So how much more complicated can this game get? What motivation do we have to study other homogeneous spaces of higher dimensions? How is combinatorics involved?
I think the first and second questions need to be answered together, as there is constant tension between the impulse to add additional complexity to mathematical structure and the need to explain to other people why they should care. Mathematicians have for the most part chosen a sort of middle path which is to build the tools that would enable the study of arbitrarily complicated spaces by classifying the components of which they can be built. Then, whenever motivation to study a particular case of these things comes along from physics, economics, computing, or other parts of mathematics (most often), the motivated researcher has some footing to make progress by putting together the pieces.
You could think of it sort of like being a plumber going to the hardware store looking for the right fittings for some pipes. Maybe you’re a physicist and you’re trying to study a system of particles that obey certain symmetries. So what do you do? You go to the group store of course, and see if they have the right group. Sometimes they don’t have the perfect one, but they have the right parts and documentation so that you can assemble the right one yourself without too much difficulty. Now that you have a sensible way to keep track of symmetries in your system, you can go back to worrying about predicting how the universe behaves.
It is on the one hand wonderfully remarkable that the business of classifying the simple pieces of which geometric spaces can be built is a tractable pursuit at all. On the other hand, it is a mathematicians job to make it tractable. If some definition of what constitutes a “simple piece” leads to an impossible classification, then it is not the right definition. While some people might have you believe that theorems like the classification of finite simple groups or semi-simple Lie algebras are miracles from gods, they have also come through people toying with and massaging definitions until the classification task began to seem sensible (though still possibly monumental).
I don’t mean to demean the examples above; I too am enamored of them, and most of my published research uses the classification of simple Lie algebras as sort of a starting point. But I think of integer factorization as the proto-problem for these deconstructive classification quests. The fact that natural numbers all break down into prime factors (they are “classified” by their prime divisors) really does seem like a miraculous piece of order in the universe, or at least an inescapable part of how human consciousness interprets it. That we don’t have a very efficient way to take a number and break it down into factors is astonishing given how fundamental this problem is.
There is a dilemma here that I’ve always sensed in mathematics but that no one really likes to speak about (if it registers in anyone else’s reality). Mathematics and academia at large are subject to economic and political realities. One of these realities is the need to have a career and to instantiate said career by publishing papers. This incentivizes the mathematician not to work on old, very difficult problems, but to invent new theories, ask questions possibly nobody else is asking, collect fruit from those fonts and write about them. This is a cynical view of what could also be romanticized as a tremendous freedom and creativity afforded to the profession by the fact that we’ve managed to convince the rest of the scientific and engineering fields that they need us to teach them calculus, statistics, and the like. (See: Mackey’s lecture on what it is to be a mathematician.) But there are also cases where solutions to old, difficult problems have eventually come through extended scenic detours.
Moreover, I don’t want to diminish the amount of work that has gone in to building up the vast amount of theory that exists today. It is not that it comes easily, but rather that it comes at all that appeals to the hungry mathematician. It can be a joyous experience to sit down with a problem, toy with it for hours, weeks or months, and then to finally resolve it. And it is absurd and incredible that human intellect is somehow at all suited to the process of assimilating information, experimenting, ordering it, and manipulating it in a way that reveals a “truer” structure. The pleasure that comes with this experience is, I guess, at least as strong a force behind the proliferation of mathematical theory. It is a good thing that we will never run out of problems.
But now I haven’t said anything about combinatorics.