Which areas of mathematics are considered the hardest?

[pivoxa15]

Broadly speaking, which major area of mathematics is considered the (or has a reputation to be the) hardest by the majority of mathematicians?

Note: I am aware that there are extremely hard unsolved problems in all areas of maths. I do not intend to be snobbish with this question but just an outsider wondering if there is an answer to this question.

 

[Office_Shredder]

whichever area you're studying. It makes you sound smarter in comparison.

Obviously, if a field of math was easy everyone would do it, and the knowledge base would expand to the point where it was hard. So the level of difficulty of a given field of math is always at a stable equilibrium point

 

[pivoxa15]

whichever area you're studying. It makes you sound smarter in comparison.

Obviously, if a field of math was easy everyone would do it, and the knowledge base would expand to the point where it was hard. So the level of difficulty of a given field of math is always at a stable equilibrium point

This case taken to the extreme would mean every field of study should be at an equilibrium. i.e. it is equally hard to do research in maths as in psychology. (I don't mean to belittle psychologists but I have a feeling resulting from a degree of personal experience that in general a mathematician can learn psychology much faster than a psychologist learn maths.) But that is not the case is it? So maybe you can think of my question as which area in maths has the most amount of 'bright' people working in. Hence that area will not have many easy questions left to answer, making it seem harder in comparison.

 

[Gza]

I think it's difficult to single out any branch of math, and assign an arbitrary level of difficulty to it. Just like in sports, you have a wide range, such as basketball, football, baseball, and skillsets: (leaping ability, height, speed); in math you have the same situation, where much different skill sets are involved in each branch (computational ability, geometrical reasoning ability, etc.) Saying that football is harder than baseball, or topology is harder than vector calculus, is a question who's answer really depends on who you're asking.

 

[mathwonk]

to me analysis is probably hardest.

 

[JasonRox]

This case taken to the extreme would mean every field of study should be at an equilibrium. i.e. it is equally hard to do research in maths as in psychology. (I don't mean to belittle psychologists but I have a feeling resulting from a degree of personal experience that in general a mathematician can learn psychology much faster than a psychologist learn maths.) But that is not the case is it? So maybe you can think of my question as which area in maths has the most amount of 'bright' people working in. Hence that area will not have many easy questions left to answer, making it seem harder in comparison.

Research in Psychology isn't easier! Where the hell did you get this from?

It looks harder in my opinion. You have to gather a bunch of data, test on people, and whatever else you have to do. There is lots to do.

I'm in mathematics, and I don't think any other area is easier than mine except for things like Business, Popular Culture, Classics, and that kind of stuff.

Things like Chemistry, Biology, Philosophy, Linguistics, and so on are just as hard a Mathematics.

 

[pivoxa15]

to me analysis is probably hardest.

Although I am only an undergrad I feel it is hard has well mainly because it always deals with the infinite. Is that also your reason?

 

[pivoxa15]

Research in Psychology isn't easier! Where the hell did you get this from?

It looks harder in my opinion. You have to gather a bunch of data, test on people, and whatever else you have to do. There is lots to do.

I'm in mathematics, and I don't think any other area is easier than mine except for things like Business, Popular Culture, Classics, and that kind of stuff.

Things like Chemistry, Biology, Philosophy, Linguistics, and so on are just as hard a Mathematics.

I studies year 12 psychology and maths. And maths was much harder. However, I got a better grade in maths than psychology. The reason is because I spent so much more time in maths. Had I spent this much time on psychology I would have memorised the whole course and got 100%. Psychology research could be a different business altogether but compared to maths research, I still think maths would be harder - i.e. if you spend 30 years on a psychology research problem, you might get somewhere with it - i.e write a reasonable report. But the same time on a difficult maths problem, you might have gone nowhwere.

Obviously there are different kinds of hardness. i.e for me the hardest subjects would be Labs and Classics. Although I believe I can overcome my incompetence in the former, the latter, I am not so sure - i.e. I could never understand a Shakespear play. The hardness in the latter I would describe as the problem of vagueness, something I hate. That is why I prefer maths even to a subject like physics.

 

[Crosson]

Artin's conjecture, involving Abstract Algebra and Number Theory, is considered a very hard field.

The Ringel-Kotzig conjecture, that every tree has a graceful labeling, is also frequently mentioned as a difficult problem (Ringel calling the attempts to solve it a "disease").

I know you are looking for areas of math, and not problems, but both of these problems drive their respective areas of math. Hundreds of people chip away at them bit by bit, every year.

 

[arildno]

to me analysis is probably hardest.

My lecturer in complex analysis said he found discrete maths to be the hardest..

 

[MadScientist 1000]

Probability... It forces me to think instead of having to work with limits and physics.

 

[pivoxa15]

Probability... It forces me to think instead of having to work with limits and physics.

Certainly in high school, probability was much harder than say calculus, although I haven't done university probability. The formulas in elementary probability are not big so it really recquires one to really undertand the maths behind it in order to solve a wordy problem.

 

[gravenewworld]

from my experiences, math logic takes the cake.

 

[Hurkyl]

Aww, logic is the most fun mathematics.

 

[Gokul43201]

Aww, logic is the most fun mathematics.

Maybe it takes the cake right in the face then?

 

[mathwonk]

sheaf cohomology seems hard to me too. and compactifications of quotients of siegel domains. and minus signs, yes minus signs are definitely the hardest thing in math, and adding fractions is hard for my students.

 

[X=7]

Classics is hard too my friend. Try translating an old Greek text! I did this for a long time before Math. But Math is harder in the sense that most Greek texts have been translated, now if you were translating an ancient text for the first time, that is a challenge. But just do not rank Greek with popular culture, business and as you call it "that kind of stuff." Deciphering old tablets-that is also a part of classics-do not not make such sweeping generalizations. If it were so easy, the linear A tablets are still not deciphered even though we have hundreds of them! To be "creatively productive" in a number of fields is hard.

Yes, lunarmansion, JasonRox is surely mistaken on that one. I too have studied Classics & Maths and, Jason, absolutely cannot agree. You are well wrong on that one!

Best wishes

x=7

 

[matt grime]

sheaf cohomology seems hard to me too.

I think the modern take on geometry is hard. Or more accurately the way geometers present it makes it seem far harder than it actually is. If only they could just agree on one set of nomenclature... it's easier than they let on, I'm sure. In fact, I've come round to the opinion that geometers are actually attempting to disguise how little they really can prove (which is distinct from what they 'know') - c.f. the need to assume something is Calabi-Yau, K3, Kahler, has log singularities, is Gorenstein,... or whatever. (Yes, I am teasing, a little).

 

[leon1127]

number theory is the hardest FOR ME because I live in a metric space. In fact, anything like combinatoric, algebra, number theory are not my friends... lol

 

[mathwonk]

I think MATT's post points up one source of difficulty, which for me is poor choice of nomenclature. why call something a K3 surface except for conceit. No one in geometry even knows what this stands for altho some say a conglomeration of names like kodaira, kahler,etc...

as james milne said in commenting on a letter about why galois fields deserve to be called such, ok he did discover them, but calling them finite fields is still more descriptive.

calabi - yau, k3, all this nonsense is just a way of naming manifolds analogous to elliptic curves, i.e. "flat" (trivial canonical bundle, and some other conditions).

aS USUAL THERE ARE THReE WORLDS in geometry, positively curved, negatively curved, and flat, (spheres, elliptic curves, and all the rest), and the flat ones are often the most interestin, while the negatively curved ones are most common ("general type")

i am being very rough here in my discussion, but not completely wrong.

on the other hand, i like terms which suggest their meaning, such as group action, and codifferential, which is the dual of the frechet derivative or differential of a map.

 

[mathwonk]

matts point is aalso illustrated by the term "classification of surfaces" which refers to that body of theory which classifies that tiny fraction of all surfaces whose clasification is known.

in a way it is accurate, but more descriptive would be "the little we know about classification of surfaces", (i.e. all but the negatively curved ones, which is most of them).

it is not alwaYS noted FoR INSTANCE that there is a surface having every possible finite fundamental group (proved in shafarevich), so classification of surfaces is no more feasible than classification of finite groups, in fact much less so.

 

[pivoxa15]

Another way to think about the hardness question is which area in maths, after a period of training and research allows the person to pick up maths in another field quickest?

 

[SeReNiTy]

I personally think algebraic topology was the hardest stuff i ever tried to study.

 

[JasonRox]

Classics is hard too my friend. Try translating an old Greek text! I did this for a long time before Math. But Math is harder in the sense that most Greek texts have been translated, now if you were translating an ancient text for the first time, that is a challenge. But just do not rank Greek with popular culture, business and as you call it "that kind of stuff." Deciphering old tablets-that is also a part of classics-do not not make such sweeping generalizations. If it were so easy, the linear A tablets are still not deciphered even though we have hundreds of them! To be "creatively productive" in a number of fields is hard.

Oh! I never knew that's actually what they did, so I take that back.

Anyways, Mathematics isn't necessarily the hardest thing out there. People have worked every subject to the bone and continue to do more, even Psychology.

You can't compare the two after only doing high school courses relating to the two. If it were high school courses, I'd mathematics is easier. High school math was a joke. Then again, it's subjective.

 

[pivoxa15]

I have to admit that nomenclature can put many people such as me off a subject and resulting in that person thinking a subject was too hard for them even though it shouldn't be the case. i.e. in junior high I always thought science was too hard because it had some huge words in them such as in geology and biology which was very sad because I ended up not doing any senior science subjects except psychology. Now I realize that those big words are all trivial.

 

[JonF]

elementary algebra, ask any 14 year old.

 

[Gib Z]

The irony is that I'm actually 14, and I wouldn't be able to say what it was exactly, but surely not elementary algebra lol

 

[pivoxa15]

I have to admit when I was 14, the hardest thing for me may well have been elementary algebra, especially factorising different expressions.

 

[Schrodinger's Dog]

The number of times I've found myself saying: well why didn't they just say that then, when looking at a particular piece of maths, leads me to the conclusion that Maths is definitely a language unto itself, mostly familiarity will get you to understand fairly quickly and the language in the method are aptly presented, but sometimes things are phrased or presented in such a way and using terms in such a context, as to be out of kilter with what you might typically understand the words to mean, which can be very confusing.

Mind you this is typical of science in general as well. Sometimes though I suppose it's difficult to remember that when your presenting material to a llevel lower than the authors, it can be easy to forget your target audience may not be familliar with the terminology, or they may be familliar with a particular phrase meaning something completely different in non-science or non-maths circles. I do think sometimes an English - maths phrase book might be in order though, because I was never any good at foreign languages anyway

 

[pivoxa15]

The number of times I've found myself saying: well why didn't they just say that then, when looking at a particular piece of maths, leads me to the conclusion that Maths is definitely a language unto itself, mostly familiarity will get you to understand fairly quickly and the language in the method are aptly presented, but sometimes things are phrased or presented in such a way and using terms in such a context, as to be out of kilter with what you might typically understand the words to mean, which can be very confusing.

Definitely and I think the only way to get around it is practising. Do many problems which contain the terminologies.

Mind you this is typical of science in general as well. Sometimes though I suppose it's difficult to remember that when your presenting material to a llevel lower than the authors, it can be easy to forget your target audience may not be familliar with the terminology, or they may be familliar with a particular phrase meaning something completely different in non-science or non-maths circles. I do think sometimes an English - maths phrase book might be in order though, because I was never any good at foreign languages anyway

The sciences may be a bit easier to get use to than maths because it is more intuitive since we live in a physical world, not a mathematical world. At the moment I am reading an intro chemistry book and is very pleased with the layout because on every page it leaves some space for definition of the terminology used on that page. A book like this may be what you are looking for.

 

[mathwonk]

pivoxa 15, yes that is my reason.

however i also think galois theory is hard. it is so complicated.
there are so many special topics and results.

to me topology was easy. when i found out about algebraic geometry i dropped topology and switched to algebraic geometry because it was hard, but not too hard.

i am still hoping to master the basics of algebraic geometry before i die, or succumb to alzheimers.

of course if anyone says a particular field is easy, you can always pose a problem he cannot solve. like compute the homotopy groups of spheres in topology, or prove the riemann hypothesis in number theory, or decide the rationality of hypersurfaces in algebraic geometry.

or,..., say, does analysis have any hard open problems? (just joking, but i do not know what they are.) please do not say the invariant subspace problem.

 

[Chaos' lil bro Order]

Anytime you are dealing with seveal awkwardly curved objects, fields arising from these objects, and calculating the effects of these objects on each other in a time-dependent fashion, you are doing the most difficult math there is. This can apply to many fields of math and physics, just add imagination, sunlight and water.

 

[Chaos' lil bro Order]

Also, (this may be entirely wrong, so sorry in advance if it is) try finding a Lagrange point for a 4-body problem.

 

[pivoxa15]

Anytime you are dealing with seveal awkwardly curved objects, fields arising from these objects, and calculating the effects of these objects on each other in a time-dependent fashion, you are doing the most difficult math there is. This can apply to many fields of math and physics, just add imagination, sunlight and water.

I can relate to that. I often wonder that physics problems may seem apparently easy after reading it but to actually solve it and then think why it is, things don't become clear. The reason is they involve movement (in space and time) so mechanics, dynamics come into it and that makes things hard for some reason. That is why subjects like chemistry can be easier even though it deals with natural phenomena as well since it dosen't recquire the mechanics behind nature. In that way it is more like maths, treaing objects i.e. atoms as perfect entities like numbers. That could also be why Rutherford said "Science is either physics or stamp collecting". Also I know a few people who are combined chemistry and maths majors and think that Physics is hard or don't like it as much. Maybe our brains are usually not very good with these things which also goes for problems in probability and abstract, rigorous maths.

But with (some) maths it is like the opposite, a problem seems hard after reading it but if you find a 'trick' to solving it than things suddenly becomes easy and the why it works also comes out.

 

[matt grime]

Anytime you are dealing with seveal awkwardly curved objects, fields arising from these objects, and calculating the effects of these objects on each other in a time-dependent fashion, you are doing the most difficult math there is. This can apply to many fields of math and physics, just add imagination, sunlight and water.

What utter BS.