Tasha the Cat received a new toy–a plastic circle containing corrugated cardboard, with a ball stuck in a track. Watch her pounce!

Given a text in two languages, is it possible to uncover the meaning of individual words?

The Bible is a particularly easy text to work with, since corresponding sentences are marked (i.e., with the same chapter and verse numbers). I downloaded a copy of the Hebrew Bible and the King James’ Version, and looked at Deuteronomy 6:4.

For each word in Hebrew, I found all the other verses with that word, and gathered together all the corresponding English verses; by picking the most popular word from those English verses (ignoring “the” and “and” and such), I found a pretty good translation of the original Hebrew word. In short, I picked the most popular English word in all those verses containing the non-English word.

So here’s Deuteronomy 6:4, with the top six English words underneath each Hebrew word:

אֶחָֽד	יְהוָ֥ה	אֱלֹהֵ֖ינוּ	יְהוָ֥ה	יִשְׂרָאֵ֑ל	שְׁמַ֖ע
one king for side all with	Lord God thy for thou thee	our God Lord for which not	Lord God thy for thou thee	Israel Lord children all his for	not Lord will heard them voice

Remember to read this from left-to-right. Pretty impressive–it didn’t quite get the verb שְׁמַ֖ע but it did well enough anyway.

It also works in Greek. Here’s Galatians 3:26 with the most popular English words underneath each Greek word.

πάντες	γὰρ	υἱοὶ	θεοῦ	ἐστε	διὰ	τῆς	πίστεως	ἐν	χριστῶ	ἰησοῦ.
all that they him for are	for that not him but unto	children shall are them your they	God that for unto not but	are you for that not shall	for that not unto God which	that for unto his which was	faith that for God but Christ	that unto for him not which	Christ Jesus are that which God	Jesus unto that him Christ said

It didn’t quite figure out διὰ is by or through.

In the end, this isn’t shocking, but it’s surprising how easy it is: the Ruby program to do this is only 150 lines long (which includes the code to print out those nice HTML tables with Unicode).

On a recent plane trip, I was reading a very abridged version of (the ten thousand page long!) Church Dogmatics by Karl Barth, and I found something totally beautiful.

Believing God to be entirely “transcendent in contrast to all immanence” and “divine in contrast to everything human,” and reading (e.g., in Philippians 2:7) that Jesus is God having emptied himself, having made himself nothing, I concluded that God somehow hid his divinity in order that he might become human and, in that form, redeem humanity.

This is wrong. Karl Barth writes:

As God was in Christ, far from being against Him, or at disunity with Himself, He has put into effect the freedom of His divine love… He has therefore done and revealed that which corresponds to His divine nature…

His particular, and highly particularised, presence in grace, in which the eternal Word descended to the lowest parts of the earth and tabernacled in the man Jesus, dwelling in this one man in the fulness of His Godhead, is itself the demonstration and exercise of His omnipresence… His omnipotence is that of a divine plenitude of power in the fact that (as opposed to any abstract omnipotence) it can assume the form of weakness and impotence and do so as omnipotence, triumphing in this form…

From this we learn that the forma Dei [Philippians 2:6] consists in the grace in which God Himself assumes and makes His own the forma servi [Philippians 2:7].

–Church Dogmatics, Volume IV, Part 1, page 185 and following.

My cutting hardly does justice to the original text, so I’ll paraphrase.

Jesus shows that God is everywhere, because God is fully in him; this doesn’t undermine omnipresence, instead, it strengthens it: the abstract “God is everywhere” is emphasized by a particular “And look, God is there–it’s Jesus.” Similarly, Jesus shows that God is all-powerful, because God triumphed in him in spite of weakness.

I had been thinking that Jesus was God with a veil over his divinity, when in fact, Jesus is God proving just how totally divine he is. For a God who is Love, the incarnation isn’t a denial of himself, but an affirmation of who he had been all along. It is often said that Jesus proved his divinity by rising from the dead; it ought to be remembered that he proved his divinity by being able to be obedient to death in the first place.

This is a beautiful perspective from which to understand the hypostatic union; the monophysites believed that Jesus’ humanity undermined his divinity, while as Barth explains, Jesus’ two natures are not only compatible, but necessary. This is another example of the sort of paradoxical argument I usually find unreasonably compelling (e.g., Chesterton’s Orthodoxy or Kierkegaard (fear and trembling appears in Philippians 2:12–a coincidence?) or Hume’s compatibilist explanation of free will).

Like most things viewed with hindsight, this perspective isn’t radical, but I (and probably a lot of people) view the divine and human natures of Christ as, essentially, in conflict when, ironically, Jesus came to reconcile those two natures, and did so first in himself.

Today, I was about to sit down and read a paper (in French–I may not speak in tongues, but apparently I can read in tongues, so to speak!), and I thought to myself about how nice it would be to have a cookie. I went to Uncle Joe’s, I went to the Classics Cafe, I went to Cobb’s coffee shop, and then I gave up, for there were no cookies in any of those places, places which so often appear to be the source of cookies.

Is there someplace else on campus that I should have looked? Admittedly, I probably would’ve settled for the biscotti in the divinity school cafe (especially with some coffee).

Someone contacted me with some questions about Bayesian document clustering; with that inspiration and a free afternoon a few weeks ago, I took a Hebrew bible and built a matrix where equals the frequency of the -th (Hebrew!) word in the -th chapter of Genesis. I calculated its singular value decomposition (supposedly this is “latent semantic analysis”), and then took some dot products (calculating the “correlation” of chapters)…

Anyhow, the result was astounding! The following table gives, for each chapter, a list of those chapters for which the given chapter is the chapter most highly correlated with it. Ah, that’s confusing; as an example to clarify this, the chapter most similar to chapters six, seven, eight, and nine is chapter one. With that, here’s the data:

Chapter 1:	2, 6-9
Chapter 5:	11
Chapter 7:	1
Chapter 10:	12-15, 34, 36, 46, 49
Chapter 11:	5
Chapter 15:	16
Chapter 21:	3, 22
Chapter 22:	4, 17-33, 35, 38, 44
Chapter 36:	10
Chapter 37:	43
Chapter 40:	41, 45, 47, 50
Chapter 41:	39
Chapter 45:	37, 42, 48
Chapter 50:	40

The shocking thing is that for 21 chapters of Genesis–for nearly half the book–the most highly correlated chapter is chapter 22–the binding of Isaac. In my mind, that story is the most powerful in Genesis, central to the message, and so it is especially remarkable that this crazy game with matrices also “detected” that most of Genesis clusters around that story.

There are usually courses at Mathcamp about surfaces; there should be courses about orbifolds! For instance, knowing that the smallest hyperbolic orbifold is the (2,3,7)-orbifold, having orbifold Euler characteristic , immediately gives that a closed hyperbolic surface of genus has no more than isometries (preserving orientation); this is “Hurwitz’ theorem.”

Just to show off this theorem, here is a cubical complex which is a surface with lots of symmetries (and the clever reader will recognize this as coming right out of Davis’ construction of aspherical manifolds): consider the -dimensional cube , and let be the edges around the origin, and be the square face containing the edges and . Define a subcomplex consisting of the squares and all squares in parallel to these. Now is a orientable surface (the link of any vertex is an -cycle, i.e., topologically an ). Don’t be fooled by the notation: has genus much larger than .

In fact, let’s calculate the genus. Every vertex of is contained in , and there are vertices in . Likewise, every edge in is contained in , and there are edges in . Finally, there are squares parallel to each of , so there are square faces in . Thus, and so is a surface of genus . This is a maybe a good exercise for someone first learning about Euler characteristic, but not especially interesting…

So here’s the punchline–or rather the punch-question–why is the genus growing exponentially in ? Because is very symmetric! And Hurwitz says to get so much symmetry, we need (linearly) as much genus. And we can find exponentially many symmetries of without any work. For starters, the group acts on by reflecting through hyperplanes, as does the group cyclically permuting the basis . If we want to be precise, let be the resulting group of order . Quotienting by these symmetries gives an orbifold , which one observes to be a square with cone points on each vertex (three with cone angle and one with cone angle ) and reflections in each of the four sides. Thus, the orbifold Euler characteristic of is , so the Euler characteristic of must be , just like we got before. One might argue that this method was “easier” than the previous method for calculating , but that misses the point—I (and probably everyone else) calculated the number of edges of by using a group action, if only implicitly.

The point is, even without doing any calculations or thinking very hard, the number of symmetries of is growing exponentially in , and therefore the genus must be growing exponentially in as well—the orbifold makes this reasoning precise.

There’s a lot of stuff left to be discovered about the number of automorphisms of genus surfaces. For instance, it’s known that the bound is attained for infinitely many genera, but there are also infinitely many genera for which it is not attained. Let be the maximal order of the automorphism group of a genus surface; Maclachlan and Accola proved (in 1968) that . This bound is sharp, too. There’s a beautiful paper [1] working out what happens in the arithmetic and non-arithmetic case. Anyway, what is known about the set of for which is attained? What is the asymptotic density of this set?

At a recent Pizza Seminar, Matt Day gave a lovely talk explaining why it isn’t possible to classify 4-manifolds.

An algorithm for deciding whether two closed 4-manifolds are homeomorphic gives an algorithm for deciding whether a closed 4-manifold is simply connected, and therefore (since every finitely presented group is the fundamental group of a 4-manifold), and algorithm for deciding when a group is trivial. Here’s the reduction: we are given a 4-manifold , and we compute its signature . By Freedman, there are no more than two closed simply connected 4-manifolds, and , having the same signature as ; we construct and , and we use the homeomorphism decision procedure to test if or .

Since there is no algorithm for deciding when a group is trivial, there can not be an algorithm for deciding when two closed 4-manifolds are homeomorphic.

There is a paper [1] discussing some of these issues. In particular, that paper discusses Novikov’s proof that cannot be recognized for .

This is a question I wandered into accidentally years ago now, which I think other people might be amused to think about (or more likely, put on an abstract algebra exam).

Let be a group, and a subgroup of . Is always isomorphic to , for some subgroups ? But beware!–I am not requiring (or expecting) any canonicity or naturality for the isomorphism: for instance sits diagonally in , and it just so happens that , so this is not a counterexample, in spite of the fact that the “horizontal” or “vertical” subgroup is not a canonical choice for the diagonal subgroup.

What is a good name for groups with this property? It’s not completely trivial: cyclic groups, for instance, have this property–not that I think this property is important, but names can be amusing…

I have examples of groups and with not (abstractly!) a product of subgroups of . My challenge to you is to find some explicit examples of and prove that doesn’t decompose.

In the end, I think this is a fun problem for a group theory final exam; I think it nicely highlights the difference between “being isomorphic” and “being equal,” though if one completes the challenge as stated, one probably already understands that distinction… So maybe the best reason for blogging about this is that chiastic title.

While on public transportation, my mind wanders… And one might assume the following about me and my buses,

• The bus travels for one unit of time,
• I will get on the bus at a random time (uniformly distributed),
• I will leave the bus at a random time (independent, unformly distributed).

Then the probability that I am on the bus at time is . So one might expect that the total number of people on the bus at time to look like for some .

I would enjoy riding a bus from the start to the end, and seeing how accurate this is, though tragically, I rather doubt it is very accurate at all. For starters, the entrance and exit times are correlated (who gets off the bus one stop after they get on?), and there are places where people are more likely to enter, and where people are more likely to exit. In fact, upon further reflection, this is a horrible model of bus ridership.

But, if you, say, averaged all the bus routes to make the entrance and exit distributions more uniform…–is there anywhere I can get this data? Wait, wait, this seems like an awful idea: I’d better stop now.

Thanks to Bryce Johnson for pointing out a mistake in my calculation of the probability above–I had forgotten to include a factor of two!

I gave a Farb student seminar talk on a lovely paper [1].

I also used some of the material in [2] which summarizes other the many applications of the "reflection group trick," and works through some examples with cubical complexes.

The main result is

Theorem. Suppose is a finite complex. Then there is a closed aspherical manifold and a retraction .

This manifold can be explictly constructed by gluing together copies of the regular neighorhood of embedded in some Euclidean space. The application of this theorem is to "promote" a finite complex to a closed aspherical manifold. For instance, we have a finite complex with non-residually-finite fundamental group: define the group , which is not residually finite, and observe that the presentation 2-complex is aspherical, so we have a finite . Then using the theorem to "promote" this to a closed aspherical manifold, we get a manifold with fundamental group retracting onto . But a group retracting onto a non-residually-finite group is also non-residually finite, so we have found a closed aspherical manifold with non-residually-finite fundamental group.

Just to whet your appetite, let me introduce a few of the main players, so as to give a sense of how to glue together copies of the regular neighborhood of .

Let be a simplicial complex, and , the vertices of .

From we construct two things: some complexes to glue together, and some groups with which to do the gluing. First, we construct the groups. Define to be the group , i.e., the abelian group generated by with . Next define to be the right-angled Coxeter group having as its Coxeter diagram; specifically, is the group with generators and relations for and also the relations if the edge is in . Note that is the abelianization of .

Next we will build the complexes to be glued together with the above groups. Let be the cone on the barycentric subdivision of , and define closed subspaces by setting to be the closed star of the vertex in the subdivision of . Note that are subcomplexes of the boundary of , and that a picture would be worth a thousand words right now.

Having the complexes and the groups, we will glue together copies of along the ’s, thinking of the latter as the mirrors. Specifically, define with provided that and , where , and is the subgroup of generated by . That is a mouthful, but it really is just carefully taking a copy for each group element of and gluing along the ’s in the appropriate manner. The resulting compplex has a action with fundamental domain . Similarly, we use to define a complex .

The topology of is related to the complex that we started with. For example, if is the triangulation of , then is a manifold. Similarly, if is a flag complex, then is contractible.

The idea, now, is to take some finite complex , embed it in , and take a regular neighborhood; the result is a manifold with boundary , and with . Triangulate as a flag complex, and call the resulting complex . Instead of gluing together copies of , glue together copies of along the subdivision of to get and . With some work, we check that is contractible because is flag, and that the contractible space covers the closed manifold , which is therefore aspherical. Since is a retraction of spaces, we have found our desired aspherical manifold with a retraction of fundamental groups.