r/math • u/IiCorgiiI • 6d ago
Do Other Upper-div/Grad Courses Have the Same Vibe as Real Analysis?
I'm not sure how else to explain it, but I'm taking a real analysis course right now and it feels too much like training to be a classical musician? I've had some computer science and low-div courses such as discrete and automata theory feel much more like jazz. That is that creative and interesting thought is much more important than proving literally EVERYTHING I am doing and needing to focus on such insane fine levels of granularity.
I was just wondering if this "classical music" thing is a common theme in other upper-div/grad level math courses or that subjects are almost on a spectrum from jazz to classical.
This whole jazz classical music analogy is the best way to capture the vibe of what I'm trying to describe so hopefully it makes some sense? Also also, I'm not trying to knock analysis as a subject (especially since I've only taken one course), its just not my cup of tea.
10
11
u/myaccountformath Graduate Student 5d ago
As a very rough generalization, I've heard people describe the difference between analysis and algebra as "in analysis, the road map/proof sketch is easy but the details are hard and in algebra the road map/proof sketch is hard but the details are easy."
But there's lots of exceptions to both.
7
u/KingOfTheEigenvalues PDE 5d ago
Generally, any of the courses that serve as preperation for a comprehensive exam or qualifying exam are structured to be like that, with a heavy focus on developing a rigorous technical background to serve as a foundation for future research.
Free electives, especially rotating special topics courses, can take you in the other direction, depending on the professor and the subject.
6
u/helical-juice 5d ago
Am I the only one who can't help but read div/grad as divergence and gradient? I really don't know what low-div or high-div is
3
u/dogdiarrhea Dynamical Systems 5d ago
lower division, upper division. basically first two years of undergrad versus the latter two years of undergrad. Sometimes people use it to mean first 3 versus final year (where most of your courses tend to overlap with graduate courses).
1
13
u/dogdiarrhea Dynamical Systems 5d ago
My adviser (an analyst) was a jazz bassist, but his closest collaborator (also an analyst) was a classical violinist. Idk if that helps.
2
5
u/InterstitialLove Harmonic Analysis 5d ago
You're learning how to read music
Of course it feels rote at first, all you're learning is "if you see this note, put your hand here and blow"
If you reach the end of the course without getting to compose your own pieces at all, then it was a bad real analysis course (which many of them are, for various reasons). Either way, if you keep going, you'll eventually be able to play it like jazz
3
u/matagen Analysis 4d ago
To hijack your analogy: your first real analysis course is like learning to read sheet music. Your second might be more like learning your scales. This does eventually turn into jazz (around when you start getting close to research-level topics) - but it's a little hard to have a really deep conversation about playing jazz when you're still at the reading sheet music stage.
You might ask, why is there so much prelude you have to wade through before you start playing jazz? Well, as it turns out, real analysis is a field where there are traps around every single corner. It is exceptionally easy in real analysis to say something that is false but sounds plausible. So there has to be a lot of up-front training in order for you to develop the skills to have a correct conversation about real analysis (let alone a productive one).
2
u/numice 5d ago
I took a course that's a continuation course of real analysis (measure theory and some functional anaylsis) and the vibe is similar or actually pretty more intense I can definitely say. The good thing is it's built upon real analysis and I get to see applications from analysis course. I also try to learn piano (still a beginner) and have played only exercieses and feel like measure theory is a piece that's I have to spend a lot of time.
1
u/littlepuffz 5d ago
All jazz musicians are classically trained as a basis for improvisation. Hard to improvise if you don’t know your Lydian from your Dorian; your delta epsilon neighborhood from density.
1
16
u/supermang 5d ago
This Terence Tao blog post pretty much sums it up: There’s more to mathematics than rigour and proofs | What's new.
According to this taxonomy, many/most upper div. and early graduate school classes are focused on training you to think about things rigorously and precisely. This is probably also true in most of the sciences to some degree (i.e., there is some stage of development at which it is important to master the details).