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Question about convergence norms February 25, 2010

Posted by Scott Carnahan in complex analysis.
14 comments

I have a question for the mathematical community about the language mathematicians use for a series of functions, and it has been bugging me for a while. I was thinking of posting it on MathOverflow, but it seems to fit the “subjective and argumentative” criterion for closing rather well.

Suppose someone introduces a countably infinite set of holomorphic functions \{ f_n \}_{n \geq 0} on some open subset U of the complex numbers, and wants to argue that the sequence sum converges to a holomorphic function. One valid way to prove it would be to show that the sequence sum converges locally uniformly absolutely (or uniformly absolutely on compact subsets of U), and then point toward the complex analysis text of choice.

Question: If someone only argues that the sequence sum converges absolutely, should I complain? (If so, how loudly?)

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Chromatic Homotopy II or how I learned to stop worrying and love LaTeXing in Real Time February 25, 2010

Posted by Chris Schommer-Pries in blog triumphalism, Latex, liveblogging.
Tags:
33 comments

I am currently taking course notes for Jacob Lurie’s class on Chromatic Stable Homotopy in real time in Latex. This is not the first time I have taken course notes live and in tex, and when people see it happening they often ask me about it.

I wanted to write an update about Jacob Lurie’s class on Chromatic Stable Homotopy and mention some of exciting and beautiful things happening in that course, but as I started writing this post I found that it was morphing into a sort advice post on how to LaTeX in real time. Since there is obvious appeal, I’ve decided to run with it and collect all the advice, tips, and tricks on how to LaTeX in real time that I’ve gathered from the wild.

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“Et al” is unethical February 21, 2010

Posted by Noah Snyder in math life.
44 comments

So apparently the AMS has a document on Ethical Guidelines. It’s actually remarkably well done. It has lots of tips that can help young mathematicians learn how to behave professionally. I was also impressed by the way that the guidelines avoid making too controversial of stands (which would go beyond the basics of ethics) while still not being milquetoast. For example, “No one should be exploited by the offer of a temporary position at an unreasonably low salary and/or an unreasonably heavy work load” is certainly an ethical obligation, but one that may be difficult to live up to.

I also thought that the guidelines about correct attribution were well phrased. For example:

To give appropriate credit, even to unpublished materials and announced results (because the knowledge that something is true or false is valuable, however it is obtained);

I have my own suggestion for a guideline on ethical use of citations: you should never ever use “et. al.” citations. Furthermore, if journal typesetters add them you should ask them to replace them with full citations.

If a bibliography just says “et al.” many readers are never going to get around to looking at the other names thereby effectively failing to properly attribute everyone. People at the end of the alphabet are already at enough of a professional disadvantage (see What’s in a Surname? The Effects of Surname Initials on Academic Success by Liran Einav and Leeat Yariv,), the use of “et al” just exacerbates this.

Hat tip: I learned about this document in a MathOverflow comment by Bill Johnson

Some fusion categories are not cyclotomic (January) February 17, 2010

Posted by Scott Morrison in Uncategorized.
5 comments

Noah Snyder and I have just uploaded to the arxiv our paper Non-cyclotomic fusion categories.

In their paper On fusion categories, Etingof, Nikshych and Ostrik asked if every fusion category can be defined over a cyclotomic field. This is supported by the following facts:

  • The representation category of any finite group has a complete rational form over a cyclotomic field.
  • The semisimplified representation category of any quantum group at a root of unity has a complete rational form over a cyclotomic field.
  • The Frobenius-Perron dimension of any object in a fusion category is a cyclotomic integer.

A fusion category is a rigid monoidal category which is semisimple and has finitely many (isomorphism classes of) simple objects. It is tempting to think of them as the common generalization of representation categories of finite groups and of representation categories of quantum groups at roots of unity (these provide many of the most interesting examples), but our result shows that this is a little dangerous. In particular, Etingof, Nikshych and Ostrik’s question can be answered in the negative:

Theorem (Morrison-Snyder): one of the fusion categories coming from the Haagerup subfactor cannot be defined over any cyclotomic field.

This shows that there are fusion categories, namely those coming from the smallest “exotic” (i.e. apparently unrelated to finite groups or quantum groups) subfactors, which exhibit more complicated behaviour than those coming from finite groups and quantum groups.

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Some Mathoverflow challenges February 15, 2010

Posted by David Speyer in fun problems, Math Overflow.
3 comments

Mathoverflow has become a roaring success! The site is getting about 20-25 questions a day, and about 95% of them get answered. But one of the problems with being so successful is that a question rarely remains on the front page for more than a day. Here are some which I believe are interesting, challenging and solvable.

Why do flag manifolds, in the P(V_{\rho}) embedding, look like products of \mathbb{P}^1‘s.
\mathcal{F}\ell_n and (\mathbb{P}^1)^{\binom{n}{2}} have the same multi-graded Hilbert functions. Is there any reason for this?

Smooth proper schemes over \mathbb{Z} with points everywhere locally,
Non-simply-connected smooth proper scheme over \mathbb{Z}?.
The condition of being smooth and proper over \mathbb{Z} appears to be very restrictive. Can you construct examples with these properties?

Pencils with many completely decomposable fibers
Does there exist a pencil of hypersurfaces in \mathbb{P}^4, which is not a cone over a pencil in \mathbb{P}^3, but has three fibers which are unions of hyperplanes?

Are submersions of differentiable manifolds flat morphisms?
This is basically a commutative algebra question: Is C^{\infty}(\mathbb{R}^{n+1}) flat over C^{\infty}(\mathbb{R}^{n})? Several people made progress, but no one finished it off.

Five Front Battle
Two generals, with armies of equal sizes, must apportion their troops between five fronts. On each front, the army with more troops will win; the nation which wins three fronts will win the war. What is the optimal strategy?

Christol’s theorem and the Cartier operator February 11, 2010

Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.
3 comments

Let’s suppose that we want to compute 2^n \mod p, and we have already been given n written out in base p as n = \sum n_i p^i. Here p is a small prime and we want to do this conversation repeatedly for many n‘s.

Remember that 2^p \equiv 2 \mod p and thus 2^n \equiv 2^{n_0+n_1 p + n_2 p^2 + \cdots + n_{\ell} p^{\ell}} \equiv 2^{n_0} 2^{n_1} \cdots 2^{n_{\ell}} \mod p. So, start with 1, multiply it by 2^{n_0}, then by 2^{n_1}, then by 2^{n_2} and so forth. When you get to the end, read off 2^n.

We can precompute the effect on \mathbb{F}_p of multiplying by 2^k, for 0 \leq k \leq p-1. Then we can compute 2^n just by scanning across the base p representation of n and applying these precomputed maps to the finite set \mathbb{F}_p.

The precise way to say that this is a simple, one-pass, process is that it is a computation which can be done by a finite-state automaton. Here is the definition: let I, S and O be finite sets (input, states and output), and s \in S (the start). For each i \in I, let A(i) be a map S \to S. We also have a map r:S \to O (readout). Given a string (i_0, i_1, \ldots, i_{\ell}) in I^{\ell}, we compute r( A(i_{\ell}) \circ \cdots A(i_1) \circ A(i_0) (s)). So our input is a string of characters from I, and our output is in O.

We can say that our example above shows that (n_{\ell}, \ldots, n_1, n_0) \mapsto 2^{\sum n_i p^i} \mod p is computable by a finite-state automaton. (In our example, the sets I, S and O all have cardinality p, but I do not want to identify them.)

This is a special case of an amazing result of Christol et al: Let f_n be a sequence of elements of \mathbb{F}_p. Then (n_{\ell}, \ldots, n_1, n_0) \mapsto f_{\sum n_i p^i} can be computed by a finite-state automaton if and only if the generating function \sum f_n x^n is algebraic over \mathbb{F}_p(x)!

We have just explained the case \sum f_n x^n = 1/(1-2x). The reader might enjoy working out the cases 1/(1-x-x^2) (the Fibonacci numbers) and (1 - \sqrt{1-4x})/2x (the Catalans).

In this post, I will use Christol’s theorem as an excuse to promote the Cartier operator, an amazing tool for working with differential forms in characteristic p.

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Grothendieck’s letter February 9, 2010

Posted by Scott Morrison in Algebraic Geometry.
comments closed

Recently on meta.mathoverflow.net, Harry Gindi pointed out that Laszlo’s webpage for an edition of SGA 4 now contains the message

Alexandre Grothendieck a malheureusement souhaité que cessent les travaux de réédition de SGA. Les pages qui étaient consacrées sont donc closes.

It has since come to light that this request came in the form of a letter, which has been circulating in the French mathematical community for the last month. I include here a link to a typed version of that letter, vouching for neither its authenticity or accuracy, along with my pathetic attempt at translating it, for those few whose French is even worse than mine. Feel free to suggest better translations (I’ll incorporate them here).

Declaration of intent of non-publication

I do not intend to publish or republish any work or text of which I am the author, in any form whatsoever, printed or electronic, whether in full or in excerpts, texts of personal nature, of scientific character, or otherwise, or letters addressed to anybody, and any translation of texts of which I am the author. Any edition or dissemination of such texts which have been made in the past without my consent, or which will be made in the future and as long as I live, is against my will expressly specified here and is unlawful in my eyes. As I learn of these, I will ask the person responsible for such pirated editions, or of any other publication containing without permission texts from my hand (beyond possible citations of a few lines each), to remove from commerce these books; and librarians holding such books to remove these books from those libraries.

If my intentions, clearly expressed here, should go unheeded, then the shame of it falls on those responsible for the illegal editions, and those responsible for the libraries concerned (as soon as they have been informed of my intention).

Written at my home, January 3, 2010,

Alexandre Grothendieck.

UPDATE: I have replaced the letter with a corrected version sent to me by Prof. Illusie, and made some changes to the translation, suggested by several people.
UPDATE: You can now see the original handwritten letter.Grothendieck's Declaration (original).

What happened to Clay Liftoff? February 9, 2010

Posted by Ben Webster in inside baseball, jobs.
6 comments

Clay has announced that in 2010, there will be no Liftoff Fellows; they say the program is suspended. The title question was asked in MathOverflow a while back, and while it was rightly shut down there, I’m still kind of curious to know the answer. Did Clay decide Liftoff was not a good program for some reason? Did they not want to spend the money? Obviously, I’m appreciative of the Liftoff program having been a Fellow myself, but its very unclear to me that it results in more math getting done, as opposed to having a few mathematicians pay off student loans faster, which I think was its main effect on me.

When fine just ain’t enough February 2, 2010

Posted by David Speyer in Algebraic Geometry, complex analysis, homological algebra, things I don't understand.
4 comments

If you use sheaves to study differential geometry, one of the basic lemmas you’ll want is the following: Let X be a smooth manifold and let \mathcal{E} be a sheaf of modules over C^{\infty}(X). (For example, \mathcal{E} might be the sheaf of sections of a vector bundle.) Then all higher sheaf cohomology of \mathcal{E} vanishes.

The proof of this theorem is basically homological algebra plus the existence of partitions of unity. This gives rise to a slogan “when you have partitions of unity, sheaf cohomology vanishes.” One way to make this definition precise is through the technology of fine sheaves.

As Wikipedia says today, “[f]ine sheaves are usually only used over paracompact Hausdorff spaces”. That means they are not used when working with the Zariski topology on schemes, for example. When I started digging into this, I realized there were good reasons: The technology of fine sheaves (and the closely related technology of soft sheaves) does not include the scheme theory cases which we would want it to.

However, there are theorems of the form “when you have partitions of unity, sheaf cohomology vanishes” on schemes and on complex manifolds. I put up a question at MathOverflow asking whether there were better formulations that included these examples, but I probably didn’t formulate it well. I think spelling out all my issues would be too discursive for MathOverflow, so I’m bringing it over here.

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