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Question: What is the hardest part of mathematics?
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Chris Budd answered on 17 Jun 2020:
Maths has lots of hard problems, many of which have been unsolved for hundreds of years. In the year 2000 the Clay Institute announced seven problems, the Millenium Prize Problems, which were considered to represent seven huge challenges to mathematicians. One of these, the Riemann hypothesis, is often considered to be the hardest problem in maths and was originally posed in 1859. It is closely related to the question of the distribution of the prime numbers. In my opinion, another of the problems, called the P versus NP problem, may prove even harder to solve. In fact this problem is concerned with the queston of working out exactly how hard it is to solve any problem in general. If we could crack P vs NP then we would know how to crack many other problems in many other fields. Another problem on this list, is whether the Navier Stokes equations have smooth solutions or not. As these are the equations of the weather, knowing the answer affects all of us every day.
Whilst all of these problems are immensly hard to solve (and anyone who can solve them will achieve immortal fame), for me the hardest part of mathematics is understanding why an abstract creation of the human mind proves to be the key for unlocking the secrets of the universe, has applications in everything that we do and is the basis of all our modern technology.

Sophie Carr answered on 17 Jun 2020:
I think the answer is that it varies – there are some problems (such as those posed by the Clay Institute) which everyone has found hard, then there are those you personally find hard. I never really understood Fourier Transforms, and I’ve really tried! I think also if you really enjoy a certain part of maths you’ll work harder on that area because you like solving the puzzles so it ends up being “easier” for you.

Katy Tant answered on 17 Jun 2020: last edited 17 Jun 2020 8:42 am
I would say research in any field is difficult because you’re delving into the unknown – the answer is often not what you would expect and, in some cases, there may not even be an answer! In fact, the branch of mathematics we call analysis often deals with the question of existence – is there a solution to my mathematical problem?

Peter Kropholler answered on 17 Jun 2020:
From a purely mathematical point of view, questions like the Riemann Hypothesis, the uniqueness of solutions to the Navier Stokes equation, and the normality of numbers like pi seem very hard and may remain unsolved for many hundreds of years.
But from another perspective, one of the hardest things about mathematics is ensuring effective communication between mathematicians and nonmathematicians: that is a hard problem in a different way and talented mathematicians with that ability to communicate have a great role.

Alexandre Borovik answered on 17 Jun 2020: last edited 18 Jun 2020 11:33 am
Each part of mathematics has some exceptionally difficult questions. However, there are areas of mathematics where long years of study are needed before you will start understand what questions are about.
My advice: forget about the hardest parts of mathematics. Mathematics is like mountaineering: up to a certain elevation, any healthy person, even children may have fun by ascending, at a relaxed pace, some friendly mountain (or a hill) — with spectacular views from the summit!. But do not try to climb Annapurna (fatality rate: 33%) or K2 without years and years of specialist training.
I personally knew people whose life was destroyed (in one case – ended in suicide) by unsuccessful attempts to prove the Jacobian Conjecture. Interestingly, its formulation can be explained to anyone who took a standard undergraduate course in multivariable calculus — see https://en.wikipedia.org/wiki/Jacobian_conjecture .

Maja Popovic answered on 17 Jun 2020: last edited 17 Jun 2020 12:26 pm
Apart from already mentioned objectively difficult parts, I would add that there is a lot of subjectiveness, too.
For example, in my first year at the university I was thrilled by infinite series and differential equations whereas many of my colleagues hated it (“very difficult”). On the other hand, they liked matrices and vectors, but I didn’t, it was somehow rather unpleasant for me.
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