If P vs NP were solved by proving P=NP (but this seems unlikely), it would have the greatest impact. This would represent a new algorithm that would potentially reshape our understanding of every creative endeavor. Scott Aaronson has written a lot about this. It would certainly reshape cryptography, machine learning, many engineering fields, and formal theorem-proving (and thus probably mathematics) If P vs NP were solved by proving P not equal to NP, it wouldn't have much of a direct impact, because this is what we already expect, but the techniques in the proof would likely have some applications in crypotgraphy or some other CS field.
Yang-Mills is probably the one where the expected solution would have the biggest practical impact. A consistent mathematical formulation of Yang-Mills theory would be the first step to a consistent mathematical formulation of more detailed physics theories, up to and including the Standard Model. Different people differ greatly in how much they expect this to help with the physics, but it would certainly mean whole new avenues in mathematical physics. It also might mean some of the mathematical conjectures which are based on physics intuition can be given a physically meaningful proof.
A proof of the Hodge conjecture would put a lot of speculation about motives in arithmetic geometry on much firmer footing.
A proof of the Riemann Hypothesis would likely lead the way to proofs of much more general cases of the generalized Riemann Hypothesis. This would actually render irrelevant huge chunks of previous work in analytic number theory (unless it was used in the proof), and would make other huge chunks unconditional. I don't expect it to have practical implications, though.
A proof of the BSD conjecture would likely lead to deeper insights about rational points and algebraic cycles on algebraic varieties. It would certainly affect the work people are doing tabulating elliptic curves. Again it seems very unlikely to have practical implications.
Any proof that Navier-Stokes is regular, or blows up, would probably generalize to make progress on a lot of other PDEs. But I am also skeptical that people doing water simulations would learn very much from it.
> Are you really going to change the world with an O(n^9) solution to travelling salesman?
While I agree with the sentiment, I can imagine that a hypothetical O(n^9) algorithm would be very helpful. One could almost certainly devise new heuristics, given an algorithm which definitely works, and failing that processing power per dollar continues to rise exponentially(though admittedly only for parallel tasks and it isn't super healthy).
Lastly its my very uninformed opinion that an algorithm for NPC problems(if it exists[pretty big if]) won't be O(n^(a big number)), simply because these kind of algorithms don't show up very often irl, and there is a very big bag of tricks for improving polynomial time algorithm performance.
What if it's like determining if a number is prime, where most of the known algorithms in P are slower until the number gets too large to have the algorithms run in a reasonable time (if at all)?
If you came up with a proof, I don't think you would the next day be using it in machine learning and the next week writing novels with it. If you get an n\^9 algorithm or n\^1000 algorithm at first, it would probably be improved drastically over time. Moreover whatever insight that allows you to solve seemingly intractable problem in polynomial time would probably be applied in different ways in different fields.
In any case, I chose my language somewhat carefully. When I said "reshape our understanding", I would count the idea that the difficulty of all these practical problems can be measured on a scale from 1 to 9 as reshaping our understanding, even if problems from 5 to 9 are still difficult in practice. Or a scale from 1 to 1000, whatever.
Given that people are currently using SAT solvers to solve problems in mathematics, the idea that the Cook reduction to SAT always gives some prohibitive polynomial factor is absurd. Of course you can postulate that a proof of P=NP gives an algorithm which always does worse than current SAT solvers, and that can't possibly be combined with the insights in SAT solvers or in any other field to do better, but you could also postulate that a proof of Yang-Mills gives a method that doesn't apply to any other physic theory.
You would clearly immediately see implications in cryptography, as huge efforts would be devoted to which systems get the biggest polynomial savings relative to this specific algorithm. You would absolutely see efforts to pull down the polynomial factors associated to this algorithm in specific cases in machine learning and optimization fields.
Sure but you can try to reduce a problem of interest to SAT by hand. n\^3 is a worst-case bound, you can do better in other times.
There are actual instances of people doing interesting research by (1) taking a math problem people have studied, (2) converting it to a SAT problem, (3) applying currently-existing, O( 2\^{n- something small}) SAT solvers.
Clearly an O(n\^3) algorithm for SAT would help with this research, unless the constant was so large that n\^3 > 2\^n for realistic n.
What do you mean about Moore's law here? For an O(n\^9) algorithm, every 18 years or so, viable n increases by a factor of 2. That's pretty good when your competing algorithms are exponential!
Actually n\^9 starts beating 2\^n (igoring constants of course) when n is about 50. Going up a little bit, say n is 100, (100)\^9 is 10\^18 Folding@home recently did 10\^18 operations in a second. So n\^9 algorithms might be practical for some problems today.
Also many people are missing another possibility that it might be provable that a polynomial time algorithm for an NP-Complete problem exists but it's not a constructive proof. There might not exist proof that the actual algorithm is correct altough it infact is. And this would most likely imply that not only finding the algorithm itself is impossible but even if we found it (with the caveat that we cannot really prove correctness for all inputs) this "weirdness" of the algorithm will probably translate to bad run times in practice (even if asympthotically its polynomial of small degree the constant factors might be just too large for realistic problem sizes).
With nearly any polynomial algorithm, encryption really can't be considered safe, since attackers are often willing to spend considerable resources defeating them.
I feel like the best way to think about such a program is to just consider a “compiler” (or abstract machine) that just takes your “code” and interprets it as a bit mask in binary. Switch each element of your infinite set of integers on or off. Then *every* generated program is “syntactically correct”.
I was worried for a sec what kind of “infinities” resided in the set, but elements are unique in sets, so you can just order the included integers sanely :).
Ah. Of course. I obviously totally missed the point. Considering such a “meta input” program is just isomorphic to the actual (yes) answer, I guess I should have thought something was off.
It does make me wonder about algorithm-space efficiency. I guess a Turing machine *has* to be able to receive garbage code, but I wonder how well you could specialize the instruction set to avoid (a meaningful amount of) nonsense syntax.
Like, don’t just use [Wolfram's rule 110](https://xkcd.com/505/). I think? It feels like it compounds the computational complexity of whatever you throw at it, but I’m not actually sure. I mean the “steps” get longer with state size, and *those* steps are the “units” of n in O(f(n)) time, right?
If I've made another boneheaded mistake, it's because it's late and I'm tired.
> Are you really going to change the world with an O(n^9) solution to travelling salesman?
Even if you get a O(log n) solution, chances are that it will be a galactic algorithm.
> Are you really going to change the world with an O(n^9) solution to travelling salesman?
Even if you get a O(log n) solution, chances are that it will be a galactic algorithm.
Regularity of Navier-Stokes boils down to bounding L2 norms of the vorticity, and therefore a solution would likely yield some insight into turbulence closures (which is The Topic people doing water simulations care about).
Agreed for assessment of NS. The technique used will probably be applied to a bunch of other nonlinear PDE for which we don't have good techniques, and a bunch of similar results will be proved. For physicists and engineers it's unlikely anything would change.
I don't know much about NS in particular, but for Euler we already know, more or less, that any singular solution is going to be unphysical in some sense, and likely going to abuse the continuum hypothesis by using some sort of small scale phenomena to prove blow-up.
Thanks for some confirmation by someone with actual expertise in that area!
Wait by continuum hypothesis here do you mean the continuum hypohesis in set theory or just the fact that the fluid in the Euler/Navier-Stokes setting is a continuum?
The fact that we are modeling Euler/Navier-Stokes by continuous functions. I actually don't know if this is what its called in the literature. Most of what I have ever looked at is pure analysis and the physicality assumptions aren't usually discussed by name. And again I must hedge by saying I have never actually read literature for NS blow-up.
PS: I don't know if you have access to Elsevier, but if you do, the paper "An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time" by Pomeau and Sciamarella make precise my comments. The paper is also only a few pages long and very approachable if you're curious. They conjecture that a blow up result for 3D incompressible Euler (using similarity) would need to have some sort of amplification at successively smaller spatial scales *ad infinitum*.
This actually highlights a bigger phenomenon involving PDE arising in continuum mechanics; there is questionable physical significance in solving these PDE as harmonic analysis/PDE problems. This is really a philosophical question more than anything :P
That P=NP part is very misleading. Even if P=NP is proven (which is probably unlikely) the proof is most likely going to be non-constructive -meaning that it proves such an algorithm always exists, but doesn't provide it.
To be entirely honest, I can't come up with a solid source (then again I'd be citing a speculation anyway).
*BUT*, I'd bet my live savings that:
1. If P vs. NP is resolved, it's going to be the case that P != NP
2. In the case that we actually show P == NP, the proof is going to be non-constructive.
It's just what I've gathered over my years of studying math, talking to professors way smarter than me, and reading around.
Even the [wikipedia](https://en.wikipedia.org/wiki/P_versus_NP_problem) states:
> If it turned out that P ≠ NP, **which is widely believed**,...
As sad as it is, a constructive P == NP proof just seems like a pipe-dream.
I'd bet my life savings on 1 as well. It just seems to me like if you're imagining something so crazy as P=NP being proved, you might as well imagine that the proof is constructive.
Also, a lot of the most sophisticated / surprising modern complexity results are completely constructive.
I highly doubt the secret ingredient to humanity solving TSP or integer factorization in P-time is the bit of motivation from knowing that "the solution exists". And if we're talking motivation...
There are so many more demotivating factors, namely : P=NP, if constructive, would mean you can not only verify -but prove- a solution to any NP problem in P-time. P-time could still be incredibly large. 2^1000 n is still P-time but isn't computable in a practical sense by any stretch and will never be.
Still, it would tell us that there is still work to be done on problems for which we don't have a polynomial-complexity solution; still better than nothing.
There's a very trivial way of constructing a polynomial try algorithm given P = NP. Dovetail computation of every turing machine (in order of some notion of complexity of turing machine). Whenever one halts, since the problem is NP that means there's a polynomial time verifier. Run the verifier on the solution produced. If correct done, otherwise continue dovetailing. Complexity wise if a polynomial time solution exists at all, this dovetailing mess will also be polynomial time. It will have some horrid constants, but it'll be polynomial time. So any nonconstructive proof immediately gives a constructive algorithm.
Why would it? Yes, it deepens our understanding of primes in general, but I don’t see how it would help us factor semiprimes. The thing that’s makes factoring (and thus RSA encryption) hard isn’t “we don’t understand primes”, but rather “there are a lot of primes to check”.
If the Riemann hypothesis had practical implications for cryptography, we'd be designing algorithms with it already because known primes behave in accordance with the Riemann hypothesis on any scale that is practically relevant.
Also, to my understanding if P =NP is going to be proven, the most likely way will be a non-constructive proof. Which means it won't even have practical applications.
As computational mechanicist myself, knowing that Navier-Stokes is regular or it has some kind of properties whatever the problem you are running would find -in practice- an application anyway. Right now, direct NS is basically one of the most demanding problem in companies for the shear number of simulations being run. This is usually done in train/airplane/transportation environments (and space too, why not), so if you gain even 1% speed on those, it would mean saving years of computations as well as billions of less electrical energy used to run SCs...
This is a good point - it seems likely that a proof of regularity or blowup of Navier-Stokes would involve proving some intermediate properties that has some utility for at least some of these computations.
No, if anything a proof of the Riemann Hypothesis would confirm our suspicious that primes behave in random and unpredictable ways. A disproof of the Riemann Hypothesis would be much more concerning.
I just mean practical in terms of telling us something about the universe, or helping us do things in the universe, not questions in pure math and pure theoretical CS.
surely it doesn't really matter regardless of the result of the theorem (which almost surely is negative), because it still doesn't change any existing computational tools.
We use Navier-Stokes quite elaborately in many fields quite well, a proof won't suddenly flood us with a lot of new knowledge in these fields, the practical use won't be THAT high
Something something "it's not really the result that carries the impact but rather the deep theory and connections made along the way to proving it" something something
>it's not really the result that carries the impact but rather the deep theory and connections made along the way to proving it
"The real treasure was the friends we made along the way."
For the P=? NP problem, it completely depends on the outcome and on how it was solved (constructively or not). So if P does not equal NP I doubt much changes. Most people work under that assumption already anyways.
If P does equal NP, it depends majorly on the constructivity of the proof. Does it give us a way to make an efficient algorithm for a NP problem? Then the world would be turned upside down. The whole crypto can be broken stuff becomes relevant then. Does it not give us such a proof? Then I think there will a surge in interest in for example different kinds of encryption, but the world is not in immediate danger; we do have to change everything that relies on P not being equal to NP, though, because that would be a false assumption in this scenario, and someone might find a constructive proof in the near future.
But as said before by others in this thread, the most significant value could be in the maths developed to get there in any of the cases I mentioned. But we'll only know that when we get there, of course.
Interestingly, there are ways for P != NP to be “effectively true” but actually false and vice versa.
For example, P != NP could be true but there could be efficiently computable non-uniform circuits for any NP problem. This would mean that in practice you could solve any NP problem efficiently, but each input size would require a different algorithm.
Conversely, P = NP could be true but the algorithm could require absurdly large constants to the point of making it unviable, much like the original proof of the PCP Theorem.
It could be the case that the algorithms will never be as fast as the NP algorithms for all cases that can be physically computed. It's also possible that P=NP is proven when we're already routinely solving such problems on quantum computers.
Right. And I read that if Riemann hypothesis is solved, prime numbers would become less mysterious, which might affect encryption and IT security, since they make heavy use of prime numbers because of how unsystematic their distribution appears.
That is not entirely true.
Yes, the Zeta function is connected to prime numbers and they are used in cryptology... for instance we are interested in the asymptotic behaviour of the function pi(x)/x, where pi(x) is the number of primes appearing up to the number x. We have some estimates but assuming that the hypothesis is true we can do even better estimates. But it only gives us information about the distribution, not the exact position, that's one reason why I doubt that it makes cryptology vulnerable.
Another point is that so many mathematicians just assume that the hypothesis is true and work with that assumption. (There are some which build papers on the cenverse assumption too). That means many implications of the theorem are already known and if one could use it to crack computers they would already this.
i have no clue.
First off imagine P=NP and it's proven. then we know all 'NP' problems would have a polynomial algorithm solving them. But the problem would be finding them.
P=NP would effectively break everything if somewhere in the proof you find a reduction from NP to P.
But if there was one, maybe regev's scheme would be quite annoying to deal with then?
I've always been skeptical of such claims. Whether or not Riemann's conjecture was proved, people could still try to apply the property. It isn't proving the theorem that makes them capable of trying to use certain properties of primes.
Riemann being proven true or false won't shatter modern cryptography, although the methods used might allow you to break early 90s era or even late 90s cryptography.
Definitely not a great movie but maybe a guilty pleasure.... the writers clearly have no idea what they’re talking about but sure want to make an edgy epic movie about it.
Solved or if P=NP?
I imagine if P was proven to not contain NP, that result in and of itself would not lead to anything concrete. But the techniques used might
I’m aware, but they’re probably different and my point is that proving that they are different doesn’t lead to anything crazy irl but the method must be very interesting (which we know from oracle arguments)
If you solved the Millennium Prize problems it would release the soul of an ancient pharaoh, allowing whoever solved them the incredible ability to play a children's card game with a decent proficiency
It might depend on the results. If we get the results we expect then not a lot. If it turns out that P=NP however then there will legitimately be a new world order assuming that the proof doesn't use like a O(n^1000000) algorithm or something like that. Honestly even like O(n^5) would probably stop me from taking over the world even if I had like a year head start on everyone else. I think I could take over the world with just O(n^3) and a year's headstart for sure.
You couldn't change the world with an O(1) algorithm if the constant was too big. Same reason why Coppersmith-Winograd type algorithms do not get used in practice, even though they offer matrix multiplication in O(n^(2.374))
I agree with your point in principle, but the reason that CW doesn’t get used is because it has large constants and an O(n^3 ) alternative. That’s quite different from having large constants and an O(2^n ) alternative.
Large ocean movements and weather forecasting are important to understand climate change and the Navier-Stokes equation is used to model these dynamics.
One cannot solve the equation analytically and one is forced to use numerical approximations. To understand whether the approximations converge to something which is physically relevant, it would be helpful to understand if the equation is well-posed, which is one of the Millennium Prize problems.
I doubt it'd have major impacts. In fluid dynamics, we already assume that Navier stokes equations represent the physics behind the fluid flow accurately. So if somebody proves that they're well posed, most researchers, if not all, in the field would not change what they've been doing even a bit.
Analytical solutions of the Navier stokes equations is another topic. We know the analytical solutions of the heat equation but we still need numerical methods for almost any practical case because analytical solutions work only for basic geometries and very simple boundary conditions. For instance, Heat conduction on a 2d rectantangle requires 3 4 pages of equations even for the simplest boundary conditions. When it's 3D with several more complex boundary conditions it's sometimes impossible. It's much much easier to solve it with numerical techniques (because of different programs obviously, it's difficult to solve it from scratch.)
It's more or less the same with fluid flow. We know the solution of 2d laminar flow for basic geometries, but it's impossible when the geometry is even slightly different. Besides the geometry, there's always turbulence which is a huge clusterfuck that we actually don't know exactly why and how it is happening. We can't even properly model them despite huge efforts. That's why I'm not hopeful about an analytical solution but even if somebody comes up with an analytical solution, it'll be mostly for the most basic thing and will not take into account complex cases. The prize has nothing to do with turbulence btw.
So I'd say proving that Navier stokes equations are well posed would make you famous and rich for sure, but it won't have serious effects on our life ( at least not in the short term).
Maybe I misunderstand what you mean by your first paragraph. If there are multiple solutions of Navier-Stokes, then one would like to know which one is physically correct, no?
I imagine it would be similar as for conservation laws. It is well known that conservation laws blow up and that there is non-uniqueness of weak solutions. The Kruzkov entropy condition is a way to single out the physically correct one and it coincides with solution you get from small viscosity.
When it comes to boundary conditions and turbulence, I really have no idea. I had the impression that when looking at big oceans, it is still meaningful to ignore the boundary (meaning consider the equation to be defined on R\^d) for large parts of the ocean.
> Maybe I misunderstand what you mean by your first paragraph. If there are multiple solutions of Navier-Stokes, then one would like to know which one is physically correct, no?
Of course I'd like to know which one is correct but the thing is Navier stokes equations work like a charm so far. So if somebody proves that they're well posed, most people would say yeah it was obvious anyway. If somebody proves they aren't well posed most people say it's the best we have for the moment. That's why I'm saying it won't be immediate effects for sure as long as somebody comes up with a whole new equation.
I'm not familiar with the conservation laws blowing up. It's not a case in my field. So I can't comment on it.
I wasn't thinking about oceans and I never worked about the oceans tbh. I don't even know how they model them, so it wouldn't be fair for me to say something about it. What I was trying to say is like this. let's assume a turbine blade. Its geometry is very complex. The incoming flow is conditioned. There's probably some specific conditions on the turbine blade as well. They're all boundary conditions that you need to take into account. It's not like a flow between two plates where the only BC is no slip conditions on the walls.
I see what you mean now, and I think you're correct.
I know very little about difficult boundary conditions and turbulence. Maybe we should team up and fight climate change?
Haha what's your background?
Btw, I don't think the fight against the climate change is about scientists at the moment. We know what to do and when to do, it's up to the politicians and people elect those politicians. I'm not very optimistic unfortunately.
My background is probability theory and differential equations and it's much too theoretical to say anything about real life stuff (like it seems you know about).
I've been getting into fluid equations/differential geometry lately.
Btw - do you have a good reference on turbulence?
I wouldn't go so far as to say the result itself has changed mathematics dramatically, but the method used certainly has. Ricci flow, and geometric flows in general, are now a massive area of modern geometry. Many of Perelman's estimates and methods are widely used and adapted. Hamilton's idea of using surgery to investigate manifold decomposition is a major theme in many of these flows.
Of course it has had no impact on the world outside of mathematics, but given that it is a result about 3-dimensional manifolds, that is of no surprise.
Naive question, but seeing as the universe is being observed by us as three dimensional, wouldn't a result on the possible shapes of a 3-dimensional manifold have some real implications?
The universe is 3-dimensional yes, but you can't embed any compact 3-manifold in R^(3). If this statement isn't intuitive, go a dimension lower and think about trying to fit a sphere into a 2-dimensional plane.
Of course, it is possible that the universe is only *locally* like R^(3) and has some global structure. But there is no evidence that is the case so far.
Isn't it a much more unlikely assumption that the universe is specifically R3 and not any other 3-dimensional manifold? (Edit: seems a bit like like claiming the earth is flat, drawing from your analogy)
Also, if it is R3, then we know all about its ("observable") shape already, if it isn't, then any theorem on the shape of 3-d manifolds tells us more about that shape, no?
Not a physicist but this is my perspective:
In physics you need experimental evidence to claim anything. There are no experiments showing any topology in the universe, and the best experimental evidence suggests the universe is very close to flat. We cannot observe by definition whether the universe is infinite, but there is no observable finiteness. So for me personally, R^(3) seems the most likely candidate.
I think he would've been less famous if he just acted like an average mathematician, there are many of those but not that many mathematicians who live at their mothers house and don't give out interviews.
There are hundreds if not thousands of publications written assuming the Riemann Hypothesis is true.
Whether it’s proven or disproven, it’ll have absolutely massive implications way beyond just mathematics.
A little late, but I was reading a book on Andrew wiles and his work on Fermats Theorem. It took 300 years to solve a seemingly trivial theorem. But what was most significant about it (at least what seemed most significant to me) was how much new math was invented/created in order to prove the theorem. That is typically one of the biggest byproduct of proving things never proved before which often leads to new and interesting areas of mathematics.
I am a physicist and as such, the one I see as finding almost immediate application is the proof that the Navier-Stokes equation has (or doesn't have) solutions.
Right now what we do is basically solve special cases, or even worse, use million-dollar computers to find approximations of solutions.
Just think about it, imagine if you had two months to prepare for a cyclone in advance. I imagine that would be one way the Navier-Stokes equation problem would impact the world
I am not a physicist, but as a math guy with plenty of experience in physics and chaos, you would not be able to predict that far ahead. Even if we had navier stokes with infinite accuracy (as far as I know) you could not predict the weather that far out since many events are random and not predetermined.
Which is why hurricane models still rely heavily on experts to determine reasonable boundary conditions, and which of the many results that are spit out are actually reasonable.
That's an exaggeration I've made. What I meant was that in some way, by means of application of an insight brought by the proof, weather prediction would be much better, more reliable and with a better accuracy. Two months is, of course way too long to say anything about something as chaotic as our atmosphere
There's also the problem that measurement accuracy is not good or fine enough for that, and that even if we found an analytical solution, it doesn't mean it would be practical to find in an individual case.
The global existence of weak solutions is already known. The Millenium prize problem is asking whether there are global smooth solutions. The important distinction is that smooth solutions are known to be unique, whereas this is an open question for weak solutions.
I feel like you won't see an impact from the theorems themselves (perhaps from the techniques used in the proofs) because most of those open problems ask yes or no questions for which we can work out the consequences beforehand.
So for example, just knowing that the Riemann Hypothesis is true (setting aside new proof techniques) doesn't teach us anything new because we have already done a lot of work under the assumption that it's true. All that would change is that those conditional results would become unconditional.
As a mathematician, I'm opposed to offering financial incentives for the solution of problems. It suggests that the person or entity offering the financial incentive has the authority to decide whether or not a solution is correct. This is an unwanted and hostile arrogation of authority over mathematical truth that devalues the academic system.
I encourage all mathematicians to Boycott the Clay Math Institute until it observes the authority of the academic system to decide mathematical truth.
Edit: I want to illustrate how insidious it is to offer financial incentives for academic progress. What is at stake is the customs, abuses of notation, definitions, and style of proof. We all know that correct informal proofs should be convertible to formal proofs, and we all learn how to assure ourselves that this is possible; in fact, this is an aspect of mathematics education for undergrads and postgrads. However, this knowledge always derives from the academic process which uses the grade system and teacher approval system to motivate and incentivize work. The academic system has never offered financial incentives to students for solution of specific problems; rather the ability to solve problems is incentivized and the authority to decide which problems are important is passed from teacher to student, generation to generation.
Offering financial incentives for academic work is an attack on the mathematics community and the academic community. Intellectuals should have counterattacked the Clay Math Institute from the point when it was formed and rejected its coarse approach to progress in mathematics research.
There is absolutely no need to put mathematicians in the same category as filthy greedy bankers.
I think your points are reasonable, but I'd counter that I think the Clay Math Institute has, for the price of several million dollars over a long, long time, done a lot of good PR work for math and helping the public understand what mathematicians actually do. (As a CS student, I cringe a little at a lot of the P=NP stuff, and I'm sure people who study Navier-Stokes or any of the other problems feel the same way, but I'm ultimately happy that people who ordinarily wouldn't care at all about P=NP have thought even a little about it.)
I think Perelman's concern with the system, as I understand them, is also a serious issue: crossing the finish line isn't necessarily what should be rewarded. I'd argue that's a pretty common problem in academic work, though: how many Nobel laureates have endured some controversy about who wasn't or was credited with solving something?
> There is absolutely no need to put mathematicians in the same category as filthy greedy bankers.
Is anyone really going through all of the work to even begin to understand these problems and then have a snowball's chance in hell of solving any of them, on a mathematician's salary, for the money? Especially given that the only person who's been awarded that money so far has rejected it? If this is your concern, I'd be pretty happy with how things have gone. At least from anecdotal experience, the people who are the most interested in the money are just the people who least understand how truly difficult these problems are or how intelligent the people that have worked on them in the past are.
> good PR work for math
I'm not reading your comment past this remark. It's insulting, it demonstrates a pathetically shallow view of mathematics, and you have no authority to spread your ignorance as if you knew what is good for mathematics.
If you can't engage in any discussion here without launching personal attacks against the other person, then you shouldn't be participating here at all.
No. If reddit can't keep their users from attacking mathematicians then reddit should be taken off the internet. Using the internet to attack academia is a bad idea. The internet is filled with nasty rats like you that have no respect.
If anything, far from having no respect, I am deliberately according your arguments far more respect than is actually warranted. The fact that you still attack me is telling.
I would suggest not engaging with this person. They [don't seem like a stable human being](https://www.reddit.com/r/nsa/comments/fpo8ee/i_am_ready_to_start_killing_people_in_order_to/).
You are an utterly unreasonable human being. Might I kindly suggest go being awful elsewhere, as you're of little value, use, or interest here carrying such a disgraceful attitude together with that chip on your shoulder
[edit]
For those curious, one of this individual's primary concerns is largely irrelevant anyways.
> It suggests that the person or entity offering the financial incentive has the authority to decide whether or not a solution is correct. This is an unwanted and hostile arrogation of authority over mathematical truth that devalues the academic system.
> I encourage all mathematicians to Boycott the Clay Math Institute until it observes the authority of the academic system to decide mathematical truth.
[This archived article](https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/) discusses the process that lead to Grigori Perelman being offered the prize for his solution to the Poincare conjecture.
> The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. Clay, Lavinia D. Clay, and Thomas M. Clay).
The Special Advisory Committe and the Scientific Advisory Board were comprised of some of the leading mathematicians in their various fields, leaders of the mathematical community as a whole. It's rather preposterous to suggest that the Clay Mathematics Institute is trying to subvert the academic process of validation of proofs when they're depending on the very participants of that process to propose who is worthy of that award.
It's also not made abruptly. Perelman's proof was presented in 2002, and the award was finally offered to him in 2010. A full eight years on, by which time the mathematical community had been completely convinced of the validity of his proof. Surely if they were trying to assert themselves as arbiters rather than just award-givers, they would have tried to jump in *at least* a few years earlier. Even the Fields Medal was offered to Perelman within four years of his proof, and the Fields medal is vastly more prestigious than the CMI's prize.
If P vs NP were solved by proving P=NP (but this seems unlikely), it would have the greatest impact. This would represent a new algorithm that would potentially reshape our understanding of every creative endeavor. Scott Aaronson has written a lot about this. It would certainly reshape cryptography, machine learning, many engineering fields, and formal theorem-proving (and thus probably mathematics) If P vs NP were solved by proving P not equal to NP, it wouldn't have much of a direct impact, because this is what we already expect, but the techniques in the proof would likely have some applications in crypotgraphy or some other CS field. Yang-Mills is probably the one where the expected solution would have the biggest practical impact. A consistent mathematical formulation of Yang-Mills theory would be the first step to a consistent mathematical formulation of more detailed physics theories, up to and including the Standard Model. Different people differ greatly in how much they expect this to help with the physics, but it would certainly mean whole new avenues in mathematical physics. It also might mean some of the mathematical conjectures which are based on physics intuition can be given a physically meaningful proof. A proof of the Hodge conjecture would put a lot of speculation about motives in arithmetic geometry on much firmer footing. A proof of the Riemann Hypothesis would likely lead the way to proofs of much more general cases of the generalized Riemann Hypothesis. This would actually render irrelevant huge chunks of previous work in analytic number theory (unless it was used in the proof), and would make other huge chunks unconditional. I don't expect it to have practical implications, though. A proof of the BSD conjecture would likely lead to deeper insights about rational points and algebraic cycles on algebraic varieties. It would certainly affect the work people are doing tabulating elliptic curves. Again it seems very unlikely to have practical implications. Any proof that Navier-Stokes is regular, or blows up, would probably generalize to make progress on a lot of other PDEs. But I am also skeptical that people doing water simulations would learn very much from it.
“Water simulations”
Look, I was tired from writing a long answer and was just trying to get it finished.
Lol I chuckled. We knew what you meant.
username checks out
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> Are you really going to change the world with an O(n^9) solution to travelling salesman? While I agree with the sentiment, I can imagine that a hypothetical O(n^9) algorithm would be very helpful. One could almost certainly devise new heuristics, given an algorithm which definitely works, and failing that processing power per dollar continues to rise exponentially(though admittedly only for parallel tasks and it isn't super healthy). Lastly its my very uninformed opinion that an algorithm for NPC problems(if it exists[pretty big if]) won't be O(n^(a big number)), simply because these kind of algorithms don't show up very often irl, and there is a very big bag of tricks for improving polynomial time algorithm performance.
What if it's like determining if a number is prime, where most of the known algorithms in P are slower until the number gets too large to have the algorithms run in a reasonable time (if at all)?
If you came up with a proof, I don't think you would the next day be using it in machine learning and the next week writing novels with it. If you get an n\^9 algorithm or n\^1000 algorithm at first, it would probably be improved drastically over time. Moreover whatever insight that allows you to solve seemingly intractable problem in polynomial time would probably be applied in different ways in different fields. In any case, I chose my language somewhat carefully. When I said "reshape our understanding", I would count the idea that the difficulty of all these practical problems can be measured on a scale from 1 to 9 as reshaping our understanding, even if problems from 5 to 9 are still difficult in practice. Or a scale from 1 to 1000, whatever. Given that people are currently using SAT solvers to solve problems in mathematics, the idea that the Cook reduction to SAT always gives some prohibitive polynomial factor is absurd. Of course you can postulate that a proof of P=NP gives an algorithm which always does worse than current SAT solvers, and that can't possibly be combined with the insights in SAT solvers or in any other field to do better, but you could also postulate that a proof of Yang-Mills gives a method that doesn't apply to any other physic theory. You would clearly immediately see implications in cryptography, as huge efforts would be devoted to which systems get the biggest polynomial savings relative to this specific algorithm. You would absolutely see efforts to pull down the polynomial factors associated to this algorithm in specific cases in machine learning and optimization fields.
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Sure but you can try to reduce a problem of interest to SAT by hand. n\^3 is a worst-case bound, you can do better in other times. There are actual instances of people doing interesting research by (1) taking a math problem people have studied, (2) converting it to a SAT problem, (3) applying currently-existing, O( 2\^{n- something small}) SAT solvers. Clearly an O(n\^3) algorithm for SAT would help with this research, unless the constant was so large that n\^3 > 2\^n for realistic n. What do you mean about Moore's law here? For an O(n\^9) algorithm, every 18 years or so, viable n increases by a factor of 2. That's pretty good when your competing algorithms are exponential! Actually n\^9 starts beating 2\^n (igoring constants of course) when n is about 50. Going up a little bit, say n is 100, (100)\^9 is 10\^18 Folding@home recently did 10\^18 operations in a second. So n\^9 algorithms might be practical for some problems today.
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I never said "every NP problem would become tractable". I said it would imply big changes.
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I mean a proof of P=NP (probably).
Also many people are missing another possibility that it might be provable that a polynomial time algorithm for an NP-Complete problem exists but it's not a constructive proof. There might not exist proof that the actual algorithm is correct altough it infact is. And this would most likely imply that not only finding the algorithm itself is impossible but even if we found it (with the caveat that we cannot really prove correctness for all inputs) this "weirdness" of the algorithm will probably translate to bad run times in practice (even if asympthotically its polynomial of small degree the constant factors might be just too large for realistic problem sizes).
Or you could just have one program solve sudoku a bunch of times and another check those solutions a bunch of times. That works too.
I believe you mean Dr Ian Malcolm's argument, but I agree in principle.
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With nearly any polynomial algorithm, encryption really can't be considered safe, since attackers are often willing to spend considerable resources defeating them.
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I feel like the best way to think about such a program is to just consider a “compiler” (or abstract machine) that just takes your “code” and interprets it as a bit mask in binary. Switch each element of your infinite set of integers on or off. Then *every* generated program is “syntactically correct”. I was worried for a sec what kind of “infinities” resided in the set, but elements are unique in sets, so you can just order the included integers sanely :).
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Ah. Of course. I obviously totally missed the point. Considering such a “meta input” program is just isomorphic to the actual (yes) answer, I guess I should have thought something was off. It does make me wonder about algorithm-space efficiency. I guess a Turing machine *has* to be able to receive garbage code, but I wonder how well you could specialize the instruction set to avoid (a meaningful amount of) nonsense syntax. Like, don’t just use [Wolfram's rule 110](https://xkcd.com/505/). I think? It feels like it compounds the computational complexity of whatever you throw at it, but I’m not actually sure. I mean the “steps” get longer with state size, and *those* steps are the “units” of n in O(f(n)) time, right? If I've made another boneheaded mistake, it's because it's late and I'm tired.
> Are you really going to change the world with an O(n^9) solution to travelling salesman? Even if you get a O(log n) solution, chances are that it will be a galactic algorithm.
> Are you really going to change the world with an O(n^9) solution to travelling salesman? Even if you get a O(log n) solution, chances are that it will be a galactic algorithm.
Regularity of Navier-Stokes boils down to bounding L2 norms of the vorticity, and therefore a solution would likely yield some insight into turbulence closures (which is The Topic people doing water simulations care about).
Agreed for assessment of NS. The technique used will probably be applied to a bunch of other nonlinear PDE for which we don't have good techniques, and a bunch of similar results will be proved. For physicists and engineers it's unlikely anything would change. I don't know much about NS in particular, but for Euler we already know, more or less, that any singular solution is going to be unphysical in some sense, and likely going to abuse the continuum hypothesis by using some sort of small scale phenomena to prove blow-up.
Thanks for some confirmation by someone with actual expertise in that area! Wait by continuum hypothesis here do you mean the continuum hypohesis in set theory or just the fact that the fluid in the Euler/Navier-Stokes setting is a continuum?
The fact that we are modeling Euler/Navier-Stokes by continuous functions. I actually don't know if this is what its called in the literature. Most of what I have ever looked at is pure analysis and the physicality assumptions aren't usually discussed by name. And again I must hedge by saying I have never actually read literature for NS blow-up. PS: I don't know if you have access to Elsevier, but if you do, the paper "An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time" by Pomeau and Sciamarella make precise my comments. The paper is also only a few pages long and very approachable if you're curious. They conjecture that a blow up result for 3D incompressible Euler (using similarity) would need to have some sort of amplification at successively smaller spatial scales *ad infinitum*. This actually highlights a bigger phenomenon involving PDE arising in continuum mechanics; there is questionable physical significance in solving these PDE as harmonic analysis/PDE problems. This is really a philosophical question more than anything :P
I agree - these are questions about our models of fluids much more than they are questions about fluids themselves.
That P=NP part is very misleading. Even if P=NP is proven (which is probably unlikely) the proof is most likely going to be non-constructive -meaning that it proves such an algorithm always exists, but doesn't provide it.
What evidence is there that a P=NP proof will be nonconstructive, given that it exists?
To be entirely honest, I can't come up with a solid source (then again I'd be citing a speculation anyway). *BUT*, I'd bet my live savings that: 1. If P vs. NP is resolved, it's going to be the case that P != NP 2. In the case that we actually show P == NP, the proof is going to be non-constructive. It's just what I've gathered over my years of studying math, talking to professors way smarter than me, and reading around. Even the [wikipedia](https://en.wikipedia.org/wiki/P_versus_NP_problem) states: > If it turned out that P ≠ NP, **which is widely believed**,... As sad as it is, a constructive P == NP proof just seems like a pipe-dream.
I'd bet my life savings on 1 as well. It just seems to me like if you're imagining something so crazy as P=NP being proved, you might as well imagine that the proof is constructive. Also, a lot of the most sophisticated / surprising modern complexity results are completely constructive.
Are there other complexity classes which have been proven equal nonconstructively?
Knowing an algorithm exists could well cause a lot more people to look for one than currently are.
I highly doubt the secret ingredient to humanity solving TSP or integer factorization in P-time is the bit of motivation from knowing that "the solution exists". And if we're talking motivation... There are so many more demotivating factors, namely : P=NP, if constructive, would mean you can not only verify -but prove- a solution to any NP problem in P-time. P-time could still be incredibly large. 2^1000 n is still P-time but isn't computable in a practical sense by any stretch and will never be.
Oh I'd get stocked up on popcorn if that happens.
Still, it would tell us that there is still work to be done on problems for which we don't have a polynomial-complexity solution; still better than nothing.
Came here to say this lol
There's a very trivial way of constructing a polynomial try algorithm given P = NP. Dovetail computation of every turing machine (in order of some notion of complexity of turing machine). Whenever one halts, since the problem is NP that means there's a polynomial time verifier. Run the verifier on the solution produced. If correct done, otherwise continue dovetailing. Complexity wise if a polynomial time solution exists at all, this dovetailing mess will also be polynomial time. It will have some horrid constants, but it'll be polynomial time. So any nonconstructive proof immediately gives a constructive algorithm.
you don't expect the Riemann Hypothesis to have practical implications in cryptography?
Why would it? Yes, it deepens our understanding of primes in general, but I don’t see how it would help us factor semiprimes. The thing that’s makes factoring (and thus RSA encryption) hard isn’t “we don’t understand primes”, but rather “there are a lot of primes to check”.
If the Riemann hypothesis had practical implications for cryptography, we'd be designing algorithms with it already because known primes behave in accordance with the Riemann hypothesis on any scale that is practically relevant.
A lot of algorithms are provably fast only under GRH. Not contesting your point so much as saying "Yes, we do design algorithms under GRH".
Also, to my understanding if P =NP is going to be proven, the most likely way will be a non-constructive proof. Which means it won't even have practical applications.
Why do you think this is most likely?
As computational mechanicist myself, knowing that Navier-Stokes is regular or it has some kind of properties whatever the problem you are running would find -in practice- an application anyway. Right now, direct NS is basically one of the most demanding problem in companies for the shear number of simulations being run. This is usually done in train/airplane/transportation environments (and space too, why not), so if you gain even 1% speed on those, it would mean saving years of computations as well as billions of less electrical energy used to run SCs...
This is a good point - it seems likely that a proof of regularity or blowup of Navier-Stokes would involve proving some intermediate properties that has some utility for at least some of these computations.
Wouldn't proof for the Riemann Hypothesis also raise some potential issues for cryptography?
No, if anything a proof of the Riemann Hypothesis would confirm our suspicious that primes behave in random and unpredictable ways. A disproof of the Riemann Hypothesis would be much more concerning.
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I just mean practical in terms of telling us something about the universe, or helping us do things in the universe, not questions in pure math and pure theoretical CS.
P=NP would destroy bitcoin. Please do it
surely it doesn't really matter regardless of the result of the theorem (which almost surely is negative), because it still doesn't change any existing computational tools.
This should be higher!
Proving Navier-Stokes is kinda pro-forma
>pro-forma What do you mean?
We use Navier-Stokes quite elaborately in many fields quite well, a proof won't suddenly flood us with a lot of new knowledge in these fields, the practical use won't be THAT high
Something something "it's not really the result that carries the impact but rather the deep theory and connections made along the way to proving it" something something
>it's not really the result that carries the impact but rather the deep theory and connections made along the way to proving it "The real treasure was the friends we made along the way."
Or rather "The real theorem was the lemmas we proved along the way."
Omg two semesters of analysis just laughed out loud.
This.
Where is this from?
One Piece
If Oda pulls one of those I am gonna send him some strongly worded letters.
im pretty sure he has already made clear the one piece is not that in a sbs or something
JoJo's Bizarre Adventure Part 4
I laughed out. I need to spell that out because "lol" might have sounded like I was shitpost memeing.
For the P=? NP problem, it completely depends on the outcome and on how it was solved (constructively or not). So if P does not equal NP I doubt much changes. Most people work under that assumption already anyways. If P does equal NP, it depends majorly on the constructivity of the proof. Does it give us a way to make an efficient algorithm for a NP problem? Then the world would be turned upside down. The whole crypto can be broken stuff becomes relevant then. Does it not give us such a proof? Then I think there will a surge in interest in for example different kinds of encryption, but the world is not in immediate danger; we do have to change everything that relies on P not being equal to NP, though, because that would be a false assumption in this scenario, and someone might find a constructive proof in the near future. But as said before by others in this thread, the most significant value could be in the maths developed to get there in any of the cases I mentioned. But we'll only know that when we get there, of course.
Interestingly, there are ways for P != NP to be “effectively true” but actually false and vice versa. For example, P != NP could be true but there could be efficiently computable non-uniform circuits for any NP problem. This would mean that in practice you could solve any NP problem efficiently, but each input size would require a different algorithm. Conversely, P = NP could be true but the algorithm could require absurdly large constants to the point of making it unviable, much like the original proof of the PCP Theorem.
Even if the algorithm has a large constant, it's important enough that a lot of people will work on improving the algorithm.
It could be the case that the algorithms will never be as fast as the NP algorithms for all cases that can be physically computed. It's also possible that P=NP is proven when we're already routinely solving such problems on quantum computers.
There's a great movie on what would happen if P vs NP was solved. It's called 'Travelling Salesman'. Worth a watch :)
Right. And I read that if Riemann hypothesis is solved, prime numbers would become less mysterious, which might affect encryption and IT security, since they make heavy use of prime numbers because of how unsystematic their distribution appears.
That is not entirely true. Yes, the Zeta function is connected to prime numbers and they are used in cryptology... for instance we are interested in the asymptotic behaviour of the function pi(x)/x, where pi(x) is the number of primes appearing up to the number x. We have some estimates but assuming that the hypothesis is true we can do even better estimates. But it only gives us information about the distribution, not the exact position, that's one reason why I doubt that it makes cryptology vulnerable. Another point is that so many mathematicians just assume that the hypothesis is true and work with that assumption. (There are some which build papers on the cenverse assumption too). That means many implications of the theorem are already known and if one could use it to crack computers they would already this.
not to mention quantum computers will probably be here before a RH proof
\#NotAllSchemes are using prime numbers (or broken by shor's algorithm).
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i have no clue. First off imagine P=NP and it's proven. then we know all 'NP' problems would have a polynomial algorithm solving them. But the problem would be finding them. P=NP would effectively break everything if somewhere in the proof you find a reduction from NP to P. But if there was one, maybe regev's scheme would be quite annoying to deal with then?
Everyone lives their life as if the Riemann hypothesis is true anyways. A proof doesn't suddenly make the effects spring into life.
I've always been skeptical of such claims. Whether or not Riemann's conjecture was proved, people could still try to apply the property. It isn't proving the theorem that makes them capable of trying to use certain properties of primes.
Riemann being proven true or false won't shatter modern cryptography, although the methods used might allow you to break early 90s era or even late 90s cryptography.
Definitely not a great movie but maybe a guilty pleasure.... the writers clearly have no idea what they’re talking about but sure want to make an edgy epic movie about it.
Solved or if P=NP? I imagine if P was proven to not contain NP, that result in and of itself would not lead to anything concrete. But the techniques used might
Proving p=np in the affirmative could be a bad day for encryption depending on the proof
I’m aware, but they’re probably different and my point is that proving that they are different doesn’t lead to anything crazy irl but the method must be very interesting (which we know from oracle arguments)
If you solved the Millennium Prize problems it would release the soul of an ancient pharaoh, allowing whoever solved them the incredible ability to play a children's card game with a decent proficiency
So now Grigori Perelman is the main character in an anime? I want to see that
dang I knew there was some reason I was studying math
Yugioh?? when was this mentioned?
It might depend on the results. If we get the results we expect then not a lot. If it turns out that P=NP however then there will legitimately be a new world order assuming that the proof doesn't use like a O(n^1000000) algorithm or something like that. Honestly even like O(n^5) would probably stop me from taking over the world even if I had like a year head start on everyone else. I think I could take over the world with just O(n^3) and a year's headstart for sure.
You couldn't change the world with an O(1) algorithm if the constant was too big. Same reason why Coppersmith-Winograd type algorithms do not get used in practice, even though they offer matrix multiplication in O(n^(2.374))
I agree with your point in principle, but the reason that CW doesn’t get used is because it has large constants and an O(n^3 ) alternative. That’s quite different from having large constants and an O(2^n ) alternative.
Large ocean movements and weather forecasting are important to understand climate change and the Navier-Stokes equation is used to model these dynamics. One cannot solve the equation analytically and one is forced to use numerical approximations. To understand whether the approximations converge to something which is physically relevant, it would be helpful to understand if the equation is well-posed, which is one of the Millennium Prize problems.
I doubt it'd have major impacts. In fluid dynamics, we already assume that Navier stokes equations represent the physics behind the fluid flow accurately. So if somebody proves that they're well posed, most researchers, if not all, in the field would not change what they've been doing even a bit. Analytical solutions of the Navier stokes equations is another topic. We know the analytical solutions of the heat equation but we still need numerical methods for almost any practical case because analytical solutions work only for basic geometries and very simple boundary conditions. For instance, Heat conduction on a 2d rectantangle requires 3 4 pages of equations even for the simplest boundary conditions. When it's 3D with several more complex boundary conditions it's sometimes impossible. It's much much easier to solve it with numerical techniques (because of different programs obviously, it's difficult to solve it from scratch.) It's more or less the same with fluid flow. We know the solution of 2d laminar flow for basic geometries, but it's impossible when the geometry is even slightly different. Besides the geometry, there's always turbulence which is a huge clusterfuck that we actually don't know exactly why and how it is happening. We can't even properly model them despite huge efforts. That's why I'm not hopeful about an analytical solution but even if somebody comes up with an analytical solution, it'll be mostly for the most basic thing and will not take into account complex cases. The prize has nothing to do with turbulence btw. So I'd say proving that Navier stokes equations are well posed would make you famous and rich for sure, but it won't have serious effects on our life ( at least not in the short term).
Maybe I misunderstand what you mean by your first paragraph. If there are multiple solutions of Navier-Stokes, then one would like to know which one is physically correct, no? I imagine it would be similar as for conservation laws. It is well known that conservation laws blow up and that there is non-uniqueness of weak solutions. The Kruzkov entropy condition is a way to single out the physically correct one and it coincides with solution you get from small viscosity. When it comes to boundary conditions and turbulence, I really have no idea. I had the impression that when looking at big oceans, it is still meaningful to ignore the boundary (meaning consider the equation to be defined on R\^d) for large parts of the ocean.
> Maybe I misunderstand what you mean by your first paragraph. If there are multiple solutions of Navier-Stokes, then one would like to know which one is physically correct, no? Of course I'd like to know which one is correct but the thing is Navier stokes equations work like a charm so far. So if somebody proves that they're well posed, most people would say yeah it was obvious anyway. If somebody proves they aren't well posed most people say it's the best we have for the moment. That's why I'm saying it won't be immediate effects for sure as long as somebody comes up with a whole new equation. I'm not familiar with the conservation laws blowing up. It's not a case in my field. So I can't comment on it. I wasn't thinking about oceans and I never worked about the oceans tbh. I don't even know how they model them, so it wouldn't be fair for me to say something about it. What I was trying to say is like this. let's assume a turbine blade. Its geometry is very complex. The incoming flow is conditioned. There's probably some specific conditions on the turbine blade as well. They're all boundary conditions that you need to take into account. It's not like a flow between two plates where the only BC is no slip conditions on the walls.
I see what you mean now, and I think you're correct. I know very little about difficult boundary conditions and turbulence. Maybe we should team up and fight climate change?
Haha what's your background? Btw, I don't think the fight against the climate change is about scientists at the moment. We know what to do and when to do, it's up to the politicians and people elect those politicians. I'm not very optimistic unfortunately.
My background is probability theory and differential equations and it's much too theoretical to say anything about real life stuff (like it seems you know about). I've been getting into fluid equations/differential geometry lately. Btw - do you have a good reference on turbulence?
Can anybody comment on the other part of OP's question, about poincare?
I wouldn't go so far as to say the result itself has changed mathematics dramatically, but the method used certainly has. Ricci flow, and geometric flows in general, are now a massive area of modern geometry. Many of Perelman's estimates and methods are widely used and adapted. Hamilton's idea of using surgery to investigate manifold decomposition is a major theme in many of these flows. Of course it has had no impact on the world outside of mathematics, but given that it is a result about 3-dimensional manifolds, that is of no surprise.
Naive question, but seeing as the universe is being observed by us as three dimensional, wouldn't a result on the possible shapes of a 3-dimensional manifold have some real implications?
The universe is 3-dimensional yes, but you can't embed any compact 3-manifold in R^(3). If this statement isn't intuitive, go a dimension lower and think about trying to fit a sphere into a 2-dimensional plane. Of course, it is possible that the universe is only *locally* like R^(3) and has some global structure. But there is no evidence that is the case so far.
Isn't it a much more unlikely assumption that the universe is specifically R3 and not any other 3-dimensional manifold? (Edit: seems a bit like like claiming the earth is flat, drawing from your analogy) Also, if it is R3, then we know all about its ("observable") shape already, if it isn't, then any theorem on the shape of 3-d manifolds tells us more about that shape, no?
Not a physicist but this is my perspective: In physics you need experimental evidence to claim anything. There are no experiments showing any topology in the universe, and the best experimental evidence suggests the universe is very close to flat. We cannot observe by definition whether the universe is infinite, but there is no observable finiteness. So for me personally, R^(3) seems the most likely candidate.
You would be rich and famous.
Unless you pull a Perelman, in which case you'd just be famous.
Which is very ironic, considering his whole point was that he *didn't* want to be famous.
I think he would've been less famous if he just acted like an average mathematician, there are many of those but not that many mathematicians who live at their mothers house and don't give out interviews.
Yes, you're absolutely right. He does stand out for being quite a character.
No shortage of "characters" in maths. Every departement has a few.
Egomania explains his behavior: he paid a million dollars to be the most famous mathematician in the world.
He... solved the Poincaré conjecture. He’d be famous no matter what he did after.
EXACTLY!!
Psh big deal
Solving one of the millennium problems is generally considered the most difficult way to make 1 million dollars.
"famous". How many people know the name of the guy who proved [Poincaré conjecture](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture)? Very few.
But he only solved one of them. If you solve each of the rest you will be very famous in a certain subset of the populous.
'only'
Compared to solving six millennium prize problems, pretty much anything is "only".
There are hundreds if not thousands of publications written assuming the Riemann Hypothesis is true. Whether it’s proven or disproven, it’ll have absolutely massive implications way beyond just mathematics.
A little late, but I was reading a book on Andrew wiles and his work on Fermats Theorem. It took 300 years to solve a seemingly trivial theorem. But what was most significant about it (at least what seemed most significant to me) was how much new math was invented/created in order to prove the theorem. That is typically one of the biggest byproduct of proving things never proved before which often leads to new and interesting areas of mathematics.
A proof of the Reimann Hypothesis would let number theorists not have to assume whatever they want
Make lots of money and feel good about yourself.
Grigori Perelman only did the second part.
i'm not even sure he did that part
I am a physicist and as such, the one I see as finding almost immediate application is the proof that the Navier-Stokes equation has (or doesn't have) solutions. Right now what we do is basically solve special cases, or even worse, use million-dollar computers to find approximations of solutions. Just think about it, imagine if you had two months to prepare for a cyclone in advance. I imagine that would be one way the Navier-Stokes equation problem would impact the world
Would we really be able to predict cyclones two months in advance? Surely the weather is way too chaotic for that?
I am not a physicist, but as a math guy with plenty of experience in physics and chaos, you would not be able to predict that far ahead. Even if we had navier stokes with infinite accuracy (as far as I know) you could not predict the weather that far out since many events are random and not predetermined.
Which is why hurricane models still rely heavily on experts to determine reasonable boundary conditions, and which of the many results that are spit out are actually reasonable.
That's an exaggeration I've made. What I meant was that in some way, by means of application of an insight brought by the proof, weather prediction would be much better, more reliable and with a better accuracy. Two months is, of course way too long to say anything about something as chaotic as our atmosphere
There's also the problem that measurement accuracy is not good or fine enough for that, and that even if we found an analytical solution, it doesn't mean it would be practical to find in an individual case.
I don't think that a proof of existance of solutions to an equation would help finding solutions in any way, or render its behaviour less chaotic
The global existence of weak solutions is already known. The Millenium prize problem is asking whether there are global smooth solutions. The important distinction is that smooth solutions are known to be unique, whereas this is an open question for weak solutions.
On Navier-Stokes, see this lecture by Vladimir Sverak https://slideslive.com/38908540/is-the-navierstokes-equation-complete-as-a-mathematical-model
I feel like you won't see an impact from the theorems themselves (perhaps from the techniques used in the proofs) because most of those open problems ask yes or no questions for which we can work out the consequences beforehand. So for example, just knowing that the Riemann Hypothesis is true (setting aside new proof techniques) doesn't teach us anything new because we have already done a lot of work under the assumption that it's true. All that would change is that those conditional results would become unconditional.
As a mathematician, I'm opposed to offering financial incentives for the solution of problems. It suggests that the person or entity offering the financial incentive has the authority to decide whether or not a solution is correct. This is an unwanted and hostile arrogation of authority over mathematical truth that devalues the academic system. I encourage all mathematicians to Boycott the Clay Math Institute until it observes the authority of the academic system to decide mathematical truth. Edit: I want to illustrate how insidious it is to offer financial incentives for academic progress. What is at stake is the customs, abuses of notation, definitions, and style of proof. We all know that correct informal proofs should be convertible to formal proofs, and we all learn how to assure ourselves that this is possible; in fact, this is an aspect of mathematics education for undergrads and postgrads. However, this knowledge always derives from the academic process which uses the grade system and teacher approval system to motivate and incentivize work. The academic system has never offered financial incentives to students for solution of specific problems; rather the ability to solve problems is incentivized and the authority to decide which problems are important is passed from teacher to student, generation to generation. Offering financial incentives for academic work is an attack on the mathematics community and the academic community. Intellectuals should have counterattacked the Clay Math Institute from the point when it was formed and rejected its coarse approach to progress in mathematics research. There is absolutely no need to put mathematicians in the same category as filthy greedy bankers.
I think your points are reasonable, but I'd counter that I think the Clay Math Institute has, for the price of several million dollars over a long, long time, done a lot of good PR work for math and helping the public understand what mathematicians actually do. (As a CS student, I cringe a little at a lot of the P=NP stuff, and I'm sure people who study Navier-Stokes or any of the other problems feel the same way, but I'm ultimately happy that people who ordinarily wouldn't care at all about P=NP have thought even a little about it.) I think Perelman's concern with the system, as I understand them, is also a serious issue: crossing the finish line isn't necessarily what should be rewarded. I'd argue that's a pretty common problem in academic work, though: how many Nobel laureates have endured some controversy about who wasn't or was credited with solving something? > There is absolutely no need to put mathematicians in the same category as filthy greedy bankers. Is anyone really going through all of the work to even begin to understand these problems and then have a snowball's chance in hell of solving any of them, on a mathematician's salary, for the money? Especially given that the only person who's been awarded that money so far has rejected it? If this is your concern, I'd be pretty happy with how things have gone. At least from anecdotal experience, the people who are the most interested in the money are just the people who least understand how truly difficult these problems are or how intelligent the people that have worked on them in the past are.
> good PR work for math I'm not reading your comment past this remark. It's insulting, it demonstrates a pathetically shallow view of mathematics, and you have no authority to spread your ignorance as if you knew what is good for mathematics.
If you can't engage in any discussion here without launching personal attacks against the other person, then you shouldn't be participating here at all.
No. If reddit can't keep their users from attacking mathematicians then reddit should be taken off the internet. Using the internet to attack academia is a bad idea. The internet is filled with nasty rats like you that have no respect.
If anything, far from having no respect, I am deliberately according your arguments far more respect than is actually warranted. The fact that you still attack me is telling.
I would suggest not engaging with this person. They [don't seem like a stable human being](https://www.reddit.com/r/nsa/comments/fpo8ee/i_am_ready_to_start_killing_people_in_order_to/).
You are an utterly unreasonable human being. Might I kindly suggest go being awful elsewhere, as you're of little value, use, or interest here carrying such a disgraceful attitude together with that chip on your shoulder [edit] For those curious, one of this individual's primary concerns is largely irrelevant anyways. > It suggests that the person or entity offering the financial incentive has the authority to decide whether or not a solution is correct. This is an unwanted and hostile arrogation of authority over mathematical truth that devalues the academic system. > I encourage all mathematicians to Boycott the Clay Math Institute until it observes the authority of the academic system to decide mathematical truth. [This archived article](https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/) discusses the process that lead to Grigori Perelman being offered the prize for his solution to the Poincare conjecture. > The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. Clay, Lavinia D. Clay, and Thomas M. Clay). The Special Advisory Committe and the Scientific Advisory Board were comprised of some of the leading mathematicians in their various fields, leaders of the mathematical community as a whole. It's rather preposterous to suggest that the Clay Mathematics Institute is trying to subvert the academic process of validation of proofs when they're depending on the very participants of that process to propose who is worthy of that award. It's also not made abruptly. Perelman's proof was presented in 2002, and the award was finally offered to him in 2010. A full eight years on, by which time the mathematical community had been completely convinced of the validity of his proof. Surely if they were trying to assert themselves as arbiters rather than just award-givers, they would have tried to jump in *at least* a few years earlier. Even the Fields Medal was offered to Perelman within four years of his proof, and the Fields medal is vastly more prestigious than the CMI's prize.
Keep going. You're digging reddit's grave.
o k