Which areas of mathematics are considered the hardest?

[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.

 

[pivoxa15]

What utter BS.

What is BS?

 

[Gib Z]

lol, BS is Bull****. Hey matt grime, remember me? :D anyway, i'd agree with matt grime, Chaos' lil bro Order is a lil outta order.

 

[Chaos' lil bro Order]

What utter BS.

Why the anger?

I'm open to you calling it BS, but you really should follow up with some reasons. Or do you like inflating your number of posts with mean, unthoughtful and stupid 15 letters comments?

 

[leon1127]

BS do not necessory mean bullcrap. When my teacher tell me to BS on the test, that means to "Be Specific".

 

[Chaos' lil bro Order]

BS do not necessory mean bullcrap. When my teacher tell me to BS on the test, that means to "Be Specific".

You are funny.

He said, 'What utter BS'. Was he saying, 'What utter Be Specific'? I don't think so.

Plus, I've read some of his other posts and he is quite the old grump.

 

[sherlockjones]

I always thought that the volumes of revolution, solids of revolution and graphical stuff in calculus were difficult.

 

[Math Is Hard]

BS do not necessory mean bullcrap. When my teacher tell me to BS on the test, that means to "Be Specific".

I can't stand it! That is hysterical!

Now I have done plenty of BSing on tests, but never at a teacher's request.

 

[Office_Shredder]

I always thought that the volumes of revolution, solids of revolution and graphical stuff in calculus were difficult.

What innocence

 

[Gib Z]

lol yes sherlock, they are pretty hard. There are harder stuff though :D

 

[sherlockjones]

i know that...thats why i said were

 

[^_^physicist]

Personally I find numerical methods of solving partial diff. equ's a hard thing to wrap my head around. Fun, and I would love, and I hope to get a chance to, despite being a physics guy, a chance to work on trying to understand numerical approximation methods and perhaps trying to see where one could go with it.

But I wouldn't say Numerical methods are the hardest thing around, I would stand by saying that their is a general level of difficulty to all things in mathematics, it just so happens everyone hits this general level doing something different from one another other.

 

[Symphony]

In my opinion , number theory is the hardest area in maths :D

 

[NeutronStar]

I think what make any area of mathematics difficult is when you have a poor teacher, or poor textbooks if you are attempting to self-learn. I've had far too many mathematics instructors who either, simply didn't know their subject very well, or they were lousy teachers, or in the absolute worse case they actually got a kick out of making it vague and difficult. On the other hand, when you find a teacher who genuinely knows what they are talking about, knows how to teach, and has a sincere interest in making it understandable to the student, then it becomes amazingly easy!

Another thing to consider also is having a solid understanding of the proper prerequisites. If a person tries to move on to some advanced mathematics without having a solid understanding of the foundational concepts of course it's going to be difficult for them. On the other hand, if they really have a good handle on the foundational concepts, they really shouldn't have all that much difficulty with the more advanced concepts.

What makes mathematics hard for the general public is the way that it is taught. It's not really the problem of the masses. It's the problem of the educational institutions for not making it easier and more interesting to understand. I love math, yet I found many math courses that I have taken to be utterly boring and difficult simply because of very poor forms of pedagogy.

I blame the school systems almost entirely for the general public's phobia of mathematics. Mathematics really isn't all that hard. Educational institutions just make it seem that way.

 

[mathwonk]

it is difficult to say which area is hardest when every area has essentially undoable problems. if you go deeply into any area you will be completely stumped. isn't that hard enough? we are moving faster though, since the independence of the parallel postulate in geometry took over a thousand years to understand, the cubic formula took maybe 600, the insolvability of quintics took maybe another 300, fermats last conjecture took over 350 and poincares problem (characterizing spheers) took only about 100. In algebraic geometry, characterizing varieties birational to projective space, (analog of poincare), is still out there. for some reason, just as in topology, the difference between rationality and unirationality is apparently deeper in dimension 3, although it was solved there first.

 

[pivoxa15]

I think what make any area of mathematics difficult is when you have a poor teacher, or poor textbooks if you are attempting to self-learn. I've had far too many mathematics instructors who either, simply didn't know their subject very well, or they were lousy teachers, or in the absolute worse case they actually got a kick out of making it vague and difficult. On the other hand, when you find a teacher who genuinely knows what they are talking about, knows how to teach, and has a sincere interest in making it understandable to the student, then it becomes amazingly easy!

That could well be true. The other factor could be how relevant the teacher's tests match what they teach. If they are very similar than the students tend to do much better hence getting an impression that they have done very well and the teacher taught well.

In university, the lecturers move so much faster and often I get lost very early so even if the lecturer was really good, I couldn't appreciate them which is depressing. The reason for this could be your next point.

Another thing to consider also is having a solid understanding of the proper prerequisites. If a person tries to move on to some advanced mathematics without having a solid understanding of the foundational concepts of course it's going to be difficult for them. On the other hand, if they really have a good handle on the foundational concepts, they really shouldn't have all that much difficulty with the more advanced concepts.

That is very important as I have come to realize from experience. I didn't have a solid maths, science background in high school and so have really struggled in university maths and science while doing the advanced subjects. Terry Tao also emphasises this point when giving advice to students.

What makes mathematics hard for the general public is the way that it is taught. It's not really the problem of the masses. It's the problem of the educational institutions for not making it easier and more interesting to understand. I love math, yet I found many math courses that I have taken to be utterly boring and difficult simply because of very poor forms of pedagogy.

I blame the school systems almost entirely for the general public's phobia of mathematics. Mathematics really isn't all that hard. Educational institutions just make it seem that way.

I didn't do well in maths in middle high which was very unfortunate because that led me to not do the advanced subjects in senior high. The fundalmental reason however, may not be at the teacher's fault but my own fault at not keeping up with the work and not doing enough excercies. From my experiences so far, whenever I have kept up with the work, I have always done well no matter how bad the teacher although a good teacher may make things even better. So as long as one is enthusiastic but I guess a bad teacher could put off students leading them to be lazy hence not do well.

 

[mathwonk]

i guess hard has several meanings, like there are hard open research problems, or the basic stuff is just hard to learn. the latter, i.e. hard to learn, is definitely related to the skill of your teacher.

but also ones own stubbornness. like many people who ask how to learn stuff here refuse to read the best sources we recommend. I also have this failing. It is so easy to kid oneself that some secondary source will somehow ease the difficulty of coming to grips with the real subject matters difficulty as it exists in the original.

often just the opposite is true. gauss proof of uniqueness of prime factorization of integers is much easier to read than mine, because he focuses only on the essentials.

 

[pivoxa15]

i guess hard has several meanings, like there are hard open research problems, or the basic stuff is just hard to learn. the latter, i.e. hard to learn, is definitely related to the skill of your teacher.

I once had a post doc teach me a first course in linear algebra and I couldn't understand a thing in his lectures. I thought it was his incompetence as a teacher. But recently I had a professor and Head of Department teach me a second course on linear algebra and intro abstract algebra and I was still lost in every lecture. This professor even lectured without looking at his notes which was really amazing.

 

[mathwonk]

your predicament poses several questions: like did you go intyo the second cousre before amstering the first course? and were you prepared for the first course?>

 

[EmilK]

Another way of looking at the question gives me the following answer. In my opinion, concepts are the hardest part of mathematics. The problem solving and the actual calculations is often pretty straight-forward if nothing strange comes up such as an undefined amount or basically errors in the computation itself. The transition going from eg. basic calculus to rotating bodies such as spheres is harder than learning just another approach to a basic calculus problem in my opinion.

 

[Atomos]

I have not done much math but combinatorics has always been difficult for me.

 

[buzzmath]

What exactly are you talking about? Do you want to know what class or what area of research has the reputation as being the most difficult? I think all areas of mathematical research have the same difficulty level. However, some classes have the reputation of being the most difficult. The mojority of students I talk to say they think that real analysis is the most difficult. I personally find abstract algebra harder. It all depends on the person I guess.

 

[pivoxa15]

your predicament poses several questions: like did you go intyo the second cousre before amstering the first course? and were you prepared for the first course?>

I was underprepared for both courses. That is why I am revising the old material getting ready for the third year algebra I will be taking next year because at this rate I will fail if I don't as my marks are getting worse each year.

 

[pivoxa15]

What exactly are you talking about? Do you want to know what class or what area of research has the reputation as being the most difficult? I think all areas of mathematical research have the same difficulty level. However, some classes have the reputation of being the most difficult. The mojority of students I talk to say they think that real analysis is the most difficult. I personally find abstract algebra harder. It all depends on the person I guess.

Broadly speaking, which general area of mathematics is considered the hardest by the majority of mathematicians is my question.

 

[Schrodinger's Dog]

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




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.

Yup good advice, I was relooking over differentiation the other day and I stopped on the different terminology which is very briefly mentioned in my textbook, in fact it gets 1 paragraph and 6 questions! Although it is explained it is hardly gone into in detail and frankly I think they didn't cover it enough, that said though just doing examples and seeing how terms relate is usually enough to get a grounding in the terminology, but it would be a damn site more helpful sometimes if they did what it sounds like they do in your book.

EDIT: I asked someone about this and they said they're all easy but specifically a 3D solution to Dirac's equation and Goldbach's postulate are particularly tricky areas of maths. I think the theory of whatever your working on holds as he gave up on Goldbach's and is working on the former ATM in his spare time.