What is the most complete area of mathematics?

This is what I mean by complete:

> You have to advance a lot into the topic to find an unanswered conjeture?
> Give any kind of explanation


What I don't want:
> Stupid brainlet
> [Insert area] because [name] said it

My guess:
Functional analysis.

Euclidean geometry

arithmetic

linear algebra

>area
You just got conned.

Math is a subproduct of your mind.

You keep looking for patterns in nature and applications. That's all.

No, that's not what math is.
Math is not physics, it is entirely based on axioms and rules.

Trigonometry

Pretty much all 1900s mathematics is fairly complete but none have fully solved problems they were initially invented to.

Like PDEs

it hasn't always been like that and there is still today debate around that
there are still people whining about the axiom of choice etcetera

I would say this is right. But for the sake of discussion: Riemannian geometry. It's still extremely active but seemingly only as it applies to other areas (topology, algebraic geometry, complex geometry) it seems that after the flurry of work done after Perlman's proof, RG as a field in its own right is dying out.

same for harmonic analysis aka fourier analysis and generalized shift invariant systems. instead of being studied in their own right new papers focus on applications to things like compressed sensing. but there are still open questions about locally compact abelian groups

point set topology

How is arithmetic (number theory) complete?

Is there any good book on it for someone who did the calculus sequence plus real analysis?

Syracuse sequence disagree.
So easy to define that a 7 year-old can run it, and yet impossible to prove.

general topology

Any homological faggotry

If you need a book on it, don't bother. You're not at that level yet.

That's not very helpful. What do you mean? People has to learn it from somewhere

>advance a lot
wat

Sure, they keep complaining but they have zero effect on the mathematical community.
The fact is, most mathematicians accept current axioms because they are extremely exact and they work extremely well.

In order to learn topology or any other field you need to work on it by yourself.
That means, you must be able to do proofs.
This is extremely difficult (and why I am not a math major).
Mathe is not something you can make a pop version of it. It is complex (pun not intended), rigorous, and logical.
But... if you still want to learn about it, this is the sequence:
> Real analysis (make sure you are good at it)
> Complex analysis (it is great for learning new mathematic techniques)
> Number theory (maybe)
> Topology

I like peter peterson's book (I know the name sounds made up). It might be a bit out there at times, but it's pretty easy to follow I think.

This nigger is dumb. If you really just want to know one little subfield, find a good book and make up the material you don't know as you go. Also, you absolutely do NOT need complex or number theory to understand RG. Topology is a plus but not actually all that important other than for making some deep connections with some of the bigger theorems (typically not intro stuff anyway) and not being scared of some of the jargon.

good one! Shame though, harmonic analysis is very pretty from what I've seen of it.

You may learn one field by your method, but you will have zero understanding.
You will have no idea of its applications and, if you actually want to prove something by yourself, it's over.

it's more because the open problems are too hard

math is eternal, it is logic described, and logic is limitless, only limited by variables.

don't listen to this fag look for your uni's standard textbook on the subject, the prereqs are usually listed in the preface of each book so you'll be able to see if you can deal with it

Math is something you need a lot of intuititon and experience for, you dont grow those overnight. Do you really expect someone to go from real analysis straight to Riemannian manifolds? You must not be in math yourselves.