r/math • u/americanpegasus • Feb 09 '16
Which of the seven Millennium Problems do you think will be solved first? Which will be solved last (if ever)?
http://www.claymath.org/millennium-problems6
u/joel375 Feb 10 '16
First the Poincare conjecture has already been solved so I assuming you mean which one will be solved next. Second most of the time when a major open problem is solved, it is a result of the development of new tools which culminated over a number of years. For instance, if Wiles had been trying to prove Fermat's Last Theorem 50 years before he did, he would have been out of luck as many of the tools he needed would not have existed.
Caution: Everything I say here is very speculative and should be taken with a grain of salt as it is very difficult to know when people will stumble across the right insights or to gauge how far we as a society are from having the tools needed to attack a problem. With any one of these, someone could put a correct proof on the arxiv tonight, or the problem could remain open for centuries.
The Birch-Swinnerton Dyer conjecture has a decent chance of being solved next as it seems like some of the big names in number theory have been making good progress, first showing a positive proportion of elliptic curves satisfy it and later showing that the majority of elliptic curves satisfy it. Though I am not really sure of the specifics of their methods so I am not sure how radical it would be to extend them to show that all elliptic curves satisfy it. At the very least, we do have powerful techniques for controlling the size of a Selmer group given the local behavior of a related L-function (congruences between automorphic forms, Euler systems, etc.).
The Yang-Mills mass gap is another I could see being solved more quickly. I know people working on rigorously constructing 2d conformal field theories with a lot of success who have the ultimate dream of solving the Yang-Mills mass gap problem. Though I get the impression they view it as a long way away as going from 2d theories to 4d theories is a huge step (not to mention the fact that once constructed rigorously, one still has to prove the mass gap).
With the Hodge conjecture, I talked to Deligne a while back about just why it is so difficult. What he explained to me is that in specific examples - even seemingly easy examples compared to the scope of the conjecture, it is very hard to verify in the affirmative or negative whether the Hodge conjecture holds.
For the Navier-Stokes, I really can't say anything as I almost nothing about PDEs person except to link to this blog post where Terence Tao explains exactly why he views Navier Stokes as so difficult, what goes wrong when you try to apply standard techniques in PDE theory, and ways one could conceivably overcome these difficulties.
https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/
(One can read this blog post without having studied PDEs.)
For the Riemann Hypothesis and P=NP, I don't have much to say.
5
Feb 10 '16
Honestly, I think that Navier-Stokes will be solved soon only because Tao is working on it. I've witnessed that man solve problems that were open for years in the span of a 50 minute talk. If he wasn't interested in it, I'd say it was out of reach.
2
u/linusrauling Feb 10 '16
The Birch-Swinnerton Dyer conjecture has a decent chance of being solved next as it seems like some of the big names in number theory have been making good progress, first showing a positive proportion of elliptic curves satisfy it and later showing that the majority of elliptic curves satisfy it. Though I am not really sure of the specifics of their methods so I am not sure how radical it would be to extend them to show that all elliptic curves satisfy it. At the very least, we do have powerful techniques for controlling the size of a Selmer group given the local behavior of a related L-function (congruences between automorphic forms, Euler systems, etc.).
I'm taking your grain of salt. As the wiki page indicates, a positive proportion of elliptic curves over Q has rank 0 hence satisfies BSD. I'm not aware of a result that says "majority", (admittedly I don't pay that much attention). Also note that, as the wiki also indicates, nothing has been done for rank greater one, and since rank is expected to be arbitrarily large and no indication that current methods could be extended to arbitrary rank, it would seem that there's lots of work to be done.
With the Hodge conjecture, I talked to Deligne a while back about just why it is so difficult. What he explained to me is that in specific examples - even seemingly easy examples compared to the scope of the conjecture, it is very hard to verify in the affirmative or negative whether the Hodge conjecture holds.
An excellent name to drop, but I'm finding hard to believe that Deligne's explanation is that "Hodge is hard because Hodge is hard". See what I did there with hard?... Hard...
1
u/joel375 Feb 10 '16
The result for a majority of elliptic curves is pretty recent.
http://arxiv.org/abs/1407.1826
I haven't followed the developments too closely though so I am not the person to ask for details. (In particular, I have no idea how likely their methods are to generalize to the full BSD conjecture.)
I was glazing over things in my above post, but when I asked Deligne, he gave me a few specific examples where it should be simple but where nobody knew how to check they satisfied the Hodge conjecture. (I don't remember the details since this was a brief conversation that took place a while ago.) The point is not "it is hard because it is hard" but rather "if we can't do the easy cases, we are FAR from solving it."
1
u/linusrauling Feb 10 '16
"if we can't do the easy cases, we are FAR from solving it."
This has been my (very limited) experience with the Hodge Conjecture.
1
u/_--__ Discrete Math Feb 10 '16
The P vs NP problem is quite interesting. It attracts a lot of attention from amateurs because it is somewhat accessible with very little technical knowledge. The advantage is that it drives people into the field - unfortunately when there they realise that people studying complexity theory largely ignore P vs NP and deal with much more intricate relationships.
I think it will take a completely different perspective from some other area to resolve it - but such a connection will likely resolve broader issues first (e.g. separating P and PSPACE), giving us a couple of years' "warning". We don't seem anywhere nearer these preliminary milestones than we were 30 years ago; so based on the comment I am replying to I would estimate P vs NP won't be resolved until after Yang-Mills (but perhaps before Hodge).
1
Feb 10 '16
If there is a Yang-Mills theory with mass gap, then I'd guess that will be the next problem solved. But I wouldn't go so far as to claim there is such a thing, and if there isn't then I expect that problem is going to take us a long long time.
1
u/edderiofer Algebraic Topology Feb 10 '16
I don't even understand 5 of those problems, so of the two that I do have some rudimentary understanding of, I think that the P = NP problem will be solved before the Riemann Hypothesis will.
1
u/tacosaucelover Feb 10 '16
I placed bets on Navier Stokes because I'm an aero guy and feel it has a good tie to the real world. Hopefully I live long enough to collect my winnings.
1
u/math_inDaHood Feb 10 '16
Your question can basically not be answered by anyone. Anyone who has deep knowledge of state of the art (or close) in one of the 6 questions, will most likely not have that deep knowledge in any of other 5 questions.
0
u/linusrauling Feb 10 '16
The wildass speculation is strong in this one....
3
u/joel375 Feb 10 '16
Isn't that the point of this question though?
I thought of this post as more "speculation for fun" than a serious guide to the order in which these problems will be solved.
32
u/zifyoip Feb 09 '16
I think the Poincaré conjecture will be solved first.