“Good mathematicians see analogies. Great mathematicians see analogies between analogies.” – Stefan Banach
The more I learn about mathematics, the more it feels like everything is connected. It’s crazy to see people define a bunch of really abstract, high-powered objects, only to later show that there is indeed a fairly natural notion of “morphism” between those objects, and moreover, you actually get a category with unbelievably nice properties (for example it turns out to be abelian or something).
Not only that, but the way certain intuitively familiar (say, geometric) concepts can be recast so naturally in completely different (say, algebraic) terms is amazing. It’s even a bit creepy, like suddenly becoming aware of the existence of so many esoteric foreign languages, most of which nobody even currently speaks, but which all have their own advantages and disadvantages. This is one of the things that lends these “analogies” and tactics of “mathematical translation” their immense power. You may have to stutter your way through unnatural, abstruse constructions in one of these “languages”, only to find a ridiculously elegant formulation of that same concept in another such language. The answer, clearly, is to learn as many languages as you can. Taking this sentence literally is also something I’ve wanted to do for as long as I can remember, but math is keeping me busy at the moment…
I was reading about the Banach-Stone theorem, which is concerned with recovering a compact Hausdorff space from the algebra of continuous functions on the space (!!!). That’s pretty remarkable if you ask me. Also in this vein is the way you can conflate points of affine -space over an algebraically closed field with maximal ideals of the corresponding polynomial ring . That is, the point can be replaced by the maximal ideal .
This leads to the definition of the spectrum, , of a ring as a locally ringed space (you take , equip it with the Zariski topology, and then define a sheaf of rings on it by using localisation at the prime ideals, more or less). After doing this, you’re pretty much inches away from defining affine schemes, and then general schemes. Grothendieck’s introduction of these objects elevated algebraic geometry to where it is today.
Even if you disregard that rippling shockwave of power, check this out: it turns out people have used very similar kinds of ideas to translate topological properties of a space into purely algebraic properties of its function algebra, for example (roughly speaking) compactness of a space corresponds to its algebra being unital. Now, the C*-algebra you obtain from a space in this sense is always commutative (of course), but with this “topology-to-C*-algebra” dictionary in hand, people decided to replace it with a noncommutative C*-algebra, which then lead to a noncommutative version of topology!
For another example, I was reading (in the Princeton Companion, for whoever’s wondering) earlier today about different ways of writing polynomials: if I write a polynomial down, say, , then this representation is clearly “centered at 0” in a certain sense. Namely, it makes it clear that if I plug in I will get . However you could also write a polynomial centered at another point; one might say is “centered at “. Because both of these representations are “biased” toward a certain point, we think of them as “local” in nature. However you could also write down a polynomial as , or something — notice how this tells you about all the roots and hence is much less biased, so one might think of this representation as “global”. Anyways, the idea is that a guy named Hasse thought long and hard about this, and tried to make an analogy between numbers and polynomials, and that’s how -adic expansions were invented: they give you information about how the number behaves locally, “at the prime “. It’s called “Hasse’s local-global principle”. The initial response to Hasse’s work was widespread skepticism. People thought it was useless until eventually it was realized one can sometimes “piece together local information” to prove a global theorem. Then they changed their minds.
All in all, I just find myself becoming even more addicted to mathematics. I would have withdrawals if I ever stopped studying this stuff. I’m definitely going to do every problem in Hartshorne, no matter how long it takes. It’s a pretty formidable task though, since I have to supplement it with multiple other books (mainly for the commutative algebra).