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.