Thursday, July 21, 2005

Math Post the Eighth: Personal Interlude with Some More Definitions

So I am now officially a doctoral candidate! Heck yeah! Also I'm now (or will soon) be attending a lengthy algebraic geometry conference in Seattle, so I'm not even in Michigan anymore. Perhaps most importantly from a blogging perspective, I'm not madly studying every second of my day anymore (merely some seconds), so I should be able to get back to blogging on a more regular basis. So nyah, say I!

I've also redone my links section, doing away with links to friends' livejournals because I couldn't think of a good way to separate them from links to friends' more serious blogs or therefore from other peoples' more serious blogs. Any complaint and I'll rethink it, but it seemed like the right way to go at the moment.

However, I've decided to defer my promised discussion of isomorphism theorems and universal properties so I can, in celebration of my prelims, explain that portion of my research that I'm currently prepared to discuss (that is, that part that shouldn't be much over anybody's head for reasons other than density). So, in broad strokes, here we go:

One first kind of assumption that mathematicians like to make is a finiteness assumption. It's usually easier to deal with things that are "small" than things that are "big," and indeed strange things can happen in the world of large objects (they can be the same as several copies of themselves stuck together, for one). One finiteness assumption we've already used is precisely that: that the number of elements in a group is finite. A lesser one harks back to my discussion of generation of a group, in particular that our group has a finite set of generators, or is finitely generated. Many (though not quite all) of the examples I've mentioned in that first group post are finitely generated (an exercise for the reader to figure out which!). Making an assumption like this is more about choosing a field of study than picking out a special case.

A second definition I'll need is the condition of a subgroup having finite index. The index of a subgroup H in a group G is the number of cosets that you can form from H with elements of G. In the integers Z, the subgroup 3Z has three cosets, the subgroup 5Z five, and so on. For finite groups, Lagrange's theorem tells us that the index of a subgroup is simply the ratio of the orders (i.e. the number of copies of the smaller group it takes to cover the large one is the number of elements in one divided by the number of elements in the other). If we consider the group of ordered pairs of integers, the subgroup generated by the pair (1,0) is of infinite index, while the subgroup generated by the elements (2,0) and (0,2) is of index 4 (consider quotienting it out and note that you get a (the) two-element group in each component).

The first big assumption I'd make is that of being just infinite. A group G is just infinite if it is infinite but has only finite quotients. I'll cut to the examples in a second, but I need to add one more assumption to get to what I'm actually studying: I don't just want G just infinite, I also want every finite-index subgroup to be just infinite as well -- a condition called being hereditarily just infinite. Our first example is the integers. Since every normal subgroup (in fact, since the integers are abelian, every subgroup) has finite index, there can be no infinite quotients. A slightly more complicated example is the group of symmetries of an infinite line with equally-spaced dots on it (like the integer points in the real numbers), written as D_\infty, where the notation means an infinity symbol. These come essentially in two kinds -- shifts some number of spaces in one direction or the other or reflection around a particular point (or combinations of the two). The group actually looks like (something I'll get to later) the semi-direct product of the integers (the line movements) with the two-element group (a reflection about some point -- you can "make" reflecting about any point by translating, reflecting, then translating back). Seeing that this group is hereditarily just infinite is most easily done by thinking hard about group actions, which I will also get to later.

Before finishing up, I need one more assumption: that the group is not something I'll call simple. A simple group has no nurmal subgroups and therefore no interesting quotients at all. The finite groups of prime order are all simple; showing any other group to be simple (that is, actually doing the proofs) is well beyond the scope of what I'll be able to consider here for some time. Computing all the finite simple groups was a major research project, popularly considered finished around 1980, and filling many thousands of pages (so many in fact that people still aren't completely sure it's all correct -- there's a major effort to write it in some comprehensible way that's still going on). Infinite simple groups satisfy the condition of being hereditarily just infinite trivially (for the mathematicians in the audience, this is because any finite index subgroup H in a group G has a finite-index subgroup H' normal in G inside it -- think about it), but aren't very interesting examples for this same reason.

(For genuine group theorists, the previous paragraph is an ersatz for simply assuming G to be residually finite, which is equivalent for just-infinite groups but harder to explain)

The last class of examples is something harder to get your hands on -- for the big-word-friendly audience, I'm talking about irreducible lattices in higher rank semisimple Lie groups -- and harder still to show in general (again for the big-words crowd, this is Margulis' stuff). For the near future, you can simply think of them as big groups of integer matrices (say 3x3 or larger). My project is to show that these are the only interesting examples. It's likely to be uphill work, but it should at least lead to some intriguing stops along the way.

Sunday, July 17, 2005

Meme-ification

Wow. I used to be behind two, now I'm behind like six or eight in terms of math. I beg preliminary exams on Tuesday, the last hurdle I have before my only goal in life becomes a dissertation. For now, though, I steal from my friend/cooking idol eclectician, and meme about food:

What is your first memory of baking/cooking on your own?

Unclear. I think I may actually have never done it until sometime during summer 2000 when I was living on my own for the first time -- and such memories are phrased purely in terms of an industrial kitchen at an MIT fraternity rather than in terms of one dish or another. Since then, more enthusiasm, more frequency, and more skill (I hope).

Who had the most influence on your cooking?

Again almost uselessly hard to say in terms of people I actually know. Looking back on it, my parents figured out cooking more or less as I grew up, so early memories are few. I suppose the most honest answer is my little brother, who started cooking seriously well before me and who I must always outdo.

Personality/reference-wise, definitely Alton Brown of Good Eats fame, who manages the right mix of improvisation, science, and art (though who can let me down when he cheats).

Do you have an old photo as 'evidence' of an early exposure to the culinary world, and would you like to share it?

Not to my knowledge, alas -- everything I know is lately acquired as these things go. For instance, everything I've yet posted about math I knew before I knew anything about food beyond how to eat lots of it.

Mageiricophobia - do you suffer from any cooking phobia, a dish that makes your palms sweat?

A dish, not really -- there's two Indian desserts of which I've yet to create even a remotely decent rendition (pedas and ras gulla, for anybody wondering), but I usually only make them for myself -- but for a more general phobia there's an easy answer: cooking for the current special someone. The girl's a vegetarian and I am so totally not. Also she has deeply serious taste (unlike many vegetarians) and obviously I'm loathe to disappoint. Between these two things I think that this particular fear has taught me more about cooking than anything else in recent memory.

What would be your most valued or used kitchen gadgets and/or what was the biggest let down?

I probably get most action either out of my chef's knife (ten inches, calphalon, $20) or my cast-iron skillet (twelve inches, I don't even know, $20) -- I suspect if you just left me these and a heat source I'd be able to accomplish 80% of my cooking without great difficulty. As to letdowns, I'd like to nail my ice-cream maker, but the real problem with that one is that I just don't have enough ice-cream consumption taking place in my house to support an ice-cream maker for anything but vanity. For real, I'd probably have to nail that jar-opener my roommate's aunts got us when we moved in -- they furnished the apartment with an impressive array of goods, but that one thing I've never even considered using.

Name some funny or weird food combinations/dishes you really like - and probably no one else!

I'm (alas) an impressively normal kid regarding food combinations. The one thing that people have been surprised about me enjoying is kheer: kheer with cocoa powder or kheer as breakfast with coffee, both of which seem utterly natural to me. After all, milk, sweet? Who cares about the rice component; sense is still made.

What are the three eatables or dishes you simply don’t want to live without?

Number one has got to be cow meat -- out in the midwest, nothing else is as cheap, as good, and as versatile. Number two is garlic, since whenever I manage to make a meal without adding at least a little bit I feel vaguely ashamed, as though I'd forgotten the existence of a family member. Number three is wine. Between the variety of flavors and the sheer utility of the alcohol (not to mention the intoxication value), when the roomie's in town I go through at least a bottle a week.

... that whole list, on reflection, actually sounds pretty bland until you've eaten at my place a few times. I do have a few decent tricks.

Your favorite ice-cream…

I'm tempted to go with my associate's mention of burnt caramel, but there's an institution back in Princeton called Thomas Sweet which produces a flavor called chocolate chip cookie. Not cookie dough, but the cookie itself -- tiny chips mixed into a base of salty-sweet butter. Every bite is pure tollhouse, but summery in a way cookies simply aren't.

You will probably never eat:

Many, many things, but at this point more mediated by my inability to get them than anything else.

Your own signature dish...

At this point, probably the steak I stole from an old New York Times recipe and modified until my roommate was weaned off a previous sauce addiction. Lamentably, I'm a much better thief than creative mind in the kitchen.

A common ingredient you just can’t bring yourself to stomach…

I haven't yet tried tounge...

Which one culture’s food would you most like to sample on its home turf?

Gujarati food. Blatantly self-serving, but these kids are the hardest veggies to impress I know of and I'd like to have a little better idea where it's all coming from.

The people I am tagging are:

Oddly, none of the bloggers I know who read this are foodies. Any are welcome to it, but if I had my choice of the whole wide world I'd pass the baton to Maggie the Mad Wine Girl over at the Wine Offensive, whose erstwhile wares I've sampled many a time and whose punk-gourmet attitude I simply adore.

More math soon, I promise!