http://math.ucr.edu/home/baez/week286.html

John Baez

I'll explain it at the end.

So, what's "rational homotopy theory"? One might naively define it as the study of spaces whose homotopy groups are vector spaces over the rational numbers.

But if you think about it, that's pretty weird!

For example, the first homotopy group of a space X, usually called the "fundamental group" of X and denoted π₁(X), consists of equivalence classes of loops in X that start and end at our favorite point. Two loops count as equivalent if you can continuously deform one until it looks like the other. If you can do this, we say these loops are "homotopic".

The fundamental group of the circle is Z, the group of integers. The reason is that two loops drawn on the circle are homotopic if and only if they wind around the same number of times - and that number must be an integer! You can walk around the block once and get back home. You can walk around the block twice and get back home. You can even walk around the block -5 times and get back home: the negative number just means you walk around the other way. But you can't walk halfway around the block and be back home!

But suppose you had a space whose fundamental group was Q, the rational numbers. Then you could walk halfway around the block and get back home. That sounds pretty weird - nay, downright impossible!

But part of why it sounds so weird is that it's not right. We really need some other "block" such that walking around that block twice is homotopic to walking around the original block once. This sounds more complicated... but also more possible.

Later in this post I'll describe a space called "the rational circle", whose fundamental group is indeed Q. Then you can see how it actually works.

Anyway: spaces whose homotopy groups are rational vector spaces are weird. Why should we care about them?

We shouldn't! In fact, the real point of rational homotopy theory lies elsewhere.

It's better not to think of rational homotopy theory as the study of weird spaces whose homotopy groups are rational vector spaces. It's better to think of it as the study of ordinary spaces - but viewed in a way that doesn't let us see their homotopy groups, only their homotopy groups tensored with Q. This process turns their homotopy groups into rational vector spaces!

This is a common theme in algebraic topology. We can think of various kinds of homotopy theory either as the completely precise study of rather strange spaces, or as the study of ordinary spaces as seen through a blurry lens. A blurry lens can be a good thing, because it simplifies a complicated picture.

However, even this way of thinking about rational homotopy theory misses the point. The real point is that rational vector spaces come from the land of linear algebra, so rational homotopy blends topology and linear algebra. So does rational homology theory, but rational homotopy theory is deeper. When we get into it, we'll take lots of important concepts from linear algebra - like commutative algebras, and Lie algebras, and Hopf algebras - and study very interesting "homotopy versions" of these concepts.

By doing this, we'll vastly generalize linear algebra. We'll wind up with a whole new perspective... and we'll see applications to physics ranging from classical field theory, to quantization, to supergravity!

And you should not be surprised that we're doing here is really categorifying linear algebra.

But more on that later. Today, I want to start with the naive viewpoint that rational homotopy theory is about spaces whose homotopy groups are rational vector spaces.

In algebraic topology, the really hard part is torsion. A group element is "torsion" if you can add it to itself a bunch of times and get zero. So, for example, every element of a finite group is torsion, but the group of integers is "torsion-free".

Look at some homotopy groups of spheres and you'll see what I mean:

 π3(S2) = Z 
 π5(S3) = Z/2 
 π7(S4) = Z x Z/12 
 π9(S5) = Z/2 
 π11(S6) = Z 
 π13(S7) = Z/2 
 π15(S8) = Z x Z/120 
 π17(S9) = Z/8 
 π19(S10) = Z x Z/8 
 

But if you were asked to make a guess about the torsion part of these groups, you'd find it a lot harder. And indeed, nobody knows the full story here.

This suggests trying to do a version of algebraic topology where we systematically get rid of torsion. We'll lose a lot of important information, but things will get easy and fun - and still far from trivial!

This is "rational homotopy theory".

How can we get rid of torsion?

Well, the nth homotopy group of a compact manifold, like a sphere, is always finitely generated - and abelian when n > 1. A finitely generated abelian group always looks like Zⁿ × T where T is finite. All the torsion is in T, so to get rid of torsion we can just throw out T.

But that doesn't work in general. In general, the nth homotopy group of a space can be any group when n = 1 - and any abelian group when n > 1.

For an arbitrary abelian group, the torsion elements always form a subgroup, called the "torsion subgroup". It's not true in general that an abelian group is the product of its torsion subgroup and some other group! But, we can still kill off the torsion by modding out by the torsion subgroup.

For a nonabelian group, the torsion elements don't necessarily form a subgroup! For example, take the free group generated by x and y, and mod out by the relations x² = y² = 1. Then x and y are torsion elements, but xy is not.

I don't know any good way to kill off the torsion for an arbitrary nonabelian group. A lot of work on rational homotopy theory sidesteps this issue by working only with "1-connected" spaces. These are spaces that are path-connected and also simply connected. That means the fundamental group is trivial - and the higher homotopy groups are always abelian, so we don't have to worry about nonabelian groups.

Now, I've made it sound like the right way to "kill off torsion" in an abelian group is to mod out by its torsion subgroup. This makes me wonder if there's a systematic way to take a space X and turn it into a space X' such that π_n(X') is π_n(X) mod its torsion subgroup. Does anyone know?

But anyway, this is not how we kill off torsion in rational homotopy theory!

Instead, here's what we do. Abelian groups are the same as Z-modules where Z is the ring of integers. Since Z is commutative, we can take tensor products of Z-modules. In other words, we can take tensor products of abelian groups. And to kill off the torsion in an abelian group, we just tensor it with the rational numbers!

I hope you see what this accomplishes. Tensoring an abelian group G with the rational numbers gives a new abelian group Q ⊗ G that includes elements like

q ⊗ g

where g ∈ G and q is a rational number. Any element g of G gives an element of Q ⊗ G, namely

1 ⊗ g

But we also get elements like

(1/2) ⊗ g

which acts like "half of g". More generally, given any element of Q tensor G, we're allowed to multiply it by any fraction.

Now, suppose g is a torsion element of G. Then ng = 0 for some n, so

1 ⊗ ng = 0,

If we multiply both sides by 1/n, we get

1 ⊗ g = 0

So, torsion elements of G get sent to zero in Q ⊗ G. We've killed the torsion.

But the great thing about this trick is that Q ⊗ G is even better than a torsion-free abelian group. It's a vector space over the rational numbers! So, we're not just killing off torsion. We're going from the world of abelian groups to the world of linear algebra, which is notoriously well-behaved.

Next let me sketch how we can take a 1-connected space X and "rationalize" it, obtaining a new space X_Q with

π_n(X_Q) = π_n(X) ⊗ Q

for all n.

Since we're doing homotopy theory, we can assume X is a "CW complex". A space of this sort is built from balls. To build a CW complex, we start with some 0-balls - that is, points. Then we take some 1-balls - that is, intervals - and glue their boundaries to the 0-balls. We get a space that's just a graph. Then we take some 2-balls - that is, disks - and glue their boundaries to the space we've got so far. Then we take some 3-balls and glue their boundaries to what we've got so far. And so on, ad infinitum. Any space is "weakly homotopy equivalent" to a space of this sort, and that's good enough for us.

So, to rationalize X we should rationalize this whole procedure! This procedure relies on balls - and also spheres, since the boundary of a ball is a sphere. So, we should define a "rational n-ball" and a "rational n-sphere", and then make sure that given a CW complex, we can build a new space where each ball or sphere we used has been replaced by a "rational" one!

I'll describe the rational n-sphere, since that's the fun part. Even though we don't need it here, let's start with the case n = 1: the "rational circle". As mentioned earlier, this is a space whose fundamental group is Q. Here's one way to build it.

First, take an ordinary circle, and take a cylinder, and glue your circle to the bottom of that cylinder. But: make sure the circle goes around the bottom of the cylinder twice! See what this accomplishes? It means that walking around your original circle once is homotopic to walking around the top of the cylinder 2 times.

This solves our problem of how walking once around the block can be the same as walking twice around some other block.

Then take another cylinder, and glue the top of your first cylinder to that. But: make sure the top of your first cylinder winds around the bottom of this new one 3 times.

Then take yet another cylinder. Glue the top of your second cylinder to the bottom of that - but make sure it wraps around the bottom 4 times.

And then do this forever...

...and then take a little rest, since it's very tiring to do an infinite amount of work. Sit back and admire your handiwork. The space you've built has Q as its fundamental group, because for any loop g and any integer n, we've created a new loop h such that g = nh.

Mathematicians call this general type of space a "telescope". An ordinary hand telescope - the kind that pirates use - is built from cylinders of metal that fit into each other:

A mathematician's telescope is similar - but it's built from infinitely many cylinders, and you can't collapse it, because they're attached to each other in a complicated way. This makes it really easy to spot a mathematician in a roomful of pirates.

We can easily generalize this telescope idea to construct the "rational n-sphere". The point is that for each integer k, there's a way to wrap the n-sphere around itself k times. So, we can use these to build an infinite telescope, just as we did for the rational circle. This telescope is a space whose homotopy groups are those of the n-sphere, but tensored with the rational numbers.

A similar trick produces a rational n-ball, but this is less exciting, since all the homotopy groups of the n-ball were trivial already - it's contractible, after all. The rational n-ball is still contractible, but it's been modified so that its "boundary" is a rational n-sphere.

Having rationalized our spheres and balls, we also need to check that the maps we used to build our CW complex extend in a canonical way from the spheres to the rational spheres. But let's skip this: in This Week's Finds we only do the fun part!

As you can see, the rationalized version of a nice little CW complex is usually a huge nightmarish space. This is a familiar tradeoff in algebra topology: spaces with comprehensible homotopy groups almost always look big and scary when we try to build them by gluing balls together. But it's a tradeoff algebraic topologists have learned to accept. There's more to life than whether a space looks nice.

In particular, this rationalization process has a very nice abstract characterization. Suppose X is any 1-connected pointed space. Then we can define "a rationalization" of X to be any 1-connected pointed space X' equipped with a map

X → X'

satisfying two properties. First, X' is a "rational space": a 1-connected pointed space whose homotopy groups are rational vector spaces. Second, the induced map

Q ⊗ π_n(X) → Q ⊗ π_n(X')

is isomorphism for all n.

It turns out that the rationalization of a space is unique up to weak homotopy equivalence. And we can construct it for CW complexes as I just explained.

Okay. So far I've been treating rational homotopy theory as the study of weird "rational" spaces. And I've showed you how to turn any space into one of these. But as I already admitted, this misses the point.

To come closer to the point, we need to recall an amazing old theorem due to J. H. C. Whitehead, which says a map

f: X → Y

between connected CW complexes is a homotopy equivalence if and only if the induced maps

π_n(f): π_n(X) → π_n(Y)

are isomorphisms for all n. This is why for more general connected spaces we define any map that induces isomorphisms on homotopy groups to be a "weak homotopy equivalence". Even better, every space is weakly homotopy equivalent to a CW complex! So, if we formally throw in inverses to all weak homotopy equivalences, we get a category called where every space is isomorphic to a CW complex. This is called the "homotopy category".

These ideas are very powerful, so it's good to generalize them to rational homotopy theory. So now suppose X and Y are 1-connected pointed spaces. And let's say a map

f: X → Y

is a "rational homotopy equivalence" if the induced maps on rational homotopy groups

Q ⊗ π_n(f): Q ⊗ π_n(X) → Q ⊗ π_n(Y)

are isomorphisms for all n. There's a nice category where we formally throw in inverses to all rational homotopy equivalences. This is called the "rational homotopy category".

In the rational homotopy category, we're studying ordinary spaces through a blurry lens, because two spaces that have a rational homotopy equivalence between them look the same.

But the rational homotopy category is equivalent to a subcategory of the usual homotopy category: namely, the subcategory consisting of rational spaces and all morphisms between those! So, we can also say we're studying strange spaces, but with complete precision - or at least, the usual level of precision in homotopy theory.

To learn more, I urge you to grab this and take a look:

2) Kathryn Hess, Rational homotopy theory: a brief introduction, in Interactions Between Homotopy Theory and Algebra, ed. Luchezar L. Avramov, Contemp. Math 436, AMS, Providence, Rhode Island, 2007. Also available as arXiv:math.AT/0604626.

For even more detail, I recommend:

3) Yves Felix, Stephen Halperin and Jean-Claude Thomas, Rational Homotopy Theory, Springer, Berlin, 2005.

I'll give more references later. In the weeks to come, we'll see two very different descriptions of the rational homotopy category, which were both discovered by Daniel Quillen back in 1969:

4) Daniel Quillen, Rational homotopy theory, Ann. Math. 90 (1969), 205-295. Also available at http://www.jstor.org/stable/1970725

It's these other descriptions that make the subject really interesting. One is based on a homotopy version of Lie algebras. The other is based on a homotopy version of commutative algebras!

In the first sentence of his paper, Quillen defines the rational homotopy category. But he does it a bit differently than I just did. This is worth understanding. He says it's "the category obtained from the category of 1-connected pointed spaces by localizing with respect to the family of maps which are isomorphisms modulo the class in the sense of Serre of torsion abelian groups".

Let me say this with less jargon. Start with the category of 1-connected pointed space. Thrown in formal inverses of all maps

f: X → Y

for which the induced maps

π_n(f): π_n(X) → π_n(Y)

have kernels and cokernels where all elements are torsion. This gives the rational homotopy category!

I'll conclude with a few remarks that would have been a bit too distracting earlier.

First: I discussed rational homotopy theory only for 1-connected spaces. This is great as far as the connection to algebra goes. But in terms of topology it's a bit sad. Sometimes people go a step further and work with "nilpotent" spaces. These are spaces where the fundamental group is nilpotent and acts nilpotently on the higher homotopy groups.

Second: the rational circle is an interesting space. As we've seen, it's a space with the rational numbers as its fundamental group. All its other homotopy groups are trivial, since that's already true for the circle.

Any space with G as its nth homotopy group and every other homotopy group being trivial is called "the Eilenberg-Mac Lane space K(G,n)". We're allowed use the word "the", since this space is unique up to weak homotopy equivalence. So, the rational 1-sphere is K(Q,1).

I've talked about lots of different Eilenberg-Mac Lane spaces in This Week's Finds, and they're all collected here:

3) John Baez, Generalized cohomology theories, Eilenberg-Mac Lane spaces, classifying spaces and K(Z,n), and other examples of classifying spaces. Available at http://math.ucr.edu/home/baez/calgary/BG.html

Now you can add K(Q,1) to your collection!

Third: in case you're wondering about Quillen's jargon: by "localizing" he means the process of taking a category and throwing in formal inverses to a bunch of morphisms. This is an important way of simplifying categories. It lets us count slightly different objects as the same.

A "Serre class" of abelian groups is a bunch of abelian groups such that whenever A and C are in this class, and the sequence

0 → A → B → C → 0

is exact, then B is in this class too. The idea is that we think of abelian groups in the Serre class as "small", or "negligible". For example: the class of finite abelian groups, or the class of torsion abelian groups. We can localize the category of abelian groups by throwing in an inverse for any morphism whose kernel and cokernel are in the Serre class.

If you like abelian categories, you can generalize this "Serre class" idea from the category of abelian groups to other abelian categories.

There's also much more to say about localization! Try this for starters:

5) nLab, Localization, http://ncatlab.org/nlab/show/localization

Besides doing rational homotopy theory, we can use localization to take homotopy theory and "localize at the prime p". This is a way to focus special attention on the "p-torsion" in our homotopy groups: that is, the elements that give zero when you multiply them by a power of p.

Finally, what about the picture at the beginning of this Week's Finds? It shows sand dunes in a region called Abalos Undae near the north pole of Mars:

5) HiRISE (High Resolution Imaging Science Experiments), Dunes in Abalos Undae, http://hirise.lpl.arizona.edu/PSP_010219_2785

The photo covers a strip about 1.2 kilometers across. As the HiRISE satellite sweeps over Mars it takes incredibly detailed photos like this. Here's the description on the HiRISE website:

The Abalos Undae dune field stretches westward, away from a portion (Abalos Colles) of the ice-rich north polar layered deposits that is separated from the main Planum Boreum dome by two large chasms. These dunes are special because their sands may have been derived from erosion of the Rupes Tenuis unit (the lowest stratigraphic unit in Planum Boreum, beneath the icier layers) during formation of the chasms. Some researches have argued that these chasms were formed partially by melting of the polar ice.

The enhanced color data illuminate differences in composition. The dunes appear blueish because of their basaltic composition, while the reddish-white areas are probably covered in dust. Upon close inspection, tiny ripples and grooves are visible on the surface of the dunes; these features are formed by wind action, as are the dunes themselves.

It is possible that the dunes are no longer migrating (the process of dune formation forces dunes to move in the direction of the main winds) and that the tiny ripples are the only active parts of the dunes today.

Addenda: The rational circle is pretty hard to draw, but Kenneth Baker did a great job of illustrating some early stages of its construction:

1) Kenneth Baker, A (reverse) rational circle?, on his blog Sketches of Topology at http://sketchesoftopology.wordpress.com/2009/12/10/a-rational-circle/

If we remove the red band we see how the orange one wrapped around it 3 times:

These pictures are done using Rhino 3D. Actually I'm using the beta version of their port to OS X. There's a function (called Flow) that lets you map a "spine" of an object to another curve to tell it how to deform the object. This is how I went from the chopped open version to the round one. It's also how I managed to make the orange wrap around the green and the red wrap around the orange.

Something that bothered me for a while about rational homotopy, as an outsider, was this phrase "the homotopy groups are rational vector spaces". A priori the (higher) homotopy groups are abelian groups. So does this mean that there exists a rational vector space structure? That there exists a unique one? That one is somehow specified?

In fact, these questions are unnecessary, for the following reason. (I think this was explained to me by the James who sometimes comments here.) Fact:

Let A be an abelian group. Then A has the structure of a rational vector space in at most one way.

So, despite appearances, being a rational vector space is a property of abelian groups, not extra structure.

The proof is fairly straightforward, I think. If A admits a rational vector space structure then

for all a ∈ A and all positive integers n, there exists a unique b ∈ A such that nb=a.

And this condition clearly determines what the scalar multiplication over Q must be. (In fact, it's an 'iff': an abelian group admits the structure of a rational vector space if and only if it satisfies this condition.)

Yes. A rational vector space is the same as a divisible torsionfree abelian group. Incidentally, an abelian group is divisible if and only if it is injective in the category of abelian groups, and is torsionfree if and only if it is flat in the category of abelian groups.

For more discussion visit the n-Category Café.

...the pursuit of science is more than the pursuit of understanding. It is driven by the creative urge, the urge to construct a vision, a map, a picture of the world that gives the world a little more beauty and coherence than it had before. Somewhere in the child that urge is born. - John Archibald Wheeler

© 2009 John Baez
baez@math.removethis.ucr.andthis.edu

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http://lambda-the-ultimate.org/node/3712

Sage is a project to create a viable free open source alternative to Magma, Maple, Mathematica and Matlab. The lead developer/manager William Stein has recently written Mathematical Software and Me: A Very Personal Recollection, a rather enjoyable story of his experience with mathematical software, especially Magma, and how Sage came to be.

One of the difficulties of writing broadly useful math software is the sheer size and scope of such a project. It is easily outside the abilities of even the most prodigious lone developer. So the focus of Sage, at least up until recently, has been on creating Python-based interfaces to existing mathematical software. For example, for symbolic calculation the Sage distribution includes Maxima (written in Common Lisp), a fork of Macsyma dating back to the early 1980s, released as open-source software by the US Department of Energy approximately 10 years ago. In addition to Maxima, Sage includes the ability to call out to Magma, Mathematica, and Maple.

There are some interesting PLT-related snippets, for example, Magma's language is frequently criticized, although it's algorithms are frequently praised. In conversations with others, OCaml and Haskell were brought up, but William Stein chose Python because he felt that it was more accessible. Also, Axiom, which includes the dependently-typed language Aldor, was rejected in favor of Maxima because Maxima was less esoteric and much more widely used.

My first is quite a hearty dish--and oft including meat or fish
My second is the mage's friend--from all assaults I will defend
My whole's a master's servant true--for all he has is my care too.
What am I?

As bitter as you find my first, to swallow it may cure the worst
Though should it not, a chill embrace meets you with bony, second face
But keep you faith! and like my whole, walk humbly 'til you reach your goal
What am I?

http://languagelog.ldc.upenn.edu/nll/?p=1953

The comments on my posting on commoner and my follow-up posting on inflected adjectives and adverbs went off in at least four directions beside the ones taken in the postings themselves. I've been trying to cope with this topic sprawl ever since and hope to get eventually to all four of these threads. Today I'm taking on two of them.

My second posting asked for reports of complaints about inflected comparatives and superlatives of adjectives and adverbs — commoner, for instance (rather than the periphrastic more common). I got many nice reports, but also a good bit of thread drift.

One set of commenters wanted to pursue the question of what the "correct" variants are, and some people responded with lists of rules. In fact, the best you can say about the matter is that there are tendencies, some very strong and some much weaker; that many of these tendencies depend on the phonological properties of the Adj/Adv base; and that there is considerable variation in actual usage, as well as in judgments about the acceptability of particular variants.

There's an immense literature on these matters. A few items from this literature are summarized in section 8 ("the phonological issue") of my paper in the 1989 Yearbook of Morphology. (Warning: most of this article is seriously technical, so trying to read the whole thing is probably not a good idea if you're not a linguist. But section 8 is less demanding.)

Other commenters noted some syntactic contexts where the periphrastic variant is required, even if the Adj/Adv normally takes the inflected variant. There are at least four of these, discussed in section 7.3 ("sketch of a syntactic analysis") of my YoM paper.

1. Parallelism in reduced coordination. Coordinated periphrastic comparatives or superlatives can have the more or most "factored out":

It's a more attractive, impressive, and ingenious idea than any other I've heard.

It's the most attractive, impressive, and ingenious idea I've ever heard.

If one of the conjuncts has an Adj/Adv that normally takes inflected degree forms (smarter), then it will nevertheless be treated periphrastically in reduced coordination; the conjuncts must be parallel:

It's a more attractive, smart, and ingenious idea than any other I've heard.

2. Degree comparatives and superlatives. Adverbs modifying Adj/Adv have only periphrastic degree forms themselves. Both variants are possible for other uses of adverbs:

Sandy dug more deeply. Sandy dug deeper [with deeper standing in for *deeplier]

But

more deeply philosophical; *deeper philosophical

3. Metalinguistic comparison. Comparison used to convey a metalinguistic judgment ('it would be more appropriate to say X than to say Y') must use the periphrastic variant:

Jan is more silly than mischievous. 'it would be more appropriate to say that Jan is silly than it would be to say that Jan is mischievous'

NOT "Jan is sillier than mischievous."

4. Absolute superlatives. A superlative used to convey merely a very high degree, without reference to a comparison class, must use the periphrastic variant:

You are most kind. 'you are extremely kind'

("You are kindest" is entirely acceptable, but doesn't convey this meaning. Instead, it makes reference to a comparison class, in this case an implicit one.)

http://languagelog.ldc.upenn.edu/nll/?p=1952

On her Fritinancy blog, Nancy Friedman has recently posted (under the heading "the tastiest suffix") an inventory of playful -licious brand names and brand descriptors, from Bake-a-Licious through Zombielicious. The -licious words come up every so often on Language Log, starting with 2006 postings by me (here) and Ben Zimmer (here), and going on with additional examples in 2007 (here) and this year (here).

dachte

Knytt Stories is a very cute game with nice music, lasting maybe half an hour. It also seems to have a community making new levels.

The outdoors are now utterly horrible to inhabit - without my powerful winter gloves, I get intense hand pain after about 5 minutes of being outdoors, and the rest of me is miserable. I may need to get a coat made of the same stuff as those gloves. I wish my apartment were not so drafty.

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I think, in general, that if it's not clear that one's job is good for society, one has the wrong job( Read more... )

denorae

• 09:09 Second day of IEEE meetings! #
• 14:02 Interesting concept #1 from today's IEEE 1722.1 meeting: connect to/configure your microphone through its onboard web server! #
• 14:04 Interesting concept #2: When we connect the Internet to the neighboring universe's Internet, need to resolve MAC address conflicts. #

http://languagelog.ldc.upenn.edu/nll/?p=1951

At the Atlantic, David Shenk mediates an exchange of letters between Mark Blumberg and Nicholas Wade about the appropriateness of calling FOXP2 a "speech gene",  about "gene for X" thinking in general, and about the nature of science journalism:

Blumberg: Trumping up FOXP2 as yet another star gene in a series of star genes (the "god" gene, the "depression" gene, the "schizophrenia" gene, etc.) not only sets FOXP2 up for a fall; it also misses an opportunity to educate the public about how complex behavior - including the capacity for language - develops and evolves.

Wade: I'm a little puzzled by your complaint, which seems to me to ignore the special dietary needs of a newspaper's readers and to assume they can be served indigestible fare similar to that in academic journals. […]

As for missing an opportunity to educate the public, that, with respect, is your job, not mine.  Education is the business of schools and universities. The business of newspapers is news.

I'm glad we got that straightened out!

Read the whole exchange between Blumberg and Wade here.

For some background, see the discussion and links in "The hunt for the Hat Gene", 11/15/2009.

And as part of my job of educating the public, let me draw your attention to some scientific news announced in a recent paper by M R Munafò et al., and as far as I know not covered by any newspapers ("Bias in genetic association studies and impact factor", Molecular Psychiatry 14: 119–120, 2009):

Studies reporting correlations between genetic variants and human phenotypes, including disease risk as well as individual differences in quantitative phenotypes such as height, weight or personality, are notorious for the difficulties they face in providing robust evidence.  Notably, in many cases an initial finding is followed by a large number of attempts at replication, some positive, some negative. Although there has been debate over the statistical arguments concerning the strength of evidence in association studies, there has been less interest in understanding why it is that some genetic associations generate such large literatures of inconclusive results. We wondered whether one source of the difficulties in the interpretation of genetic association studies might lie with the journal that published the initial finding. Studies published in journals with a high impact factor typically attract considerable attention. However, it is not clear that these studies are necessarily more robust than those published in journals with lower impact factors. […]

Data were analysed using meta-regression of individual study bias score against journal impact factor. This indicated a significant correlation between impact factor and bias score (R²=+0.13, z=4.27, P=0.00002). Our results are presented graphically in Figure 1. We also note that journals with high impact factors tend to publish studies with high bias scores and small sample sizes (as indicated by the smaller circles in the figure).

Here's Figure 1 and its caption:

Meta-regression of individual study bias score and journal impact factor. Bias score is plotted against the 2006 impact factor of the journal in which the study was published. Meta-regression indicates a positive correlation between journal impact factor and bias score (R²=+0.13, P=0.00002), suggesting that genetic association studies published in journals with a high impact factor are more likely to provide an overestimate of the true effect. Circles, representing individual studies, are proportional to the sample size (that is, accuracy) of the study.

In other words, the more prestigious the journal (as measured by its "impact factor"), the less likely the genetic association studies it publishes are to be replicated.

If I were merely in the business of news or entertainment, I'd observe at this point that the particular FOXP2 study behind the Blumberg/Wade discussion was published in one of the highest-impact-factor journals in the world, Nature, and thus is statistically somewhat more prone to fail to replicate than if it had been published (say) in Prof. Blumberg's journal, Behavioral Neuroscience.

But this would be unfair. Details aside, the paper's conclusion (that the two different amino acids in the human-specific version of FOXP2 cause "differential transcriptional regulation in vitro" of a very large number of other genes) is surely true; and the detailed claims about the genetic networks involved may well turn out to be helpful in understanding how the capacity for language develops and evolves.

However, we can also be fairly confident that calling FOXP2 a "speech gene"  — and the whole "gene for X" style of thinking that this exemplifies — will become more and more clearly a source of confusion. In my earlier post, I quoted Simon Fisher (the scientist who first discovered the connection between a FOXP2 mutation and a syndrome that includes some speech-related disabilities):

[T]he deceptive simplicity of finding correlations between genetic and phenotypic variation has led to a common misconception that there exist straightforward linear relationships between specific genes and particular behavioural and/or cognitive outputs. The problem is exacerbated by the adoption of an abstract view of the nature of the gene, without consideration of molecular, developmental or ontogenetic frameworks. […] Genes do not specify behaviours or cognitive processes; they make regulatory factors, signalling molecules, receptors, enzymes, and so on, that interact in highly complex networks, modulated by environmental influences, in order to build and maintain the brain.

At some point, I guess, this will become not merely truth, but also news.

http://jimwoodring.blogspot.com/2009/12/sea-strudel.html