Which areas of mathematics are considered the hardest?

[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.