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!