Cam's Blog

October 31, 2009

The next two weeks

Filed under: Math — cfranc @ 4:29 pm

I’ve been thinking about what I’d like to write about on this blog, and I have some ideas to keep me busy for the next little while. Yesterday I introduced the notion of symmetry and mentioned the notion of an abstract group. Over the next few days I’d like to discuss a related topic that is close to my heart: the symmetries of numbers. For this I’ll need to introduce groups and number fields, and I’ll try to explain some aspects of Galois theory in an accessible fashion. I’d like to follow this up sometime in the future with a discussion of how the Monster group can be realized as a Galois group over the rationals.

The Monster group will have to wait for awhile, as Vladimir Berkovich is visiting Montreal for the next two weeks. He’s giving a series of 3 lectures, to be followed by an open discussion session. The first lecture takes place on Thursday November 5 at Concordia University, as part of the Quebec-Vermont Number Theory Seminar. An abstract hasn’t been posted yet, but I believe this will be a more advanced research level talk on p-adic integration of one-forms. On the following Thursday November 12, Berkovich will give two introductory lectures on his analytic spaces, assuming background in rigid analytic geometry and Raynaud’s approach to p-adic geometry via formal schemes. Finally, on Friday November 13, there will be an open discussion session on Berkovich spaces. As I’d eventually like to understand how these spaces arise in Harris-Taylor’s work on the local Langlands conjecture, it seems like a good time to learn some more about Berkovich’s work. So expect some posts on this as well!

Oh, and if you’ve never heard of Berkovich or his work before, you should definitely check this out from his personal webpage. It’s taken from the AMS volume on the 2007 Arizona Winter School on p-adic Geometry, and describes Berkovich’s discovery of his analytic spaces. Some interesting personal details are included about the discrimination that Berkovich faced in Russia at the time. In particular, despite his brilliance as a mathematician, as a Jew he had a difficult time getting an academic position in mathematics. He had to settle for a job as a computer programmer, and learned to work on mathematics during odd free hours of the working day, or else very early in the morning (his evenings were devoted to his family!). These personal reflections make for an entertaining read for mathematicians and nonmathematicians alike, although you may wish to skim through some of the mathematics.

October 30, 2009

Warren Ellis and Planetary — “It’s a strange world. Let’s keep it that way.”

Filed under: Comics, Math, Warren Ellis — cfranc @ 7:18 pm

This first post is about one of my favorite comic book authors, Warren Ellis. I haven’t yet read a work by Ellis that I failed to enjoy. Now, you should be warned at the outset that I’m not a comic expert. I’ve only recently returned to reading them. As a kid I certainly had an interest in comics, but with three other siblings, it would have been ridiculous for me to expect my parents to buy me all the comics that I would have liked. The occasional copies of Ghost Rider, Spawn and Spiderman where enough, though, to give this return to the medium a pleasant sense of nostalgia.

The pending release of the Watchmen movie last year prompted me to rush out and finally read the graphic novel before it could be spoiled on the big screen (gladly, it wasn’t). Anyone that’s read Moore’s work will attest to the fact that he is very gifted. If you’ve only experienced the Watchmen film, or haven’t experienced either the film or the comic, then you really must read it. But Warren Ellis didn’t write the Watchmen, and this is a post about Ellis. So how does this relate to him? Well, while taking my teenage “hiatus” from comics, my friend Jeff spoke extremely highly of both Watchmen and a comic series called Planetary. Ellis wrote Planetary, and since the suggestion to give Watchmen a shot was such a success, I opted to give Planetary a chance as well. It’s thanks to this that I’ve returned to the comic medium.

What is Planetary? Clicking here will give you a detailed answer. In brief, it’s a story about a group of four (five, really) individuals with incredible powers investigating the mysteries of the universe. There is a group of four baddies, a sort of evil version of the Fantastic Four, bent on using these secrets for their own nefarious purposes. I don’t want to give too much away, but one of the fundamental secrets about the Planetary universe is that it is in actuality a multiverse. Ellis is more specific:

This is the shape of reality. A theoretical snowflake existing in 196,833 dimensional space. The snowflake rotates. Each element of the snowflake rotates. Each rotation describes an entirely new universe. The total number of rotations are equal to the number of atoms making up the earth. Each rotation makes a new earth. This is the multiverse. — Planetary issue 1

Let’s take a minute to digress about the mathematics being alluded to in this quotation. For this we need introduce the concept of a group. Since this is a post about comics and not mathematics, we’ll be quite pedestrian in our discussion. A group is a mathematical gadget that abstracts the concept of symmetry. Imagine some concrete geometric object in your mind, for instance a square. A symmetry of that object is a transformation from that object to itself which preserves the object. For instance, you can rotate a square through an angle of 90 degrees, and you’ll wind up with the same square. Or you can reflect the square through one of its two diagonal axes. Here are two examples of transformations which are not symmetries: (1) rotate the square by 45 degrees. You still get a square, but it’s oriented differently from the one you started with. (2) Fold the square in half, so that you wind up with a squared off C (in this discussion a square has no interior). This doesn’t look at all like a square, so it is also not a symmetry. A square has eight distinct symmetries, and they fall into two categories.  First there are the rotations by 90, 180, 270 and 360 degrees.  Then there are the reflections through the four axes of the square; the two diagonals, the horizontal and the vertical. Note that a rotation by 360 degrees just leaves the square as is — it’s a somewhat privileged symmetry, called the identity transformation. The collection of these symmetries is the symmetry group of the square. Mathematicians refer to it as the dihedral group D_4 (read Dee-Four). In general, the symmetries of a regular n-gon define the dihedral group D_n. Other geometric objects will generate other symmetry groups.

If one extracts the “essense” of these various symmetry groups, and expresses it in mathematical terms, then one obtains the modern notion of an abstract group. Luckily for this post, Ellis thinks of groups in terms of symmetries, so we can get by without this more abstract (but extrememly elegent) definition. Now, there are very many groups. Infinitely many in fact. However, similarly to how every integer can be factored into prime numbers, there is a certain class of groups, called simple groups, which are the building blocks of all groups (to be a bit more precise, we really mean finite groups here). These simple groups can be categorized in a nice way. Barring some trivial exceptions, they can be arranged into 16 distinct types. Each type is an infinite family of simple groups sharing many characteristics; these common characteristics are what puts them in the same family. What is somewhat surprising is that there are exactly 26 sporadic simple groups which don’t fit into this classification scheme. The largest of these is called the Monster group; it is a group containing

246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 — Wikipedia

symmetries. This number is approximately equal to 1054

The Monster can be realized as a subcollection of all the symmetries of a 196,833 dimensional space (more precisely, the smallest faithful complex representation of the monster is 196,833 dimensional). This explains the number in the Planetary quotation above. Ellis is imagining the Monster group as a symmetry group of some intricate geometric “snowflake” in 196,833 dimensional space. The quotation seems to imply that there is one universe in the multiverse for each symmetry of this snowflake, which is simply poetic. It also suggests that there are as many symmetries of the snowflake as there are atoms in the earth. Common estimates put the number of atoms in the earth in the ballpark of 1050, which ain’t too far off from 1054, at least by comic standards. (Now to be a little objective, I should point out that the number of atoms on earth is not a constant number. We blow up atoms, we toss atoms into space and comets donate to our deficit every once in awhile. I personaly am willling to lend Ellis some poetic liscence here and assume he means them to be approximately constant.)

It was quite satisfying to discover this bit of mathematics figuring so prominently in Planetary. And the inclusion of the multiverse in the story is not just some gimmicky sci-fi nonsense, employed to entertain math guys like me. Planetary borrows from and revises other comic universes — DC and Marvel, most notably — and even some classic characters of literature make appearances. The Planetary story is a literary multiverse in a very real sense. For those unfamiliar with these other sources, Planetary may not have quite as much to offer. However, I do feel that even if you are completely clueless about comics, there’s still a lot to be gained from reading this 27 (a perfect cube!) issue series. The storytelling is great, as is the artwork, and it gives a nice overall introduction to many of the themes of superhero comics.

Wow, the mathematical digression has padded this out a bit, so I’m going to stop this short. I’ll close by mentioning that you can read a webcomic by Ellis, called FreakAngels, for free. I just found this today, and it’s badass. So don’t pass up this free introduction to the wonder of Warren. If you enjoy it, then you may want to check out Planetary, Global Frequency, Gravel, Transmetropolitan, or really anything else by Ellis. I most recently read Gravel, and it really appealed to the H.P. Lovecraft fan in me. It’s very seasonal reading material, I assure you.

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