When people find out that I'm a graduate student in math, they usually react by asking something like, "what research is there in math?" which sort of baffles me. In my view, it's easy to see that not only is there a bunch of stuff left to be learned, but that unlike in other fields, there will always be more stuff to be learned. If anything, it's fields like biology and chemistry that should induce these sorts of questions because those seem like very finite systems, where we're just trying to figure out how a relatively limited number of objects act and interact. At times it seems crazy that we don't know yet how any reaction would work. Math truly is infinite in the sense that there is no limit to the objects under study, so it's a bizarre concept to me that people think there's nothing left to do.
I guess it's because most people take math classes that make it seem like a bunch of methods and never think beyond that. The other day, a guy asked me if I was taking things like "Calculus VI." I had to keep myself from laughing, but I guess it's reasonable if you just keep your head down and are taught:
-how to solve one step equations
-how to solve two step equations
-how to solve quadratic equations
-how to solve trigonometric equations
-how to row reduce a matrix
-how to find a limit
-how to differentiate
-how to find a Riemann sum
-how to integrate
-how to find a gradient
-how to integrate multiple variables
Note that none of those things make it clear where problems come from or even what the objects you are dealing with are, so it seems like there is just a finite set of stuff to solve and it's been done by other people and this is how you do it. It's such a strange mindset, though.
Anyway, other math people sometimes like to ask what I do and by this point that almost makes me laugh, too, because I know that almost nobody will understand the answer, or at least not have the patience to understand it. If you do research in any field, you are probably familiar with the phenomenon that is hyper-specialization. We tend to look at people in a field as having a sort of homogeneous area of knowledge, and it's probably true to an extent, but when you're in a field, it seems so heterogeneous and disparate that you would never have any chance of knowing what the guy down the hall actually does beyond being able to specify a general subfield.
It's funny because grad students don't know anything (and I include myself here) but we're starting to specialize and so we learn that these things are important and these other things can be ignored, but our friend is learning just the opposite. I have a friend whose office is right near mine who studies compositions. I actually had to look up whether it was compositions or partitions the other day because I had a sum that looked like it was over compositions of an integer and I wanted to know if that was the right term. For reference, she deals with ways of adding integers to get a certain integer. For example (2,1, 1) and (3,1) are compositions [I think, I don't deal with these objects and have little interest in them] of 4. She deals specifically with random compositions and the distributions of things related to them. I don't really know, and though she is good at what she does, I can't ever bring myself to read what she writes because it looks like what I like to call "the wrong kind of math," which is just page after page of algebraic manipulation of sums and occasionally "Big Oh"-notation, which is generally just a sign that I won't enjoy it.
She won't even ask what it is I do because it makes no sense to her.
Another friend likes graph theory and wants to do research with that, though it's hard because almost nobody does that here. He's asked me a few times how categories or diagrams work because they're objects he's never dealt with and I've tried to explain it in terms of directed graphs, to limited success. Any other approach is like Chinese to him, but I think he gets some of it.
So what do I do? I am supposed to be working on a theory of curved A-infinity algebras that parallels non-curved A-infinity algebras, which I guess are pretty well understood and are rather important to string theory, though the physical systems they model are beyond my knowledge (this is another peculiar phenomenon of mathematical research). Of course, this is just words to almost everybody. It doesn't help that it is a rather abstract algebraic setting and most of the grad students I know hate abstract algebra to a certain extent.
I've often thought of people, at least up through the undergraduate level, who self-identify as "math people" as being in two camps, the ones who liked algebra but not geometry in high school and the ones who liked geometry. Nobody likes trigonometry, by the way. In my view, people in the first camp don't actually like math; they like being told how to do things and then doing them. Unlike in things like literature where it is evident that some thought will be necessary because "there is no right or wrong answer," it's not clear that some math classes (algebra, as taught in high schools) just require application of techniques to many similar problems, and some require a higher level of deductive reasoning (geometry, as taught in high schools). Almost needless to say, "real" math people tend to look down on "fake" math people.
Recently, though, I've come to think of it more like a spectrum, as those liberal arts people are so eager to append to human sexuality, or like some other, more complicated object (it almost irks me to use the word spectrum like this, since it has two specific uses in math that aren't like what "spectrum" probably makes you think of, but that's math for you). There are some people who like having some rules to work with, and some people like to have more rules, and some people like to have less. Some people want more tools, and others like to get by with as few as they need. To make it more concrete, I'll make it more abstract. Some people hate abstract algebra because they don't like not being able to commute variables, or not being able to say if a product is 0, then one of the factors is, too. Some people are comfortable with real numbers and are content to deal with the analytic properties and whatnot and others only like the integers and the concept of modding out or gluing spaces together frightens and confuses them.
Anyway, my point is that in order to understand what I'm learning about, you have to be ok with the concept of an algebra, and then with the concept of a graded algebra, and then a differential graded (dg-) algebra, etc. Some people don't like this concept even if they are ok with vector spaces, which is really basically what they are, minus the grading, maybe. So if I explain it, I always have to explain it starting there. So, to make a long story short, an A-infinity algebra is a graded algebra which has a bunch of "higher multiplications," which are really linear maps from the tensor powers of the algebra back to the algebra, which satisfy a certain equation stated in terms of a sum of all the possible ways of mapping from the n-th tensor power to the algebra. What it really means is that the higher multiplications are "associative up to homotopy." This means that while you maybe can't group any way you want and get the same answer, you'll get the same answer up to homotopy, which is a concept I don't even want to explain. Needless to say, this concept scares a lot of math people.
An A-infinity algebra A, by the equations it must have a linear map of degree 1 b_1:A -> A that squares to 0, so it is naturally a dg-algebra. To be curved, the algebra just needs an extra map that takes the 0-th tensor power of A back to A. The 0-th power is understood to be the underlying field. Anyway, this means that all the defining equations can now include this map, so the sums are different, and this means that in general b_1 won't square to 0, so it's no longer a dg-algebra in the natural way. This messes up all sorts of category theory type conclusions that were true in the non-curved case, and so it needs to be investigated, I guess by me. I guess also that that's enough for now, so later!
Sunday, August 28, 2011
Sunday, August 14, 2011
New Hat
If right wing ideologues are going to destroy this country, the only reasonable response seems to be pushing further left. Go carryin' pictures of chairman me, and you ain't gonna make it with anyone, anyhow.
The hat is a souvenir from a Chinese girl in my department who just got back last week. I gave her a ludicrously jingoistic USA hat in return.
Monday, June 20, 2011
In Response to a Math Question
I got a comment a few posts back about Graham's number and whether it larger than a googolplex of towers or some such thing. Let me first say sorry for not replying to that; I thought I had, but apparently just thought about it for a little and forgot to write anything up. The answer is I have no idea. I don't even know where you would begin trying to prove something like that. If you read the wikipedia link about Graham's number, then you actually know more about it than I do, since I didn't finish reading it originally and then forgot all about it until now.
If you don't feel like reading about what it is, suffice it to say that it is a positive integer that was given as an upper bound to a problem in Ramsey theory, which is a branch of graph theory. I don't really know much about it, but it seems like an interesting branch of mathematics, albeit one that I don't think about much. As for how large it is, I think only the word unfathomably suffices to describe it. Something like there aren't enough particles in the universe to save it in digital form.
In order to keep this post slightly less low-content, I'll mention that what amazes me is that even though numbers like Graham's number are basically impossible to get a handle on, in math we routinely deal with things that are much, much larger than that. For as large as it is, it's still a finite number (which is the same as a finite ordinal [sort of]!), and we deal with sets that are infinite all the time.
I think we sort of lose track of how big infinite is because we think of sets like the natural numbers, which are infinite, but we think of them in forms like
{1,2, ...}
which sort of hides the fact that there are huge things in there and it's almost easy to forget that.
Now, to blow your mind a little bit, which are there more of, non-negative integers, or non-negative even numbers? Intuitively, there are more non-negative integers since all even non-negative integers are non-negative integers. But consider the function f(n) = 2n. It turns each non-negative integer into an even non-negative integer, and it does it in such a way that we don't repeat anything and every even non-negative integer gets hit. So there must be the same number of them.
So it's natural to think that maybe there are only finite and infinite things, and in the literal sense that is true: something is either finite or NOT finite, but in a sense it isn't true. There are lots of infinities. For example, consider the real numbers. If you don't know what that is, suffice it to say consider the set of all sequences you can make of 0's and 1's. Now, if that set is the same size as the positive integers (or non-negative integers, or just integers, or even rational numbers), then we can count them, and we can make a new sequence by starting with the first sequence and setting the first entry of the new sequence to be the opposite of whatever the first sequence is. That is, if the first sequence has a 0 in the first place, choose a 1, and if it had a 1, choose a 0. Then move on to the second sequence and do the same thing with the second entry of the new sequence.
If you keep doing this for each sequence, you'll get a new sequence that differs from every sequence in one place, so can't be an element of the original set of sequences, but is clearly just made of 0's and 1's, so must be an element of the original set. So, there's a contradiction and you must have a bigger set than the natural numbers. In math terms, we say that the real numbers form an uncountable set.
Can we get even bigger? Yeah, just consider the set of all subsets of the real numbers. Then consider the set of all subsets of that set, and so on and so forth. Cantor proved that you always get a bigger set by doing this, so that there is no largest set. It gets even crazier than that, but I guess that's enough food for thought.
If you don't feel like reading about what it is, suffice it to say that it is a positive integer that was given as an upper bound to a problem in Ramsey theory, which is a branch of graph theory. I don't really know much about it, but it seems like an interesting branch of mathematics, albeit one that I don't think about much. As for how large it is, I think only the word unfathomably suffices to describe it. Something like there aren't enough particles in the universe to save it in digital form.
In order to keep this post slightly less low-content, I'll mention that what amazes me is that even though numbers like Graham's number are basically impossible to get a handle on, in math we routinely deal with things that are much, much larger than that. For as large as it is, it's still a finite number (which is the same as a finite ordinal [sort of]!), and we deal with sets that are infinite all the time.
I think we sort of lose track of how big infinite is because we think of sets like the natural numbers, which are infinite, but we think of them in forms like
{1,2, ...}
which sort of hides the fact that there are huge things in there and it's almost easy to forget that.
Now, to blow your mind a little bit, which are there more of, non-negative integers, or non-negative even numbers? Intuitively, there are more non-negative integers since all even non-negative integers are non-negative integers. But consider the function f(n) = 2n. It turns each non-negative integer into an even non-negative integer, and it does it in such a way that we don't repeat anything and every even non-negative integer gets hit. So there must be the same number of them.
So it's natural to think that maybe there are only finite and infinite things, and in the literal sense that is true: something is either finite or NOT finite, but in a sense it isn't true. There are lots of infinities. For example, consider the real numbers. If you don't know what that is, suffice it to say consider the set of all sequences you can make of 0's and 1's. Now, if that set is the same size as the positive integers (or non-negative integers, or just integers, or even rational numbers), then we can count them, and we can make a new sequence by starting with the first sequence and setting the first entry of the new sequence to be the opposite of whatever the first sequence is. That is, if the first sequence has a 0 in the first place, choose a 1, and if it had a 1, choose a 0. Then move on to the second sequence and do the same thing with the second entry of the new sequence.
If you keep doing this for each sequence, you'll get a new sequence that differs from every sequence in one place, so can't be an element of the original set of sequences, but is clearly just made of 0's and 1's, so must be an element of the original set. So, there's a contradiction and you must have a bigger set than the natural numbers. In math terms, we say that the real numbers form an uncountable set.
Can we get even bigger? Yeah, just consider the set of all subsets of the real numbers. Then consider the set of all subsets of that set, and so on and so forth. Cantor proved that you always get a bigger set by doing this, so that there is no largest set. It gets even crazier than that, but I guess that's enough food for thought.
Sunday, June 19, 2011
Hey Look What I Made
It's noodle salad. It's the first time I made this kind of noodle salad, which uses Italian dressing and not mayonnaise. So that's pretty good. My buddy Phill had a cookout yesterday so that is what I made and people liked it. Not much else happening.
Wednesday, June 8, 2011
Take a Load Off, Fannie
I realize I haven't posted in weeks, but I've been busy. Fortunately, that should all be done now and I get to go down, Miss Moses and wait on the judgment day. Won't you stay and keep Anna Lee company? So, maybe there will be a content post soon. Looking forward to weddings and Illinois.
Sunday, May 8, 2011
Planet Waves
It looks like I will have to follow through on my threats of pointlessly reviewing semi-obscure Dylan songs because the comments have been Slow Train Coming (it sounds like slow in coming; expect more terrible puns if I don't get satisfactory comments).
So, I've decided to write a brief review of Dylan's 1974 album, Planet Waves. It will be brief partially because there isn't that much to say about it. It's nice sounding because he's got The Band backing him up again and their version of "roots rock" is always pleasant and interesting, with them switching instruments and having multiple moving parts all at once, but lyrically there isn't much there. Most of the songs feel like they were written on a lark and they don't really address anything deep or have the layered meanings and references of John Wesley Harding. Anyway, here is my track by track review:
"On a Night Like This" - A strong opener because the band knows how to use accordion and have fun with a lighthearted song.
"Going Going Gone" - Sort of nondescript. The best line is probably "all that's gold isn't meant to shine," which is hardly up to his usual standards.
"Tough Mama" - Another upbeat number that The Band gets to have fun with. There's a little bit more imagery here. In a way it reminds me of his earlier song "Love Minus Zero/No Limit" in that it's a step up from most love songs which are bland and not descriptive. You would think if you were so in love with somebody to write a song, you would have plenty of reasons to enumerate, but usually all you get is blah about nice hair or eyes or something. Dylan steps it up here by painting you a picture of his "Tough Mama," though it doesn't come off as genuine as the earlier work. He gets bonus points for this line, though : "Today on the countryside it was a-hotter than a crotch."
"Hazel" - He's played this one in concert, so I have a live version, I think from The Band's Last Waltz concert, but other than them having played it together, I can't think of why he chose this song over any of his others. It's just a typical love song to "Hazel," who never really comes off as a real person to me. There's some nice piano driving the song, though, so that's fun.
"Something There is About You" - More fun stuff going on in the background from the Band, and this time it's under something at least fairly interesting. I don't know what's with the strange structure of the title, but something there is about it that I like. Also sort of notable for a mention of Dylan's childhood in Minnesota, which he never really talks about, "rainy days on the Great Lakes, walking the hills of old Duluth." :)
"Forever Young" - I'm sure everyone has heard this song. Most people are fans of the first version that appears on this album (the next track is another arrangement of the same song), but I like the second one better. Lyrically, it's a decent track and maybe the album's strongest, but I can't see it actually working as a lullaby, which is supposedly the intention. If you're going to write an ineffective lullaby, you might as well have the Band go crazy behind you, I figure.
"Dirge" - "I hate myself for loving you" and whatnot. It's pretty good, and more Dylan imagery. "Doom Machine," "just a painted face on a trip down Suicide Road," etc.
"You Angel You" - a song with "dummy lyrics," in Bob's own words. It's catchy, at least.
"Never Say Goodbye" - this song and its bass line keep getting stuck in my head, so there's that.
"Wedding Song" - Closing the album on a down note seems kind of odd, but it's not as if he didn't do that with "Highway 61 Revisited," "Blonde on Blonde," and "Desire." On the other hand, those albums were far less bouncy, so it's kind of odd. This closing track is ok, but not nearly up to the closing tracks of those albums. Meh
Alright, well that's it. Maybe the next post will be a bunch of nonsense about model categories.
So, I've decided to write a brief review of Dylan's 1974 album, Planet Waves. It will be brief partially because there isn't that much to say about it. It's nice sounding because he's got The Band backing him up again and their version of "roots rock" is always pleasant and interesting, with them switching instruments and having multiple moving parts all at once, but lyrically there isn't much there. Most of the songs feel like they were written on a lark and they don't really address anything deep or have the layered meanings and references of John Wesley Harding. Anyway, here is my track by track review:
"On a Night Like This" - A strong opener because the band knows how to use accordion and have fun with a lighthearted song.
"Going Going Gone" - Sort of nondescript. The best line is probably "all that's gold isn't meant to shine," which is hardly up to his usual standards.
"Tough Mama" - Another upbeat number that The Band gets to have fun with. There's a little bit more imagery here. In a way it reminds me of his earlier song "Love Minus Zero/No Limit" in that it's a step up from most love songs which are bland and not descriptive. You would think if you were so in love with somebody to write a song, you would have plenty of reasons to enumerate, but usually all you get is blah about nice hair or eyes or something. Dylan steps it up here by painting you a picture of his "Tough Mama," though it doesn't come off as genuine as the earlier work. He gets bonus points for this line, though : "Today on the countryside it was a-hotter than a crotch."
"Hazel" - He's played this one in concert, so I have a live version, I think from The Band's Last Waltz concert, but other than them having played it together, I can't think of why he chose this song over any of his others. It's just a typical love song to "Hazel," who never really comes off as a real person to me. There's some nice piano driving the song, though, so that's fun.
"Something There is About You" - More fun stuff going on in the background from the Band, and this time it's under something at least fairly interesting. I don't know what's with the strange structure of the title, but something there is about it that I like. Also sort of notable for a mention of Dylan's childhood in Minnesota, which he never really talks about, "rainy days on the Great Lakes, walking the hills of old Duluth." :)
"Forever Young" - I'm sure everyone has heard this song. Most people are fans of the first version that appears on this album (the next track is another arrangement of the same song), but I like the second one better. Lyrically, it's a decent track and maybe the album's strongest, but I can't see it actually working as a lullaby, which is supposedly the intention. If you're going to write an ineffective lullaby, you might as well have the Band go crazy behind you, I figure.
"Dirge" - "I hate myself for loving you" and whatnot. It's pretty good, and more Dylan imagery. "Doom Machine," "just a painted face on a trip down Suicide Road," etc.
"You Angel You" - a song with "dummy lyrics," in Bob's own words. It's catchy, at least.
"Never Say Goodbye" - this song and its bass line keep getting stuck in my head, so there's that.
"Wedding Song" - Closing the album on a down note seems kind of odd, but it's not as if he didn't do that with "Highway 61 Revisited," "Blonde on Blonde," and "Desire." On the other hand, those albums were far less bouncy, so it's kind of odd. This closing track is ok, but not nearly up to the closing tracks of those albums. Meh
Alright, well that's it. Maybe the next post will be a bunch of nonsense about model categories.
Wednesday, May 4, 2011
Can You Guess Who?
Can you guess who I saw tonight? The answer is Salman Rushdie. This wasn't like that episode where Kramer thought he saw Rushdie but it was just some dude named Salbas. He came here to talk about writing and novels and crap. He was kind of hilarious and almost made me want to read one of his books. Too bad I'm busy doing a bunch of math. But it's alright.
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