Journal stuff
Jan. 12th, 2011 09:54 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo time for some "random stuff from Harry's life"-type stuff...
[Note, I began writing this Tuesday, so all references to "last class" or "today's class" mean Monday.]
So the other day in commutative algebra Hochster used the fact that HomR(M,N)P=HomR_P(MP,NP) if M is finitely presented. (Here R is a commutative ring, P is a prime... yeah, you know.) I didn't recognize this fact and made a point to prove it later. This is pretty easy, but what's noteworthy is that when I was doing it I noticed, wait, actually I think I've seen this exact proof before, back when I was taking Chicago's 1st year graduate algebra sequence with Nori and May. Either that or something *very* similar; I imagine a lot of easy proofs using finite presentation must look a lot like that. (Actually, IIRC, we did like one proof that used finite presentation, and thereafter to keep things simple we just assumed everything was Noetherian, with Nori just noting that for a general ring you'd have to replace finitely generated by finitely presented.)
Actually finite presentation is kind of a weird condition, isn't it? I mean it's a finiteness condition, yeah, but if we're already assuming the object in question is finitely generated, then in some sense it's a finiteness condition that keeps the object from being too *small*.
Also that reminded me that I didn't know offhand whether a module being finitely presented meant that, given any finite set of generators for it, it could be presented with that generating set and only finitely many relations. (I remembered that this was true for groups from that teach-yourself-the-very-basics-of-geometric-group-theory class with Miklos. But of course the notion of "finite generation", and "finite presentation" even more so, can mean very different things depending on what sort of objects you're working with.) I tried to prove this but ended up looking it up, finding an answer on MO here; I felt a bit silly then as not only is it pretty easy, but I even drew that same exact diagram and just didn't even think of trying the snake lemma.
(Sudden thought: What about for A-algebras? Or commutative A-algebras, that should be easier. I'll think about that later. Maybe look it up or ask MO. I'm reminded of when I found that paper that answered the question "are epimorphisms necessarily surjective?" for a number of different categories; ideally there'd be something like that for this.)
So, that was a silly mistake. What can I say, I haven't done this in a while. Where "this" can be construed pretty broadly. To be honest, I have had a pretty shitty (demoralizing, unfun) past 2 semesters. So it is good to be really back in the saddle again. Hochster's commutative algebra class is a real blaze, and I like it. First day: Injective modules, every module embeds in one. (I'm not confident I would have followed this if I didn't already know it.) Second day: Essential extensions and injective hulls. Third day: Structure of injective modules over (commutative) noetherian rings. (He stated it at the end of the second day, and I figured the main part would basically just be Zorn's lemma, but I couldn't figure out why it would be E(R/P) instead of just E(R/I). If it's radical, well, E(A×B)=E(A)×E(B), that's pretty straightforward... but while there will be only finitely many minimal primes over I, they won't be comaximal. And what if it's not radical? Would we get E(R/I)=E(R/√I)? Then today he actually did it, and it's just, any nonzero module over a noetherian ring has an associated prime (contains some R/P)! D'oh! I wasn't anywhere near thinking of that. Yeah, it's been a while.) (Here E stands for injective hull.)
Meanwhile I'm taking multiple complex variables with Barrett. Not many people in that one. Here's something I realized I don't know: On the first day, as an example of something that doesn't generalize to multiple dimensions, Barrett stated the fact that for U⊆C open, S a subset of U having no limit points in U, then for any assignment of values to the points of S there is some holomorphic function on U taking the specified values on the specified points. At the time I was just thinking like, "Oh yeah, that's a fact, weird that it doesn't generalize", but later I realized "Wait a minute! I'm not sure I've ever seen that fact before!" Funny that I accepted it so quickly. I'm kind of at a loss for how to show this, actually. May want to just see if I can look it up.
Last class we proved that yes, in multiple dimensions too, holomorphic functions are analytic, via the Cauchy integral formula for polydiscs. That was a surprise, that there's going to be more than one Cauchy integral formula once we go to multiple dimensions. Actually I just find using the Cauchy integral formula to be funny. It's one of those things I've never been able to remember. I realize it's fundamental to, say, proving the whole calculus-of-residues-thing works, but if I think of complex analysis I tend to just think in terms of the calculus of residues, so when I see the Cauchy integral formula I tend to pause for a second while I note how it follows from the calculus of residues. Yeah, really the chain of implication goes the other way around, but which is it easier to think in terms of? (I seem to recall when doing some representations-of-finite-groups thing once, reproving some basic but useful fact from the orthogonality relations that I later remembered was actually how you go about proving the orthogonality relations in the first place... but really, don't you just tend to think of the orthogonality relations as fundamental?)
Meanwhile, I'm not actually teaching this semester; instead they've put me in the MathLab tutoring / helping kids with homework 10 hours a week. (The MathLab is a room in the basement of the math building where kids go to do math homework and get help from the math people on duty there; all GSIs have to do 1 hour of MathLab duty a week.) I had no idea this was even an option - I didn't ask to be assigned to this - but now I certainly will each semester, because it is so much easier than teaching a class! (Or the terribleness that is TAing 215 or 216.) It's amazing, get to actually once again just spend my free time on, you know, schoolwork and such. (Which reminds me: Starting tomorrow, no more reading Reddit. I started reading it over break when I got bored, but now it's eating up too much of the free time I have finally regained. May stop reading Hacker News too if I find that isn't sufficient.) It's also kind of nice to know that if for some reason I should miss work one day - well, that's bad, and I'll get chewed out for it, but it's nowhere near "I missed teaching a class" bad.
Meanwhile in Truth House I'm just doing work around the house this semester, no more ICC work for me - cleaning the kitchen, organizing food, cleaning the hallways. I asked for ICC work last semester because I had thought I would be scribe for the ICC board again. (Does anyone else think "scribe" sounds a lot cooler than "secretary"? I'm gonna say I was the scribe. Actually they officially list the position as just "minutes taker". So, yeah, scribe.) I hadn't expected that there would be massive confusion about what I was actually supposed to do. This resulted in me missing some work and so now I owe the ICC a bunch of money (which I'm not contesting because some of it was actually my fault, after all). Nothing I can't afford, but damned annoying. So I am *not* asking for ICC work again.
Dan is actually trying to start a D&D group in Truth House. Actually the immediate motivation for this seems to be a bet with Maddy - he's not playing video games, she's not eating chocolate, see who breaks their promise first. (Come to think of it I don't actually know how much the bet is for.) He was pretty shocked to hear I wasn't actually that interested in playing D&D... now if we could get board game nights going again... (Actually, the first day of this, Dan and I played a whole bunch of Flash Duel. So far I like DeGrey best. Then I realized I was supposed to have started cleaning the basement an hour ago. Luckily Chad didn't care.)
There are some new people but having missed the big initial house meeting I can't claim to really know any of them so far; they don't seem to be around a lot. (Then again, I haven't necessarily been around a lot either - though certainly better than previously.) Missing this also meant I was very confused about our new house officers! Chad is work manager all of a sudden? Etc.
Apparently Jeff Lagarias is actually around now? I say this because I saw his office door open the other day. I do have to wonder when this sabbatical of his is actually supposed to be. So I should go bug him sometime. (I missed him when he was last around, due to not reading my email carefully.) He wanted a hard copy of our draft, so, well, that's what school printers are for. Meanwhile I have finally started on my computation of A_{22d(2)}, though I've done very little of it so far. (Also, I think our intro needs to be seriously rewritten, but, eh. Will worry about that some other time. Meanwhile Josh still hasn't finished writing up his proof of his improved upper bound.) No idea if Mike Zieve ever got around to taking a look at what I sent him...
What I suppose I haven't mentioned here at all is the recent death of Dr. Ostfeld. Man... that was a real shock. There's really not much I can say about it that hasn't already been said, of course. He really was the face of the school (to a greater extent than I realized, actually). Since I was in New Jersey I went to the funeral service... I remember I was thinking, I meant to return to the Academies, but, you know, not like this... shit. (I mean, a large part of the reason would be to *see* Dr. Ostfeld...) Well, so it goes.
(I do have to say the idea that Dr. Ostfeld was at all religious was a surprise to me. Actually to be honest that whole aspect seemed kind of ridiculous. The pastor gave a speech that was filled with bad reasoning - and to be clear, by that I don't mean "ha ha religion", I mean entirely secular bad reasoning. Given the context I guess it was supposed to instill some sort of hope, but it's kind of hard to do that when what you're saying just doesn't follow.)
...well, enough about that. Like I said - other people have all said it already, and better. Here's something tangential. One of the people I encountered afterward was Fan Zhang (now studying economics, which, apparently, nobody actually understands), who asked me who from high school I still kept in contact with at all. Not that many, I replied (though at the time undercounted). Still... not that many. Tom comments here and sometimes updates his own LJ. Mike Baumstein and I still meet up sometimes when we're both in New Jersey. Molly and I follow each other on Twitter. I have various back-and-forths with Ingrid and Linda on Facebook (and of course last year there was Appendicitis). (Actually one day towards the end of last semester Ingrid called me on the phone out of nowhere trying to identify the game "No Thanks!".) And of course Sara Raftery but she went to Chicago with me. Oh! And of course Dave Putnins, seeing as he's at Michigan with me now; way to overlook the obvious, Harry. And uh I guess Jon Pinyan's still on LJ. Watson doesn't count as he was never actually in school with me. I mean, I guess that isn't strictly speaking all, but (IINM) that accounts for the vast majority of what little there is. I'm not really sure what the point of this exercise was, to be honest, but having made this inventory I felt compelled to report it.
MIT Mystery Hunt is this weekend. Not forgetting about it this year. I'm afraid I'm going to have to betray Youlian, though; while I could remote solve for Manic Sages, I'd be by myself, unless I wanted to try to organize other people, which I am terrible at. And these puzzles are not really things to be worked on in solitude. (Though I did manage to pretty much solo I'm Your Magic Man two years ago, getting the key insight despite missing how the puzzle hinted at it! What was also funny about that puzzle was how successful we were at identifying many of the hidden cards from tiny snippets of their art, even though this was of course entirely irrelevant.) And Kevin, who's organizing a remote solving team in East Hall, solves for Codex. So having no particular loyalty to any team (having just been introduced to the hunt two years ago by Youlian and having hardly any idea of who the teams actually are), I guess that means I'm Codex this year. (Also, this will be a good time to reset my sleep schedule, especially with the 3-day weekend. Though I don't need to so much, because towards the end of last semester I started getting a lot better at waking up early. Still, I figure it'll help.)
Finally, some links to silly things from MO:
Hendrik Lenstra quotes. (I particularly like the statement that "Recreational Number Theory is that branch of Number Theory which is too difficult for serious study.")
Mathematical apocrypha (apparently there's *books* of these?).
And links to nonsilly things not from MO:
And here's something cool - not new, but neat - handling the problem of strings representing different things by taking our intuition that these are different types, and actually implementing that with actual types in Haskell, so the Haskell compiler can (mostly) check for you that you've sanitized your input properly.
Speaking of things with the abbreviation "MO", Heidi insists I come visit her over my spring break. So, uh, guess I'll be doing that then? This leaves the problem of actually finding a practical way of getting to Rolla...
-Harry
[Note, I began writing this Tuesday, so all references to "last class" or "today's class" mean Monday.]
So the other day in commutative algebra Hochster used the fact that HomR(M,N)P=HomR_P(MP,NP) if M is finitely presented. (Here R is a commutative ring, P is a prime... yeah, you know.) I didn't recognize this fact and made a point to prove it later. This is pretty easy, but what's noteworthy is that when I was doing it I noticed, wait, actually I think I've seen this exact proof before, back when I was taking Chicago's 1st year graduate algebra sequence with Nori and May. Either that or something *very* similar; I imagine a lot of easy proofs using finite presentation must look a lot like that. (Actually, IIRC, we did like one proof that used finite presentation, and thereafter to keep things simple we just assumed everything was Noetherian, with Nori just noting that for a general ring you'd have to replace finitely generated by finitely presented.)
Actually finite presentation is kind of a weird condition, isn't it? I mean it's a finiteness condition, yeah, but if we're already assuming the object in question is finitely generated, then in some sense it's a finiteness condition that keeps the object from being too *small*.
Also that reminded me that I didn't know offhand whether a module being finitely presented meant that, given any finite set of generators for it, it could be presented with that generating set and only finitely many relations. (I remembered that this was true for groups from that teach-yourself-the-very-basics-of-geometric-group-theory class with Miklos. But of course the notion of "finite generation", and "finite presentation" even more so, can mean very different things depending on what sort of objects you're working with.) I tried to prove this but ended up looking it up, finding an answer on MO here; I felt a bit silly then as not only is it pretty easy, but I even drew that same exact diagram and just didn't even think of trying the snake lemma.
(Sudden thought: What about for A-algebras? Or commutative A-algebras, that should be easier. I'll think about that later. Maybe look it up or ask MO. I'm reminded of when I found that paper that answered the question "are epimorphisms necessarily surjective?" for a number of different categories; ideally there'd be something like that for this.)
So, that was a silly mistake. What can I say, I haven't done this in a while. Where "this" can be construed pretty broadly. To be honest, I have had a pretty shitty (demoralizing, unfun) past 2 semesters. So it is good to be really back in the saddle again. Hochster's commutative algebra class is a real blaze, and I like it. First day: Injective modules, every module embeds in one. (I'm not confident I would have followed this if I didn't already know it.) Second day: Essential extensions and injective hulls. Third day: Structure of injective modules over (commutative) noetherian rings. (He stated it at the end of the second day, and I figured the main part would basically just be Zorn's lemma, but I couldn't figure out why it would be E(R/P) instead of just E(R/I). If it's radical, well, E(A×B)=E(A)×E(B), that's pretty straightforward... but while there will be only finitely many minimal primes over I, they won't be comaximal. And what if it's not radical? Would we get E(R/I)=E(R/√I)? Then today he actually did it, and it's just, any nonzero module over a noetherian ring has an associated prime (contains some R/P)! D'oh! I wasn't anywhere near thinking of that. Yeah, it's been a while.) (Here E stands for injective hull.)
Meanwhile I'm taking multiple complex variables with Barrett. Not many people in that one. Here's something I realized I don't know: On the first day, as an example of something that doesn't generalize to multiple dimensions, Barrett stated the fact that for U⊆C open, S a subset of U having no limit points in U, then for any assignment of values to the points of S there is some holomorphic function on U taking the specified values on the specified points. At the time I was just thinking like, "Oh yeah, that's a fact, weird that it doesn't generalize", but later I realized "Wait a minute! I'm not sure I've ever seen that fact before!" Funny that I accepted it so quickly. I'm kind of at a loss for how to show this, actually. May want to just see if I can look it up.
Last class we proved that yes, in multiple dimensions too, holomorphic functions are analytic, via the Cauchy integral formula for polydiscs. That was a surprise, that there's going to be more than one Cauchy integral formula once we go to multiple dimensions. Actually I just find using the Cauchy integral formula to be funny. It's one of those things I've never been able to remember. I realize it's fundamental to, say, proving the whole calculus-of-residues-thing works, but if I think of complex analysis I tend to just think in terms of the calculus of residues, so when I see the Cauchy integral formula I tend to pause for a second while I note how it follows from the calculus of residues. Yeah, really the chain of implication goes the other way around, but which is it easier to think in terms of? (I seem to recall when doing some representations-of-finite-groups thing once, reproving some basic but useful fact from the orthogonality relations that I later remembered was actually how you go about proving the orthogonality relations in the first place... but really, don't you just tend to think of the orthogonality relations as fundamental?)
Meanwhile, I'm not actually teaching this semester; instead they've put me in the MathLab tutoring / helping kids with homework 10 hours a week. (The MathLab is a room in the basement of the math building where kids go to do math homework and get help from the math people on duty there; all GSIs have to do 1 hour of MathLab duty a week.) I had no idea this was even an option - I didn't ask to be assigned to this - but now I certainly will each semester, because it is so much easier than teaching a class! (Or the terribleness that is TAing 215 or 216.) It's amazing, get to actually once again just spend my free time on, you know, schoolwork and such. (Which reminds me: Starting tomorrow, no more reading Reddit. I started reading it over break when I got bored, but now it's eating up too much of the free time I have finally regained. May stop reading Hacker News too if I find that isn't sufficient.) It's also kind of nice to know that if for some reason I should miss work one day - well, that's bad, and I'll get chewed out for it, but it's nowhere near "I missed teaching a class" bad.
Meanwhile in Truth House I'm just doing work around the house this semester, no more ICC work for me - cleaning the kitchen, organizing food, cleaning the hallways. I asked for ICC work last semester because I had thought I would be scribe for the ICC board again. (Does anyone else think "scribe" sounds a lot cooler than "secretary"? I'm gonna say I was the scribe. Actually they officially list the position as just "minutes taker". So, yeah, scribe.) I hadn't expected that there would be massive confusion about what I was actually supposed to do. This resulted in me missing some work and so now I owe the ICC a bunch of money (which I'm not contesting because some of it was actually my fault, after all). Nothing I can't afford, but damned annoying. So I am *not* asking for ICC work again.
Dan is actually trying to start a D&D group in Truth House. Actually the immediate motivation for this seems to be a bet with Maddy - he's not playing video games, she's not eating chocolate, see who breaks their promise first. (Come to think of it I don't actually know how much the bet is for.) He was pretty shocked to hear I wasn't actually that interested in playing D&D... now if we could get board game nights going again... (Actually, the first day of this, Dan and I played a whole bunch of Flash Duel. So far I like DeGrey best. Then I realized I was supposed to have started cleaning the basement an hour ago. Luckily Chad didn't care.)
There are some new people but having missed the big initial house meeting I can't claim to really know any of them so far; they don't seem to be around a lot. (Then again, I haven't necessarily been around a lot either - though certainly better than previously.) Missing this also meant I was very confused about our new house officers! Chad is work manager all of a sudden? Etc.
Apparently Jeff Lagarias is actually around now? I say this because I saw his office door open the other day. I do have to wonder when this sabbatical of his is actually supposed to be. So I should go bug him sometime. (I missed him when he was last around, due to not reading my email carefully.) He wanted a hard copy of our draft, so, well, that's what school printers are for. Meanwhile I have finally started on my computation of A_{22d(2)}, though I've done very little of it so far. (Also, I think our intro needs to be seriously rewritten, but, eh. Will worry about that some other time. Meanwhile Josh still hasn't finished writing up his proof of his improved upper bound.) No idea if Mike Zieve ever got around to taking a look at what I sent him...
What I suppose I haven't mentioned here at all is the recent death of Dr. Ostfeld. Man... that was a real shock. There's really not much I can say about it that hasn't already been said, of course. He really was the face of the school (to a greater extent than I realized, actually). Since I was in New Jersey I went to the funeral service... I remember I was thinking, I meant to return to the Academies, but, you know, not like this... shit. (I mean, a large part of the reason would be to *see* Dr. Ostfeld...) Well, so it goes.
(I do have to say the idea that Dr. Ostfeld was at all religious was a surprise to me. Actually to be honest that whole aspect seemed kind of ridiculous. The pastor gave a speech that was filled with bad reasoning - and to be clear, by that I don't mean "ha ha religion", I mean entirely secular bad reasoning. Given the context I guess it was supposed to instill some sort of hope, but it's kind of hard to do that when what you're saying just doesn't follow.)
...well, enough about that. Like I said - other people have all said it already, and better. Here's something tangential. One of the people I encountered afterward was Fan Zhang (now studying economics, which, apparently, nobody actually understands), who asked me who from high school I still kept in contact with at all. Not that many, I replied (though at the time undercounted). Still... not that many. Tom comments here and sometimes updates his own LJ. Mike Baumstein and I still meet up sometimes when we're both in New Jersey. Molly and I follow each other on Twitter. I have various back-and-forths with Ingrid and Linda on Facebook (and of course last year there was Appendicitis). (Actually one day towards the end of last semester Ingrid called me on the phone out of nowhere trying to identify the game "No Thanks!".) And of course Sara Raftery but she went to Chicago with me. Oh! And of course Dave Putnins, seeing as he's at Michigan with me now; way to overlook the obvious, Harry. And uh I guess Jon Pinyan's still on LJ. Watson doesn't count as he was never actually in school with me. I mean, I guess that isn't strictly speaking all, but (IINM) that accounts for the vast majority of what little there is. I'm not really sure what the point of this exercise was, to be honest, but having made this inventory I felt compelled to report it.
MIT Mystery Hunt is this weekend. Not forgetting about it this year. I'm afraid I'm going to have to betray Youlian, though; while I could remote solve for Manic Sages, I'd be by myself, unless I wanted to try to organize other people, which I am terrible at. And these puzzles are not really things to be worked on in solitude. (Though I did manage to pretty much solo I'm Your Magic Man two years ago, getting the key insight despite missing how the puzzle hinted at it! What was also funny about that puzzle was how successful we were at identifying many of the hidden cards from tiny snippets of their art, even though this was of course entirely irrelevant.) And Kevin, who's organizing a remote solving team in East Hall, solves for Codex. So having no particular loyalty to any team (having just been introduced to the hunt two years ago by Youlian and having hardly any idea of who the teams actually are), I guess that means I'm Codex this year. (Also, this will be a good time to reset my sleep schedule, especially with the 3-day weekend. Though I don't need to so much, because towards the end of last semester I started getting a lot better at waking up early. Still, I figure it'll help.)
Finally, some links to silly things from MO:
Hendrik Lenstra quotes. (I particularly like the statement that "Recreational Number Theory is that branch of Number Theory which is too difficult for serious study.")
Mathematical apocrypha (apparently there's *books* of these?).
And links to nonsilly things not from MO:
And here's something cool - not new, but neat - handling the problem of strings representing different things by taking our intuition that these are different types, and actually implementing that with actual types in Haskell, so the Haskell compiler can (mostly) check for you that you've sanitized your input properly.
Speaking of things with the abbreviation "MO", Heidi insists I come visit her over my spring break. So, uh, guess I'll be doing that then? This leaves the problem of actually finding a practical way of getting to Rolla...
-Harry
