Mammon, You Been on My Mind

Here you go, an unfinished product, but a product nonetheless. Forgive the eye rhymes and whatnot; it's just a goofy little song that resulted from posts/comments. Also, I'm not sure if "My" should be capitalized or not:

Never met a man named

Nebuchadnezzar

I guess he was

One of a kind

I ain’t sayin’

It can’t happen never

That Mammon, you

Been on my mind

Maybe Judas’s little sister

Or a long lost cousin

Needed a new heart

And a little more time

Thirty pieces of silver

For a simple operation

Or maybe, Mammon, you

Been on my mind

I can’t be the only one

To think the golden calf,

And I ain’t sayin

It’s a sign

Looks like a chargin’ bull

By little more than half

But Mammon, you

Been on my mind

The joke is really just a reference to a Bob Dylan song, "Mama You Been On My Mind." I figured it was obscure enough that nobody would get it. I was busy writing better songs, so this one is kind of a throwaway. It's good to write throwaway stuff sometimes, I figure, especially if it gives me practice in different keys. This one's written in F, which is not so difficult to play in, but sort of hard to sing in. I recently wrote a (better, in my opinion) song in D (!!!), which means two sharps for anyone who doesn't know. More sharps is generally more difficult, but D is a sort of particularly strange key for me because I don't have a D harmonica, and playing the blues in D, besides maybe being kind of weird historically, would require a more functional G harmonica than the one I have, which is held together only partially by a liquefied rubber band.

In terms of lyrics, choosing a structure where the last line is always the same or just a slight variant on the same line is a convenient way of churning out a song but makes it difficult to fit in what you want to fit in. The third verse demonstrates that, I think, since there's no natural way to phrase it that doesn't require the other lines to end in something like "calf" or "bull" and almost nothing rhymes with the former, while the latter seems only to rhyme with three+ syllable words, like "syllable," which make for awkward lines. Anyway, I challenge you to do better, since I think it's pretty clear how to structure a verse.

Excellent

Olbermann owns Bloomberg. Hilarious, Keith. Too bad it's on a channel nobody has.

Songwriting Potential

I can't have been the first one to notice how reminiscent of this this guy is. Feel free to build your songwriting career as the poet of OWS on this observation.

Well

Don't that just beat all. Literally.

Wow

The best headline I've seen about it: "You can't spell 'Cardiac' without 'Card'"

Motte

What the what? On a night when Carp can't locate pitches, this guy steps in and does that? Not just him, yes, but that's just electric pitching.

Sports Writing

There's a blog I read regularly which has a few posters and covers politics, sports, literature, all sorts of stuff. Today there is a lengthy piece detailing the Red Sox's late season collapse, and I thought people who like quality sports writing would enjoy it. That's all.

Lekach

Since nobody seems to like my lengthy, math-heavy posts, I'll make a short, non-math post. It was recently Rosh Hashanah, so to celebrate, I made this Lekach, a Jewish honey cake. I found the recipe online. It's got raisins and walnuts in it and is pretty good. Think more like zucchini bread than actual cake. My non-practicing Jewish housemate made matzoh ball soup, which for him is really more like chicken noodle soup with matzoh balls added because that's way better than just chicken broth with doughballs. Regardless, it was pretty good. L'chaim!

Continuing the Discussion

There's an old saw in mathematics that a mathematician is a machine for turning coffee into theorems. There's a newer saw that then says a comathematician is a machine for turning cotheorems into ffee, but that's a bit of a category theory joke. So, I've posted a picture of my new coffee cup for reference into how much of a functor I am lately. Anyway, I thought Pops raised some interesting points in the comments to my second to last (is "penultimate" appropriate here?) post, and for lengthy discussion, the blog post format seems more apt. I had been using this cup to hold change, by the way, but I decided it would be more convenient just to drink two of these instead of five normal cups, and it amuses me.

First, abelian groups. Or maybe Abelian groups. I'm not sure if that should be capitalized, as it's a common noun if we compare it to similar terms, such as simple group or finite group, but it's named after Niels Henrik Abel, who was quite proper. Regardless, I'll catch you up on what that means. A group is a set with a well-behaved "multiplication," that is a set G, and a function m: G x G -> G, that is associative, and the set is equipped with an identity and inverses. Generally, we drop the m notation and just write it like multiplication, so that all we require is that for all a,b,c in G, we have

a(bc) = (ab)c

an element e in G such that

ae = ea = a

and an element a^(-1) such that

aa^(1) = a^(-1)a =e.

You'll recognize this as being pretty normal, since all the sets of numbers you've ever worked with should meet these conditions for addition, and if you remove 0, they'll work for multiplication, too. Square matrices of a given size provide another convenient example for both operations, as long as you restrict to invertible matrices in the multiplicative case. If you don't know what that means, it just means matrices that meet that last condition. Certain sets of functions also provide common examples, but I won't go into too much detail.

The matrix example is nice because if you've ever done multiplication with them, you'll notice that AB = BA is generally not true, so we have an example of a non-abelian group. Abelian groups are groups where the multiplication does commute, so they are especially nice. Real numbers, rationals, integers, and anything you can come up with built around those will probably be abelian.

A group is called finite if, unsurprisingly, the underlying set G is finite. A nice example would be integers modulo an integer, where the "multiplication" is given by addition. To make that concrete, take all the integers (positive and negative) and divide by a fixed integer, say 4, but stop at the remainder step. You'll notice that you can only get 0,1,2, or 3. Now, choose two integers and add them, and then take their remainder when divided by 4. You'll notice that you get the same result whether you took the remainders first and then added or if you added first and then took the remainder, as long as you interpret adding remainders 2 + 3 = 5 to mean the remainder of 5, which is 1, etc. So it makes sense to add remainders, and you get a nice operation that maybe you never thought of before. You get weird looking relations like 2 + 2 = 0 and 3 + 3 = 2, but that's group theory for you.

The point I was making about finite abelian groups is that even though there are an infinite number of them (in fact the class of finite abelian groups is a proper class, not a set, unless you mod out by isomorphism), they are easily classified. Every finite abelian group it turns out is just a direct sum of this type of group, where the remainder is taken after division by a prime power. A direct sum just means an n-tuple in this case. It seems to me that chemical reactions should behave in a similar way, that more complicated molecules could be thought of as sums of simpler molecules, so that we should always be able to do arithmetic in higher molecule spaces using arithmetic in lower molecule spaces, if that makes any sense. Maybe not, though.

Now, on to the story that amuses me. As I mentioned in the comments, a lot of my students seem incapable of thinking, or at least content to go on avoiding it for as long as possible, so they just shotgun a bunch of terminology every time they come to a question they don't immediately understand. I like to use this to amuse myself and see if I can get them to write unnecessary things that they otherwise would never include in an answer. So, for example, one time we had a lab that focused on a non-parametric test, a permutation test. This type of test basically works on data that is in two groups, and we want to see if permuting which group each datum is in affects anything. If not, then we can attribute any differences we see in the groups to the randomness of the sampling. So, it requires a computer to generate random permutations of data, which is something computers are notoriously bad at doing. In fact, they are incapable of doing anything random by the deterministic nature of their operation, so programmers get around this by making the computer take strange inputs like the number of milliseconds we're at, and running them through bizarre functions that should produce seemingly random outputs. Of course, it's not random, so numbers generated in this fashion are called pseudo-random. None of this knowledge is necessary to explain how the test works because you could do it manually by generating random permutations yourself, but unless you have a few years to kill, I wouldn't suggest it. So, I decided to talk all about pseudo-random numbers, and lo and behold, students' lab reports contained all sorts of usage of the terms "pseudo-random" and "deterministic," which made me chuckle. Keep up the good work, bio department!

Mirepoix

Happy labor day to all. I was sitting around doing very little, wondering what to make tonight, when I noticed I had the requisite onions, carrots, and celery to make mirepoix, though I couldn't remember the name and had to do some Wikipedia-ing to find it. Regardless, Google didn't fail me in finding an easy recipe which used it. So, I present the fruits, or rather, meat and vegetables, of my labor:




How is it? Pretty good, but not mind blowing, as I expected. I like it, and I have to hand it to the French for coming up with mirepoix, which seems like it would be a good base for many a delicious dish, but I also understand why we order out for Chinese food instead of French. That is, when you order food, you are hungry now and don't want to wait for food to simmer for two hours. Also, bay leaves? A grand conspiracy if I have ever seen one. At best some iota of flavor only after waiting hours. I confess I didn't have the thyme (haha) to do this recipe right, but I imagine that that didn't really change the outcome. On the whole, though, I was pleased with the outcome and would vote a solid would make again if I have the time. I accompanied the meal with Shock Top instead of the recommended sauvignon blanc, but that's thanks to the archaic alcohol laws here in Pennsylvania, which make it possible to buy wine only at liquor stores, of which there aren't any within walking distance. Also, despite Andy's best efforts, all red wine just tastes like weird grape juice to me, and all white wine just tastes like, well, slightly different juice. Shock Top is a kind of meh (we said "meh," M-E-H) Belgian-style witbier (wheat beer) with some orange in it. It's alright, but I can only drink about two before it starts to act strangely in my stomach. On the plus side, it's brewed in good ol' St. Louis. Alright, that's it

Generalized Abstract Nonsense

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!

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.

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.

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.

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.

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.

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.

Comment!!!

Comment away or I will start reviewing obscure Dylan songs!!!

Ain't No Good

Every other Wednesday there is a seminar that lasts about an hour at most by a grad student in our department. The only people that come are other grad students because faculty have better things to do than listen to us talk about research or often just definitions of stuff that they probably know better/don't work with, and who else would come to a talk about math?

Anyway, I haven't given a talk and don't really plan on it, at least not any time soon, despite the girl who is in charge of it asking me multiple times to present just to avoid some of the less...enjoyable...people from talking more than once. I've only recently actually considered doing it at all since I've produced exactly sin(0) work until now, and the idea of presenting old problems or just introducing people to some area they aren't familiar with seems somehow more pointless than category theory (get it, because we generally want to move away from thinking of objects like sets, etc. by what "points" they contain and move towards thinking about the maps between such objects...?). But this quarter I am taking an independent study with a professor, pending official approval from the department head and whatnot on a form I had to fill out (for the question about how my grade would be evaluated, he said to write "by professor." haha).

Forgive my abuse of punctuation.

Anyway, it isn't research, at least not yet, but it's sort of close in a way, and may get there if I can master the small object argument (don't ask), so it seems more worthy of presenting. The main allure of it, though, is that if I talk about what I am doing to any other student, literally no one knows what I am talking about. Here's an example:

If a functor from a model category to a category with a class of weak equivalences that is closed under the 2-of-3 axiom takes trivial cofibrations between cofibrant objects to weak equivalences, then it preserves all weak equivalences between cofibrant objects.

That's called Ken Brown's Lemma, apparently. The wonderful thing about it is that it simultaneously says so much and so little. So, I may tell her that I'll do it, but I have to consider it more and make a bit more progress first. Also, I'm always wary of her invitations; she'll pull out your feathers for her brand new hat, and when she's done that, she'll feed you to her cat.

I've Been Listening To This Song Obsessively Lately

In keeping with my obsession with Dylan and his seeming obsession with adverbs, I thought I'd title this post something like the above. I keep listening to "Buckets of Rain" over and over again. It's the closing track to "Blood on the Tracks," for those of you less versed in Dylan's repertoire, and one of Eric's favorites, if I recall correctly. There's something quiet and brilliant about it, like the sermon of a broken-hearted Minnesotan Buddha. Here's a gem:

I been meek
And hard like an oak
I seen pretty people disappear like smoke
Friends will arrive, friends will disappear
If you want me, honey baby
I’ll be here

Cool.

New Quarter

I've got a shiny new quarter! We are on the quarter system, so this is kind of a pun.

So, what is on tap for the next few months? Class #1 is Complex Analysis, which is not just normal analysis but harder, but specifically analysis (think calculus) in the field of complex numbers. I don't know why but it seems like complex numbers are a thing that everybody who's had at least Algebra II should know about, but nobody seems to know by name. They're just numbers of the form a + bi, where a and b are real, and i is the familiar square root of -1. There are bunches of ways to think about them, but usually people think about the real numbers as being a line and the complex numbers being a plane where the real numbers are one axis and the imaginary numbers are the other axis. Other times people think of the complex numbers, with a "point at infinity" attached, as a sphere. Then the real numbers would form a great circle on the sphere, as would the imaginary numbers, or any other line, for that matter. The most important thing, though, is that this is the last of my "required" courses.

Class #2 is part two of Applied Functional Analysis, which I am taking just to fill out my credits and because the first quarter was almost laughably easy, so I'm assuming we'll continue on in that fashion for the next ten weeks.

Class #3, which I haven't gotten officially recognized as a class yet, but only because I haven't talked to the secretary over break, is a reading class with the professor who taught my Topology I & II courses over fall and winter terms. He asked me if I would want to finish (the relevant parts of) the book we had been using and then read some stuff about model categories and other nonsense, probably because I was the only one who liked all this abstract nonsense and diagram chasing and wasn't busy doing research about numbers in boxes or some such. So, I'm keeping busy reading and re-reading stuff until I can't think any more.

As far as teaching goes, I was expecting to be doing a recitation section or two for what I call "math for dummies," but the first years don't have any teaching experience yet, so they got all those, which means I am relegated to the tutoring center for nine hours/week. Until now, the most I've had in there is six hours/week, so I am trying to get my hours scheduled to days that aren't before large exams. Those are the worst days because all these people who have never gone to class and can't do basic algebra come in and ask you to basically teach them everything in a few hours, and it is frowned on to just tell them that that is stupid and they are stupid and should just drop their class because they will assuredly fail, even if it is true.

I should specify: there are a few types of students who come for help.

1. Students who come in periodically for help with specific problems/concepts that they don't understand - Most of these students are good and nice to work with because you can't do problems for them and then let them do similar problems, or just guide them through the problems step by step, or sometimes just casually watch over them as they work. Usually these students come from one of the many calculus classes or linear algebra, which is understandable, since linear is the first time that most people have to think abstractly at all, and sometimes calculus problems just involve seeing some clever transformation that isn't obvious the first time, but also sometimes they come from lower classes, trying to figure out when to apply what formula.

2. Hopeless students - These students are usually continuing education students or sometimes low-level people trying (often for the 2nd or 3rd time) to get the math requirements for their major done. These students are a huge pain to work with because while they try, they will never succeed, and you can't just dismiss them, because at least they are trying and aren't putting it off till the last minute. I don't know whether it speaks to the absolutely horrendous state of math education K-12 in the U.S., or if there is just a segment of the population that is completely incapable of reasoning or the use of symbols to further that reasoning. One of the biggest hurdles is people that, as I mentioned earlier, can't do even basic algebra. I really think that if you graduated high school without the ability to do everything in a typical Algebra I (I am talking h.s. or junior high level) class without any real thought or effort, you are basically on the same level as someone who is still sounding words out when they read and your school completely failed you.

The reason I can't figure out whether it is the fault of the educational system or of just some people is that people seem to lack a fundamental understanding of what they're doing, rather than the specifics of how to do it. For example, you are probably familiar with the quadratic formula, and most people that come in have at least that formula drilled into their heads. But if I give you the equation of a parabola and tell you to find x-intercepts, would you know to use that formula? People don't make the connection between an equation and a graph, that the graph is a set of points with coordinates (x,y) that satisfy a given equation. They don't seem to even understand what that means, i.e., that an equation is a statement which is either true or not. They can't distinguish between an equation and an expression, and instead just memorize how to solve equations and are baffled by what an equation, or any mathematical expression, for that matter, is saying. I don't know whether this is something that most people can be taught or if it is just something you have to work out for yourself.

3. Awful students - These are the previously mentioned students who don't care enough to do anything but care about getting passing grades. Another thing they love to do is try to make you do their take home quizzes/homework for them. Some of them realize that we won't just do their problems for them, but will do similar problems, so they "cleverly" copy their problems to another sheet of paper and then ask us to do those problems. I don't know why people think this will work. It is insultingly dumb and really annoying.

So that does it for my rant, and since this is already a basically impenetrable wall of text, I'll just end the update here.

Not Much Happening

As the title would leave you to believe, there's not much happening. The term is almost over, which is nice because I won't have to deal with all the work that teaching (teaching assisting?) Data Analysis entails. It's not a horrible class or anything, but the grading takes forever since it's all explanations, not math. I often have to read through paragraph+ length answers despite the fact that all the questions can be answered properly in AT MOST two sentences. The problem is that the class is for bio major seniors who have spent the last four years learning to regurgitate terminology mindlessly, and who have taken to heart the lesson that the less you know, the more you should write. So, there are all these practically novella length completely wrong answers, which I have to read through. The kids that are good at it have started to learn how to be concise and right answers are generally easy to grade, anyway, because I can just comb through them for the right combination of words, but wrong answers take forever because I have to try to figure out what they are thinking (which is usually nothing) in order to give partial credit, and then I have to write something about how they are wrong or what they should have said.

They just had an exam this last week, which means the grading was even worse because I have to be more careful and the length of their answers has been stepped up War and Peace levels. One girl, who is almost surely the worst student of all time, actually wrote a whole page of nonsense that didn't even answer the question, and then a request for at least partial credit for addressing something tangential to what was asked. I wrote "this does not approach an answer" and gave her 0 points for that. They lose points with me for wasting my time with stuff like that. Other things I have written on exams include "don't write me a novel," "answer the question," "do this [with an arrow pointing to the directions]," and "baffling." That last one actually made some other TAs laugh. They are generally sympathetic, but not sympathetic enough to help grade.

Most of the classes have many sections, taught by a few instructors/TAs, and there are usually one or two TAs who just grade, which means when the low level classes have an exam, there's an all day grading session in the common area/break room/whatever it is called where the mailboxes and coffee are. I have been part of a couple for a class last spring, and it's almost nice because it's social and there is pizza. Of course, the social aspect can be not so great if you don't like the people you are working with, but sometimes I get jealous, since when I am grading exams, it is just me and the one professor who teaches the class in our separate offices for nine hours at a time and there is no pizza :(

On the learning side of things, there's not much happening, either. I have three courses this term, as I am trying to finish up all my course requirements this year, which means three courses per term. They're not bad, though.

One of them is applied functional analysis, which is probably the easiest course I have had here. It's like baby analysis, definitely a step down in difficulty from the analysis course I had here last year, which was intended to be preparation for the qualifier, so it was fairly rigorous. This course is for people who have already had analysis, but the book we use is for someone who never has, so a lot of the exercises are nothing new if you have a solid grasp of things like vector spaces and metric spaces. Since this post is already rambling and fairly low-content, I guess I will explain.

A vector space is just a set of vectors that meets certain basic algebraic conditions. This may sound daunting, but it isn't. It just means that you can add and subtract vectors and stay in the space, as well as multiply them by scalars (which basically just means real numbers or complex (!) numbers). The addition and multiplication have to behave nicely, too. They have to be associative and commutative and whatnot.

A metric space is just a set that has a distance function on it. I think I've talked about them before, and it's not super hard to figure out what is meant by a distance function, anyway. It just needs to tell you the distance between two points in a way that a distance does. For example, d(x,y) = |x - y| is a metric for any Euclidean space [n-tuples of numbers].

In this class, we are specifically interested in normed vector spaces, which is basically a vector space that is also a metric space, and the metric behaves like an absolute value, and even more specifically in inner product spaces, which are normed spaces where the norm (absolute value) comes from something that acts like a dot product, if that means anything to you. So, it's not highly interesting from my point of view, but it's alright.

My second course is abstract algebra, which is concerned with abstract algebraic (duh!) spaces, such as rings and groups and whatnot. I had a class in it in undergrad, and I haven't picked up a whole lot here, but it's nice to get the review and to learn Sylow's theorems, which are super useful.

My third class is my most interesting, so I arbitrarily numbered it third. It's algebraic topology, which I can only define in an abstruse way. Topology is generally the study of topological invariants, and algebraic topology is just a branch where these invariants are algebraic structures like groups. There's more to it than that, but I can't really explain everything. It's the field of study that primarily lead to category theory, which is probably the most abstract form of math. It's also very difficult to understand in my experience. A category in the mathematical sense is just a collection (not even just sets!!!) of objects with morphisms, which are basically just arrows between the objects that you can compose if the head of one points to the tail of the other. This makes everything basically into a category, and they can usually be thought of in tons of different ways if you just change what you want to be the morphisms, and ultimately you sort of don't care about the objects at all. It's very bizarre. Here's an example:

A group is an algebraic concept. It's just a set where there's a "multiplication," which is associative [ (ab)c = a(bc) ], and there's an identity element ,e, which acts like a 1. That is, ea = ae = a, and every element has an inverse, which basically means you can divide. Anyway, we can think of a group as a category with only one object, which is the group itself, and all the morphisms are multiplication by one of the elements of the group, so that all the arrows in the category point from the only object back to itself, so it's a very simple category.

For a totally different example, you can think of the collection of all sets being the objects of a category, and the morphisms just being functions from one set to the other. Since if f:X -> Y and g:Y -> Z, then we can compose the functions like g(f(x)), this makes sense as a category. Technically, we also need identity morphisms, but for each set X, the function i(x) = x suits works fine. So, this is a gigantic category. In fact, it is so big that the objects don't form a set at all, but rather a proper class. IT'S CRAZY.

Risk-y Business

In case anyone was wondering, I tried to let that dude down easy, and he hasn't replied, so I assume he's feeling a little (-_-;)

Anyway, my students have an exam on Tuesday, and most of them had their last lab on Wednesday, so they generally wouldn't be getting the graded labs back until after the exam, but since some kids asked me for them, and since I am such an excellent TA, I came in today (Sunday) just to grade them. It's not particularly long or anything, but I will make sure they know how I slaved and slaved for their benefit, of course.

Grading took at most an hour, since I already had one of the two sections graded and since I am very lenient and tend to grade people that I know work together just once and replicate the grade across the group. So I had a bunch of time left over, and the math grad organization recently came into possession of a bunch of board games for the purposes of a board game night, which I didn't go to in order to play board games in Virginia, but I have the combination to the locker in which they are stored, so I decided to simulate some world conquest in my office. As secretary (maybe???) of the organization, I have a lot of pull as to what we do with our money (note: I took the job because there is no work and I am not even required to go to meetings), so I insisted we get Risk. There was plenty of money, so this was no issue. The president (my officemate) picked out the original version of Risk, as per the advice of the department board game aficionado. So here is what is different:

Not much.

There aren't III or V pieces, just oblong pieces that represent X's. The I's are in this version just wooden cubes. Interestingly, there's no brown, but there is pink. I am guessing that they realized that guys do not like to play with pink things in general. I assume that most people who enjoy Risk are men, as well as nerds. I also couldn't find in the instructions, which seem to be pared down for the re-release, how many armies each side starts with, but fortunately I remembered. If there are six players, each gets 20 armies, and for each fewer player, each side gets 5 more, by the way. If you only have two players, there's some sort of dummy third player, but I'm not clear on how that works because who plays Risk with two players?

I think a big part of the fun is the unofficial diplomacy that occurs when you actually have 3 or more (ideally at least 4) people playing that know how to play and don't get all butthurt when you inevitably break your alliance by smashing through their continent's defense. Some more unorganized thoughts about the game:

It's interesting how taking Australia first almost always ends not just in disaster, but in a particular kind of disaster. Almost every time I play, there ends up being a color that is almost forced to try to take Asia because there simply aren't enough continents for everyone to take a safer one at the beginning, and somebody has to get stuck with a bad position, including a couple territories in northern Asia. That guy basically has to try to take over all the undefended territories in a way that lets him maximize the armies along the border, which means for slow playing. Trying to take Asia in one fell swoop at the beginning is essentially suicide, so you get this weird thing where there's one guy with Australia whose goal is basically to get out of Asia, but has no choice but to go through Asia, and one guy who is stuck with most of Asia but no army bonus because he doesn't have Siam/India/Middle East, which is the Australian's path to expansion. But, because he's always taking these little undefended countries, the Asian guy gets to collect a lot of cards, which means he'll eventually get a set, and it tends to be at a time after the Australian guy has his continent but doesn't want to expand north because it would be trying to take Asia from the guy who has been trying to take it all game. What usually happens is that the guy with the set smashes through the border because the set will by that point outweigh the continent bonus, and likely will just take over Australia in a single turn. Maybe it is only due to using the usual rules for sets, which highly favors taking territories because the set values quickly outweigh everything else. I'm not sure what happens if you use the rules where sets only increase in value by 1 each time, but I do sometimes play that way, and it seems like a similar thing happens, in that I don't think I've ever played a game where the color that take Australia first wins.

Since I usually have to play by myself, I have to make sure to preserve features of the game that come from different people playing, and I try not to bias it towards any color or anything by making one guy do stupid things, but of course that isn't perfect. I always have a soft spot for the color that ends up being what I want to term the "roving empire," or maybe the huns of the game. Usually everyone's best strategy at the beginning of the game is to take a continent that that are set up to take, but as I mentioned, if you play with six players, there really aren't enough continents for this to work, so one player will inevitably end up losing this bid. This player never, in my experience, wins, but he's (pardon my androcentric pronouns, but generally all sides are me) also generally isn't really knocked out until later because eliminating another player requires a lot of armies, which nobody will have early in the game. So, he's just this guy who inhabits a non-claimed continent but can't take it from a more powerful player, meaning that his best chances lie in trying to survive long enough to get cards and hopefully establish himself again once the superpowers start fighting over the Bering Strait or something. In effect, he becomes this wandering army, trying to take weak territories, but not really holding onto them because the cards matter more than any individual territory if the continent is impossible to get. I think it's natural to root for this guy and much more fun to play than the guy who has to take the other strategy for a guy who lost his continent bid, which is to sit like a rock, collecting armies and hoping that somebody gets messed up. Probably just counting out armies is annoying.

Some places in the game inevitably get contested more than others, particularly ones that are on the boundary of two continents, and it's funny that for every one that mirrors some historically conflicted area, such as the whole Mediterranean sea, which reminds me of Rome vs. Carthage, or Greece and Turkey, or the middle east, or even 30 Years War-esque battles for Europe, there is an oddball, like Iceland vs. Greenland, two places which have probably never had a historic war, or North Africa vs. Brazil, which seems almost laughable if you look at an actual globe.

As much as I love Risk, the end game isn't very fun, and I usually just give up because I know how it's going to end and counting out 50+ armies for a set becomes tedious. I think the +1 rules are better for sets in general and probably should have started playing that way today but didn't, so I got stuck counting out too many armies, which is especially tedious without III's and V's.

Alright, well, that's probably enough for now.

Gaydar

So, I went with my friend who happens to be gay to a meeting for a grad students' gay rights thing because he didn't want to go alone and because there was a free happy hour after the meeting. The meeting was alright except there was a Chinese kid and an Indian kid, neither of whom understood the icebreaker of saying which muppet you would be. I chose one of the two old dudes, by the way. The happy hour was fun and saved me having to make my own dinner. Maybe I should have specified that I am not gay, though, because some dude sent me an email, which I got this morning, asking me out. Oh, dear. (~_~;)

Takoyaki party!

Since it's been almost a week since my last post, and since the trickle of comments has dried up on that one, I've decided to post some more pictures of my trip. One day, Mie's older sister, who has two kids, invited us over to make takoyaki. Before that, we passed the time playing sugoroku, which is a generic name for a type of boardgame that generally just involves rolling a die or spinning a spinner, etc. to decide how many pieces to move. So, like Candy Land, except without the candy theme. This particular game was all about ojisan gyaggu, old puns and jokes that older men tend to say. An ojisan is an uncle, but it's also the term for any older guy who isn't old enough to be an ojiisan, a grandpa. So, you would play like normal, and on certain squares, had to say one of these gags, which are very corny and dumb, so of course I love them. The little girl in the picture is Misato, Mie's cute-as-a-button niece, whom we often call Mi-chan. Her younger brother was with his dad, so it was just the three of us and Mie's sister (not in the picture), Kaori. I don't remember who won, actually.


Here's us making the takoyaki. Tako means octopus, and yaki comes from yaku, which means to fry or burn or bake or whatever. The character is 焼く, if you are interested. The character for octopus is 鮹　or 蛸, but neither is used very much. Anyway, as you can probably guess, takoyaki is cooked octopus. Specifically, it is little balls of dough with octopus (sometimes other things like shrimp, too) that are cooked in this special maker thing that looks like a hot plate. It has little half-sphere indentations, and you fill those up most of the way with batter, and then add the stuff that goes inside. You have to turn the balls to make them cook evenly and get a nice shape to them, which requires a bit of skill.

Misato is quite the little cook, so hers were probably the best. Apparently, I don't turn them fast enough, so mine get cooked more thoroughly, but I think it's better that way. We divided the hot-plate into territories for which we were each responsible for one, except Misato, who was in charge of her mother's territory, too. Despite that, she still had time to intervene in mine because from her and Mie's point of view I ran my territory rather like a third-world country. Regardless, they were delicious!

Misato and her mom gave us these matching hats for Christmas, which turned out to be pretty convenient because it was easier to find each other on the ski slopes with them on. But, that's a post for another time!

Belated Pictures of Sand

I got pictures of my winter trip to glorious Nippon a while back, but I haven't gotten around to posting them yet, obviously, since it's one of those big projects, at least as blog posts go. So, I'll try to get them up with some sort of commentary about what we were doing, but it won't end up in chronological or logical or morphological or any other kind of -logical order because I've no doubt forgotten details in the time since and organizing stuff isn't really my forte.

Here we are celebrating our ascent to the top of the Tottori Sand Dunes. When I heard about the sand dunes, I figured it was just going to be like a long beach, but it turns out that the dunes are quite sizable and climbing up them is rather like hiking up a hillside except that the hillside is made of sand and thus more easily deformable under your feet. Forgive me for the topological terminology but I've been computing fundamental groups lately.

Perhaps this picture will give you an idea of how large the dunes really are. We hiked up to the top of the biggest dune we could see, which seemed to be the thing to do. It was cold and clear that day, so we were wearing winter clothing, which seems sort of odd against the backdrop of sand and sea.
My outfit, incidentally, was almost 100% Japanese made. The pants, jacket, and neckwarmer were all gifts from Mie, who apparently has a different idea than most westerners of what looks good on me. The shoes I bought last year in Japan, and the hat was a gift from Mie's niece, who should show up in a later post.


We weren't alone on the trip. Mie's friend accompanied us there. She lives in Tottori (the city, which is on the opposite end of the prefecture from Yonago, where we stayed), so she acted sort of as a tour guide. Her name escapes me at the moment because most of Mie's friends have names that end with -ko, which is very typical for Japanese girl names, but makes remembering them an exercise in frustration.

After tromping around on the dunes for a while, we decided to take a ride on a camel. But there was no real reason to do that, and it was cheaper just to sit on the camel and get our picture taken, so we opted for that. Also, I think we were sort of cold and tired at the time, so it was more practical. When we were sitting on the camel, it started making some weird noises, so the dude there had to calm it down, or at least we thought so. Mie's friend told us after that it was really just pooping and he was probably trying to get us not to notice.

Anyway, after that we headed back and I bought some pear-flavored soft serve because I had never had it before. Pears are a specialty of Tottori prefecture, so we never had it in Shimane. It was really good, even better than green tea ice cream, which is what I would normally get if I were eating ice cream in Japan.


Finally, here's another picture of me. I ran down to the ocean while Mie and her friend stayed up at the top. I suspect they didn't want to climb back up the hillside since it was rather a lot of work. Can you tell which one is me?

I was originally planning on posting more here, but there are too many pictures and I don't feel like spending the next two hours crafting a blog post, so it will have to wait. Until then, it's time for more category theory!

Eric

I should have posted something earlier, but I didn't. I don't know what else there is to say about Eric, but here is a fitting song from an album we both liked. Maybe it's a little obvious, especially given the first lines of the song. It's weird because neither of us had this music during college and we both came upon it afterwords, so it's just another one of those things that we only got to share indirectly.

I was in Japan when Eric died, which was sort of worse. Japan was a thing we both got to experience for a while, but never together and I had hoped we would get to some day. He loved talking about what he did when he was studying in Tokyo and I think it was some of his best times. He loved going to Coco Ichibanya Curry, so I went there with Mie as a sort of tribute to him. I never really understood why he liked it so much, since it's kind of just a chain restaurant and curry is such an easy thing to make. So I had level 4 hotness curry because I think that's what he recommended. It was pretty good. Later I walked around looking for a vending machine that sold Boss coffee because he liked to talk about that, too. Well, I guess that is my tribute to Eric.

He was a friend of mine.

Sorry I couldn't find the Bob version.