I'm not going to focus too long on this, but I'm sure you have read somewhere that Levon Helm of The Band died a couple days ago, which is sad. Here's a link which is probably unnecessary for anyone who is either American or Canadian, which should cover pretty much my entire audience, since you've no doubt heard the song, but it is still one of the best ever, and no doubt it is getting tons of views lately.
Anyway, what's new with me? Well, not a whole lot. There's a lot more going on over at JLink's blog, so why not head over and check it out? I'm pretty much just grinding through math and getting up to my usual boring hijinks when I'm not doing that. In my math life, I am taking a class in algebraic geometry while slowly progressing on research, so have two more paragraphs on that:
Algebraic geometry is all about schemes and varieties and whatnot. It's very terminology heavy, as you can probably guess from that last sentence. For example, a scheme is the spectrum of a ring, equipped with the Zariski topology, and its structure sheaf. If you don't know what those are, don't worry, I won't explain, but suffice it to say that each will require a couple more sentences just to tell you what they even are. It's quite complicated to say it all, but in some sense is just formalizing geometric notions as algebra. It's also highly unintuitive to just look at and I doubt anyone would be interested in it, but tell me this isn't a convoluted way of building stuff:
ring -> prime ideals on the ring -> topological space on the set of these ideals -> sheaf of rings over this space
My research is, well, my research. I'm currently trying to show that the cohomology of the cone of the inclusion of a certain module category into another module category is acyclic, which will show that the two categories' homotopy categories are equivalent. That seems quite out there, but the actual calculation is basically just trying to find the preimage under a certain operator of a given pair of sets of linear maps. Even that sounds confusing, but it's really just saying if I know d(f,g) = 0, how can I find h and k so that d(h,k) = (f,g). So, it's a lot like guessing at stuff and seeing what happens when you apply d. I won't explain further.
Outside of that, not much is happening. I substituted for a class last week and that earned me $100. My one friend bought Risk 2210, which is like Risk with extra territories, a currency system which is used for buying cards and special units that affect the game, and a 5 turn limit. It seems quite complicated at first glance, even compared to usual Risk, so I think it will be hard to find many people who would be willing to learn to play it, but it turns out that it is actually quite simple, and also quite fun. It's his birthday soon, so I bought Risk: Godstorm off the interwebs, and we'll see how fun that is to play. Don't worry, he doesn't read this blog, so I'm not giving anything away.
My roommate and I recently finished watching Star Trek: Voyager, so I'm kind of out of stuff to watch. I don't really like getting involved in tv dramas and they end up being boring to me after a while, and there's only a small subset of comedies that aren't terrible and I'm already watching/have watched all of them, I think. I may post some more thoughts on Trek, since probably one person out there enjoys those shows.
I originally started watching The Next Generation when it popped up on Netflix, and it was quite enjoyable, if very dated and full of stupid crap that makes no sense all the time. I also basically had to skip watching seasons 1 & 2 since they are painful, but from there on out, it is mostly pretty great, with a few egregious examples in the final season. Then we started watching DS9, just to keep watching something. It's another one of those shows that takes a while to really find its groove. I wasn't really much of a fan of it for the first couple seasons, but by the time Sisko shaves his head, it's fairly consistently watchable, if not good. It's also extremely dated and sort of strange to watch now, since apparently tv drama has improved quite a bit since then. Things that seemed good at the time now come off as really cheesy or bland. Another thing I noticed is that Jadzia is one of the most annoying characters of all time and every time she refers to something Curzon did as something she did, I just want to throw her out the nearest airlock, but I don't want to spend a whole post talking about this show.
Then we finished that show (incidentally, despite the huge run-up to the finale of DS9, it feels quite underwhelming and simultaneously rushes and drags, while All Good Things is a standalone classic of an episode), I was going to stop forcing my roommate to watch Trek, since even TNG doesn't really hold up to the test of time and Voyager wasn't even considered any good when it was on. But, my roommate decided to watch it anyway, so we plowed through that as well. I think a lot of it was made more enjoyable by my tendency to be doing something else while it was on because its ultimate failing is that it's really just not compelling at all. The characters are almost uniformly featureless and most episodes are just a forgettable technobabble problem with a technobabble solution. Like I said, I may post about this again, but I do want to mention that Kate Mulgrew is such a terrible actress that it's actually sort of interesting to watch her. It's like she doesn't really know how people actually behave, so she's constantly mugging inappropriately to the point where I sometimes would just laugh at the random reaction shots that seemed to be almost omnipresent in the show. And since I mentioned the finales for the other two sequel treks, the finale of Voyager is pretty much complete garbage, like the guys making the show realized that it didn't matter what they did at that point, so they just crapped out something that barely qualifies as wrapping up the story.
Alright, now that I've probably lost everyone, I'll end the post with note that the last time I posted, I posted twice in one day, so unless you are observant, you may not have seen the first, much more substantial post. I only mention it because nobody commented on it, but one person (thanks!) commented on the shorter post.
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4 comments:
I had a wonderful comment on the barriers to entry of math terminology (for example, don't tell anyone, but I forgot what a prime ideal is strictly by name, and that temporarily kept me from following your ring-to-sheaf-by-arrows definition) all written out, and I got a timeout error when I clicked to post it. I guess it didn't fit in the margins. Anyway, it's a shame how much math people don't know they know because there so many layers of definitions.
I had not heard about Mr. Helm. Bummer.
Is there a map on sheaves to bring them in? Maybe a toss metric? har har. Thanks for the gratuitous plug. Keep Risking.
Thanks for the comment. Blogger seems to have undergone some weird layout stuff recently, so maybe the comment thing is related to that? Incidentally, I could never keep straight what a prime ideal is, either, until taking algebraic geometry, where the primality comes up all the time. It's actually easy to remember if you just use the prototypical example as your guide: the multiples of a prime number as an ideal in the ring of integers. The multiples of any integer, say, n, form an ideal since nk - nl = n(k-l) and k(nl) = n(kl), so that the ideal is indeed a subring that "absorbs" things outside of it, but if n isn't prime, like say, 4, you can get to the ideal by multiplying things that aren't in it, like say, 2(2)=4. 2 isn't in the ideal, but 4 is. If n is prime, then if ab is in the ideal, ab = nl, so n|a or n|b, so either a or b is in the ideal. The ideal (0) is also prime here, but my advisor likes to say "0 is prime, don't let anyone tell you it's not."
Right, the "absorptive subring" definition pretty much covers it. I always have to go through "an ideal is a subring that..." followed by "a prime ideal is an ideal that..." Hence the layers of definitions. Many more layers to get to a sheaf.
Math professors are the quotable best. There's only one that covers fecal shower ideals, though. Classic.
(I think my first post fail was simply how long I took to write it (slow typing, plus stopping half-done for a while) so that I appeared to be inactive and it did an auto-logout. Shucks.)
My apologies for the apparent comment pattern: when I comment, mathematically or not, it seems to shut down the thread for anyone else to join in. Not cool.
While I agree that it's double plus bad that the comment stream gets dammed up, I think it isn't a fault of your comments. I actually am not sure if anyone else is even skimming through to the end, since my posts are not thought out and tend to run on forever.
Here's another (warning: category theoretic) notion of ideals that I like: they are the things you can mod out by. In groups, it's normal subgroups, in abelian groups, it's subgroups, and in topological spaces, it's, well, anything, but be prepared not to understand where you're standing at the end. Prime ideals can be characterized as being the kernels of some homomorphism, so I guess there's your category theory way of explaining them: they're kernels.
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