Thread: what the hardest topic in maths.

Hi, have been wondering what's the hardest topic in maths can someone tell me.

There are plenty of difficult field of mathematics- some people may find one field the most difficult, whilst other find others the most difficult. Difficulty is relative to each person.

Originally Posted by maxking

Hi, have been wondering what's the hardest topic in maths can someone tell me.

The Riemann Hypothesis is considered by many to be the most profound and difficult open problem in mathematics.

The hardest part is always coming up with new and useful areas of math, and those that pull it off are always remembered.

all the problems in the millenium prize proposed by the clay institute of mathematics, is all very very hard. Like Dr.Rocket said, The Riemann Hypothesis is one of the 7problems.

P versus NP problem
Hodge conjecture
Poincaré conjecture (solved)
Riemann hypothesis
Yang–Mills existence and mass gap
Navier–Stokes existence and smoothness
Birch and Swinnerton-Dyer conjecture


the solutions, and the ideas required to solve these problems will have a significant effect on the topic and many other areas of mathematics as well.

Can you or the good Doctor give us a little background on these unsolved problems?
I am familiar with the Poincaire conjecture, I'm sure I could educate myself on the Yang-Mills problem ( if its the one I'm thinking of ), and I assume Navier-Stokes are the fluid flow equations. As to the rest, I have no clue, and a layman's explanation would be greatly appreciated. Educate us, please.

I can explain P vs. NP as that falls under computer science (and math).

P and NP are two sets and the question is whether the two are equal. The difficulty comes in how the two sets are defined.

P is the set of all problems solvable by a deterministic Turing machine (or any of its equivalent formulations) in an amount of time that's polynomial in the size of the input. Turing machines are basically what we think of when we think of computers. Polynomial time means that when the problem gets bigger, the time doesn't increase too drastically. So P is basically what problems are feasible to run on computers.

NP is the set of all problems solvable by a non-deterministic Turing machine. That is, one that can take all possible computation paths (both branches of an if, for example) and then choose the one that eventually works. It does this by magic really, but we can simulate a non-deterministic machine on a deterministic machine. The problem is, doing so takes an exponential amount of time and space, at least as far as anyone currently knows. No one has proven that it has to though, and that's the $1,000,000 problem.

So, if you can show a to solve an NP-complete problem with a deterministic algorithm in a polynomial amount of time, you'll be famous.

Originally Posted by MigL

Can you or the good Doctor give us a little background on these unsolved problems?
I am familiar with the Poincaire conjecture, I'm sure I could educate myself on the Yang-Mills problem ( if its the one I'm thinking of ), and I assume Navier-Stokes are the fluid flow equations. As to the rest, I have no clue, and a layman's explanation would be greatly appreciated. Educate us, please.

There are detailed descriptions by world class experts here: http://www.claymath.org/millennium/

Thanks for the explanation of the N vs. NP problem MagiMaster. And thanks for the link DrR, very easy to understand even for someone with my ( very ) limited understanding of advanced mathematics.

Originally Posted by maxking

Hi, have been wondering what's the hardest topic in maths can someone tell me.

According to a Cambridge university maths graduate I once met on an Ada course. She said it was omega (Ω).

That's the topic, I don't know what the problem was.

Application of Godel's incompleteness theorem. What does it mean for time, math, and meaning itself.

Originally Posted by rock

Application of Godel's incompleteness theorem. What does it mean for time, math, and meaning itself.

Gödel and time
Gödel find a solution to Einstein's equations that allowed backwards time travel.

Gödel and maths
Gödel showed that for sufficiently complex formal systems (ordinary arithmetic with addition and multiplication will do), there
is an information hole. This is the mathematical 'equivalent' of Heisenberg's uncertainty principle in physics.
What this means in the maths domain is that there is no algorithm which can churn out all the theorems.

Gödel and meaning
You don't need Gödel here for 'nothing has any meaning except the meaning you give it'.

Let me tell you about the polynomials an algebraic expression which consists of two or more terms, is called a polynomialExample: 5x-2, 3x+7y
this is the basic definition of polynomial if you want to learn more then go to link.

The estimation of intrinsic difficulty is confused by the fact that intrinsically easier topics have been pushed farther into complexity or profundity because they are easier to push.

As when comparing the difficulty of playing musical instruments, the standards of play are higher for many of the easier ones - so you have to put as much effort into the piano as you would into the guitar, say, to meet minimal standards of performance, despite the piano being intrinsically easier to play both physically and mentally.

In my various tutoring as well as discussion experiences, probability seems the most difficult concept to handle and use, think with or argue from, mathematically.

As evidence, look at the relative sophistication or complexity of the situations that can be easily comprehended by people new to the topic, in basic probability compared with basic geometry or basic topology or basic number theory or the like. People struggle with, say, Baye's Theorem - even experienced and expert pros have to be careful with it in even simple situations - and it's "objectively" rudimentary.

As i think there are many topics in math which are complicated and hard in which irrational numbers are also one of those typical topics which looks some different and are interesting .So one should know what are irrational numbers and there use .

Originally Posted by iceaura

The estimation of intrinsic difficulty is confused by the fact that intrinsically easier topics have been pushed farther into complexity or profundity because they are easier to push.

I would never have guessed, but I am going to ask Dave Spart for his opinion!

Isn't prime distribution an unsolved difficult problem? Prime numbers have been known for 2000 years and we still don't fully understand them.

Number theory problems are usually the easiest to understand, but the hardest to prove.

Originally Posted by mathman

Number theory problems are usually the easiest to understand, but the hardest to prove.

From what I've read, it's strange to me how simple the concept of prime numbers seems but they're even involved in that Riemann Hypothesis (which I don't really understand at all). Can someone please explain, in laymen's terms, what it is? I know it has to do with complex numbers, primes, and proving zeros.

hi all ,
Math problem .A subject having combination of complex questions and topics is math .So we must see all these and i am not expert like all of you dear .So please help me clear my doubts .i want to ask many topics as diffrential , integral , ratio as what are ratios and topics related to it .