iff the $(0,1)$ vector fields on $M$ are closed under the Lie bracket.

This statement is reminiscent of the Frobenius theorem on the integrability of distributions on a real manifold and I always knew there was a relation but it was never clear to me what exactly the relation was. The basic question is: on which manifold do the $(0,1)$ vector fields give a distribution?

After reading in Voisin’s excellent book “Hodge theory and complex algebraic geometry”, I know understand the answer to this question and I can give an outline of an (incomplete) proof for the Theorem.

Let me start by assuming that the manifold $M$ is real analytic (this is source of the incompleteness).  This means that it has an atlas whose transition functions are all real analytic — ie given by power series.  Then I can complexify my manifold to obtain $M_{\mathbb{C}}$.  This is a complex manifold, whose complex dimension is the same as the real dimension of $M$.  It is given by an atlas whose transition functions are the same power series as those for $M$, except they are now viewed complex analytic functions.

Now the complexified tangent bundle of $M$ is the actual tangent bundle of $M_{\mathbb{C}}$ and so the subspace of $(0,1)$ vector fields gives a holomorphic distribution on $M_{\mathbb{C}}$ (here we also assume that the original complex structure $J$ was real analytic).  Then we apply a holomorphic version of the Frobenius theorem to show that locally we can find a holomophic map $U \rightarrow \mathbb{C}^n$ whose kernel is this distribution.  Then we show that the restriction of this holomorphic map to $M \cap U \subset U$ is a diffeomorphism. By definition, this map will be $J$-holomorphic and thus we build an atlas on $M$. This completes the proof.  Pretty nifty, eh?

1. Ben Wieland - March 2, 2009

The proof’s not incomplete; it’s just a proof of a weaker theorem! The weaker theorem may be all you need.

Whitney proved that all (compact?) smooth manifolds are canonically real analytic. Namely, embed it in euclidean space and express it as (a component of) the intersection of the vanishing loci of polynomials. It’s a real algebraic variety. (Decades later, Grauert showed that there are no exotic real analytic manifolds that don’t embed in euclidean space, strengthening “canonically” to “essentially uniquely.”)

Of course, this just pushes the real analytic question from the manifold to the almost complex structure. For some kinds of structures, it really matters, like the Reeb foliation which is C^oo but not real analytic. I don’t know if people want to use bump functions when building their involutive almost complex structures, but the existence of the strong NN theorem suggests not.

2. Kevin Lin - March 7, 2009

Great post! A small correction: the complexified tangent bundle of M is not the tangent bundle of M_C, it is the pullback of the tangent bundle of M_C along the inclusion of M into M_C.

3. John Hubbard - April 11, 2009

May I suggest you look at appendix A4 of the book

Teichmuller theory, Vol 1, by John Hubbard, Matrix editions, 2006. It contains the complete proof sketched above.

With one of my students (Milena Pibiniak), I am working on the C-infinity version, which I believe can be reduced to generalities in “global analysis”, i.e., calculus on Banach manifolds.

If anayone is interested, I can keep them informed about progress on that front.

Sorry comments are closed for this entryNo. This is a rumour that has been circulating for a number of years. It is believed that the rumour started due to the consideration of Niagara Falls as well as other sites as possible locations for the park in the initial planning stages. The destination varies in different versions of the tale, but unfortunately Canada's Wonderland is not coming to a city near you! This has been clearly stated by the park on many occasions, including by the park's General Manager in 2002, Barry Jones.
Not Likely. As the story goes, a maintenance employee was working on the track when the train ran the course, cutting off the workers head. This story is most likely false, as there are no indications of media coverage from the time of the "incident". There would surely have been coverage of such a tragedy if it were true, but there is always a possibility.
Yes. In 1988, a man jumped into the water underneath Victoria Falls to retrieve his frisbee and drowned. This is the only death ever to have occurred inside the park. Click here to read about this in the Toronto Star.
Yes, but not in the park. A guest was shot in August, 2003 in front of the park entrance. The incident was believed to have been gang related and was not tied at all to Canada's Wonderland. However, metal detectors were added at all of the Paramount Parks the following season to prevent firearms and other weapons from being taken into the park itself. In addition, a heavier security presence is now present at the park.
Yes. You were able to walk behind the falls when the park opened but this was discontinued some years later. Although the exact year it was closed is unknown, it was believed to have been around the early 1990s. You can see the sealed off former entrance to the pathway on either side of Victoria Falls. If you look closely enough at the falls, you will see that the pathway behind the water is still in existence.
Yes. You were able to walk up the mountain when the park opened, but this was discontinued some years later