This mainly goes out to the professional mathematicians, but what would be your assessment of the Millennium Problems? In the sense of which would be the most difficult, which might be solved first, the current status of the problems within the community itself. Not necessarily all of them, maybe the one or two that pertain to your area. I ask because I was listening to an algebraist and topologist at my university talk about the PoincarĂ© conjecture and found the discussion fascinating. In essence I'm looking for a discussion on the problems, focusing on their current status and opinions of mathematicians in the relevant fields of what their future will be.
Have you read the book "The Millenium Problems" by Keith Devlin? It does a great job of summarizing the problems for the non-expert. From what I remember, Devlin claims that the P vs. NP problem is the "easiest" by which he means that it is the most likely to be solved by a non-professional, however unlikely. As for the most difficult, he spends a lot of time on the Hodge Conjecture explaining to the reader that they will basically never understand the details, though he does make an attempt. If you like this book, I would like to recommend "Prime Obsession" by John Derbyshire also. It is an in depth look at the history and mathematics behind the Riemann Hypothesis.
Read it a while ago, it's good, but I'd prefer a more in-depth look at the problems. Unfortunately I can't seem to find any literature dealing with the problems and their place in the community. (Of course there is no problem finding literature concerning the problems themselves.)
I seem to remember reading somewhere (possibly in the book itself) that Devlin's "The Millennium Problems" book was to be the forerunner of a larger work describing the problems at a more advanced level. However it looks like the later work never materialised.
That's true, it mentions it in the foreword. I went looking for the larger tome but never found it. Shame, because it would have made for a great read. The actual problem descriptions themselves are worth sitting down with and reading, very reminiscent of the Hilbert problems in how they are stated although you can see that it's a different generation. Good for a contrast between 19th and 20th century mathematics, or at least I found so.
More accurately it's an in depth look at the history and a simplistic look at a tiny part of the mathematics. You couldn't expect much more given the target audience. I found it a nice read though. Edward's Riemann Zeta Function text is one of the more accessible introductions to the mathematics, and follows the historical development well, explaining Riemann's paper (there's a must read translation in the appendix). A nice survey article: http://www.ams.org/notices/200303/fea-conrey-web.pdf
My money goes on the Rhiemann Hypothesis not bc I understand it but because it made it to the otp of the lsit from the 19th century list into 20th.
From what I understand, the latest research suggests a Cantorian style revolution is needed before P=?=NP can even be touched. The mathematics that we have simply isn't sophisticated enough to get near it.
-Riemann Hypothesis solution could be easy to solve for a Physicist if Hilbert-Polya operator is constructible...so is "equivalent to a Hamiltonian [tex] \zeta(1/2+iH)|n>=0 [/tex] [tex] H|n>=E_n |n> [/tex]
http://arxiv.org/ftp/math/papers/0607/0607095.pdf A very curious paper on RH Thermodynamics and Chebyshev explicit formula...
I think they call it the "easiest" problem because it's probably the easiest to understand and can be tackled by nearly anyone. This doesn't mean that the solution is easy, which it certainly isn't, but i think i could describe the problem in a short post such that anyone at all would understand.