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I am working on a project relating to- https://en.wikipedia.org/wiki/Millennium_Prize_Problems

I wanted to list the effects of them being proved or disproved in different aspects of science and maths. It was fairly easy to find such results on physics and maths as the questions primarily come up from them however I couldn't find any reference to effects it would have on our understanding of chemistry.

Is it because there will be no effect at all and that they are the least concern to chemists (in terms of applicability) or if there are then what are they?

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    $\begingroup$ To understand just how absurdly far-fetched this may come across, ask yourself the same question: what effect will it have on your life? Say, you get up, brush your teeth, have your breakfast, then BANG! $P\ne NP.$ So what? $\endgroup$ – Ivan Neretin Sep 19 at 10:25
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    $\begingroup$ I suspect that two problems (p-np and Navier–Stokes existence and smoothness) would have repercussions, in defining some problems as either bounded or unbounded. Guaranteeing that you can find a solution to a certain level of accuracy in a given amount of time is akin to guaranteeing that you will come out winning in a lottery if you buy enough tickets. However I admit to speculating - I am not directly familiar with the problems. $\endgroup$ – Buck Thorn Sep 19 at 10:49
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    $\begingroup$ @IvanNeretin I know it wouldn't be of any importance to me in my general life but i wanted to know it's impact on a discipline or a field of study such as Chemistry $\endgroup$ – StackUpPhysics Sep 19 at 14:37
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All right, let's sum it up.

  • P vs NP, if solved the way that pretty much everybody expects (that is, $\rm P\ne NP$), will have no far-reaching consequences, because that's what we were thinking for quite a while now. If it would miraculously happen to be otherwise (that is, $\rm P=NP$), that would be quite a shock to many fields, especially to computer science, and by extension, to computational chemistry. But I don't believe in miracles.
  • Navier–Stokes (no matter how it will turn out) will have consequences in fluid dynamics, and by extension, maybe in some relatively narrow areas of chemical technology.
  • Yang–Mills will cause repercussions in the areas of physics which are the most distant from chemistry.
  • The rest of the problems, as far as I understand, have no bearing whatsoever on our material world.
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    $\begingroup$ "The rest of the problems, as far as I understand, have no bearing whatsoever on our material world." - Not true. The Hodge conjecture, at the very least, would impact string theory (mathoverflow.net/questions/221892/…), and the Birch-Swinnerton-Dyer conjecture and Riemann hypothesis have cryptographics ramifications, which would affect money, which is material (or at least is convertible into something material). The Poincare conjecture is also relevant for certain GR spacetimes. $\endgroup$ – probably_someone Sep 19 at 18:49
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    $\begingroup$ String theory has no bearing on our world either, and won't have for a while. Poincare... well, if our universe is closed (which we are not sure about), then it is applicable and may have some consequences, though I don't see any. As for the rest, well, point taken: Riemann and probably BSD will affect the world, if only through computer algorithms. But then again, it is much like P vs NP: if what we think true is proven true, then some tentative estimates become exact, and things continue running as usual. And if it is proven not true, that would be a miracle. $\endgroup$ – Ivan Neretin Sep 19 at 19:05
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    $\begingroup$ ... and possibly the worst version of $P = NP$, a completely nonconstructive proof of that equality. So then we know that many problems are "easy", but we're ignorant of the easy way. $\endgroup$ – Eric Towers Sep 19 at 23:55
  • $\begingroup$ @IvanNeretin If string theory has no bearing on our world, then Yang-Mills surely doesn't either. Likewise, if P vs. NP affects our world, then Riemann and BSD do too. I'm just confused about the distinction between the things you say do have some bearing on "our material world" and those that you say have "absolutely no bearing on our material world." $\endgroup$ – probably_someone Sep 20 at 1:25
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    $\begingroup$ That's why I said "point taken". Admittedly, Riemann and BSD do have some bearing on our world, just not on chemistry. $\endgroup$ – Ivan Neretin Sep 20 at 6:35

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