What are the most important and interesting questions in current mathematics?

Answer 1

The P=NP and the Riemann Hypothesis are probably the most important problems right now. The Poincare conjecture has been solved. The other Millenium problems are important for mathematicians, but probably less important for the average person.

P=NP has to do with quick computability of several important problems, while the Riemann Hypothesis is related to questions about the distribution of prime numbers.

Other good ones:
Goldbach's conjecture: Is every even number more than 4 the sum of two prime numbers?

Twin Prime Conjecture: are there an infinite number of primes p where p+2 is also prime?

Odd Perfect number: Is there an odd number which is the sum of all its divisors excluding itself? (All the even perfect numbers have been categorized: for example 28=1+2+7+14 is perfect). This one is interesting, but probably not important.

Invariant Subspace Problem: Does every bounded operator on a Hilbert space have an invariant subspace?

Answer 2

1

Answer 3

I'd say the Millennium Problems. The main problem with these problems is that these problems may not be understandable to the general public, and many are not in my mathematical field. Here they are:

P versus NP

Are easy-to-check problems always easy to solve (in a certain precise way)? Most mathematicians and computer scientists think not, but no proof has ever been given of that.

The Hodge Conjecture

Concerns something called algebraic varieties.

The Poincaré Conjecture

Is a 3D manifold such that each closed path can be shrunk into a point a sphere?

The Riemann Hypothesis

Do all the zeroes of the zeta function have real part = 1/2?

Yang-Mills Existence and Mass Gap

Must all of a certain collection of particles have positive mass?

Navier-Stokes Existence and Smoothness

Concerns Navier Stokes equations which governs gases and other media.

The Birch and Swinnerton-Dyer Conjecture

About the abelian group of an elliptic curve. This is distantly related to Fermat's last theorem, which has been proved.

See Wikipedia or other references on the Web for more information on these problems.

Answer 4

To Brittainy:
No, there is no end to the decimal expansion of Pi (or any other irrational number, for that matter)

To Laura N:
Have you been living in a cave? Back in '94 one of the most famous problems of all time (called Fermat's Theorem) was finally proven (solved) by Andrew Wiles. It was *so* big it even made the 6:00 O'clock News and CNN ran a special on it.

And there are quite a few of David Hilberts list of problems that haven't been solved. See
http://www.mathacademy.com/pr/prime/articles/hilbert_prob/index.asp
for details.

And there are a couple of simple looking problems that have stood the test of time.

Everyone knows (or should know) what a prime number is. And they should, probably, know that the list of prime numbers is endless (that is, there is no 'largest' prime number) The proof is fairly simple, but I'm not going to reproduce it here.

The question is: 'Twin Primes' (prime numbers whose difference is 2 such as 11 and 13 or 29 and 31) are even more scarce than prime numbers. But are they endless? Or is there a largest twin prime pair? It's a question that has been around since antiquity, its origins lost in the mists of time. But nobody has ever given a proof one way or the other. But it may not be far off. See:
http://www.sciencenews.org/articles/20050716/mathtrek.asp


Doug

Answer 5

A note about previous comments:

1) Pi is an irrational number in base 10. That means that it never ends (and one can prove that it never ends).

2) Mathematics is a very active field of research.

3) Riemann's hypothesis (please note spelling) is an important problem, but not necessarily the most important.


One particular problem that is immediately useful to everyday life is the P=NP problem. P class problems are essentially those that can be solved "in a reasonable time" on a computer. A classic example is the process of alphabetically sorting a list. Other types of problems, classified as NP, can apparently only be solved through time-prohibitive methods, but this is the P=NP question: Are there computer algorithms that can, "in a reasonable time," solve NP class problems? A variety of companies are wishing that this were true (FedEx, UPS, Intel, etc.) for it would (theoretically) allow these companies to substantially reduce their costs of operation by finding the most efficient methods for operation.

Answer 6

Your question is tricky. Questions in mathematics require time to become recognised as important or interesting, and hence will not be current.

If you haven't already, take a look at the millenium prize problems posed by the Clay Mathematics Institute - with one million dollar prizes attached.

Answer 7

One of the most important questions in math right now seems to be finding a way to factor huge numbers quickly. I've read that the difficulty involved in doing that is the basis for the security of most modern encryption, so if it were possible to do quickly factor huge numbers we would have to find other ways to communicate secretly. Another problem is finding a way to generate pseudorandom numbers of extremely high quality. Many forms of encryption rely on pseudorandom numbers. There are many other problems that have yet to be solved, but the most important one is to enable secure communication among peers. Without that, it would be difficult to get anything done in business or government, and it would also be impossible for citizens to have complete privacy.

Answer 8

Riemman's Hypothesis<<<<<< Goldbach's conjecture
The Twin Prime conjecture

Answer 9

I wouldn't consider mathematics a current field. What possible developments could be made?

Answer 10

the questions concerning prime numbers