This may be bordering on IMHO, but I’ll put it here in hopes that there is some kind of educated concensus by mathematicians as to what is the hardest subject in math (thats proven, and taught at the highest levels.) By this I don’t mean something broad like “Calculus.” Something more specific, like the name of the topic (as in, if it was in a text book, what the chapter(s) would be titled, and what it is (just a little explanation, I don’t expect a lesson.)
The hardest thing I’ve heard of is Schrodinger’s Wave Equation, which is so hard that it can only be solved for one case.
I think this would be the wrong place, I belive this is all very relative any given persons area of study. Some one who knows Schrodinger’s Wave Equation fairly well might not be able to pass the first actuarial test and vice versa.
The “Four Color Theorem” went a long time without proof, and Fermat’s last theorem is still baffling the best and brightest. But I don’t know what you’d call that branch of math. Number theory?..TRM
We covered Fermat’s last theorem in number theory in my school. BTW from what I remember even if we do prove it, it won’t really do anything, we just want to see if we can prove it. OTOH if we could find non-trivial zeros in the Riemann Zeta Function it would mean quite a bit.
Algebraic geometry is considered by many mathematicians to be the hardest mathematics. One reason is that it draws on so many other fields. What Wiles did was essentially algebraic geometry.
The four-color theorem states that any map drawn in the plane requires at most four colors to color the regions in such that no two neighboring regions have the same color. It was proved a few years ago, although the proof was controversial as it relied extensively on computer analysis, and there was some question as to the correctness of the program used. I don’t know if the controversy has been decided.
The zeta function evaluated at n is the sum over k from 1 to infinity of 1/(k[sup]n[/sup]), I believe. The value of zeta(3) is unknown, and there are some other conjectures about it.
FLT was proved a few years ago by Andrew Wiles. While it has no direct applications, it’s related to a conjecture in elliptic curve theory, which has extensive applications in cryptography.
Now for the OP…
There will be no general consensus among the mathematicians here, except possibly that the more abstract branches tend to be harder. Personally, I could never get my head around maps of maps (i.e., a map that takes one function to another function–that’s overly simple, but you get the idea). Other than that, perhaps the best measure is the number of open questions. Logic and set theory have a couple (just how big is the continuum, and what do we do about the Skolem paradox?). There are a lot in theoretical computer science, both in computability (are Turing machines really all that?) and in complexity (does P = NP?). That last one really is the million dollar question–there’s an actual million dollar prize attached to it. All you have to do is prove your answer and have it stand up for a couple years.
There are a couple of other modern theories that are highly abstract. Category theory is one–instead of studying a class of objects, we study the class of classes of objects and the structure-preserving maps between them. Topos theory is another one I’ve heard about, although I’d be lying if I said I know exactly what that is. Higher-order logic, and just about anything dealing with self-reference, are also up there in terms of abstraction.
Well, Group theory is pretty much despised by all mathemeticians who dont work on group theory as being “too abstract”. And this is from a bunch of mathematicians so thats saying something.