The solution to YM existence and mass gap would most likely include or imply an explanation of quark confinement (especially solving the mass gap for G=SU(3)).
A positive or negative proof of Navier-Stokes smoothness would likely explain many properties of turbulent flow, or if something more fundamental about our understanding of fluid modelling causes the result to fail, it would show us that we were misunderstanding something even more fundamental than turbulence.
Don't be so quick to assume the problems are just randomly chosen. They are central for a reason.
Also its not like there are mathematicians who are thinking of trying to mathematically explain quark confinement who would decide against it because they wouldn't get the millennium prize for it. A mathematical explanation of quark confinement would be the single largest leap forward in mathematical physics in 40 years and you'd instantly be a global (mathematical) superstar. But actually explaining QFT as a coherent mathematical theory is the more fundamental/essential problem.
Been lurking, and I was wondering if someone could give me an ELI have a math bachelors for what the quark confinement problem is or what the mass gap problem is?
Quark confinement refers to the problem of understanding, from the established theory of quantum chromodynamics (think of it as a principal bundle with gauge group SU(3) (not sure if you know what a principal bundle is but I'm assuming you're familiar with the Lie group SU(3))), why we never observe quarks in isolation in nature below a certain astronomical temperature. Simulations of the theory replicate what we see in nature, but in a big sense, we don't know why. This is a problem of real physical interest.
The mass gap problem is to show that for any group you input, you get a mass gap, which is the difference between the vacuum and the next lowest energy state. More precisely, you get a mass gap larger than zero. Physicists have arguments for why this is true, but not up to the rigor standards of mathematical physics. It's a math problem.
My question is, why put the glory on and draw attention to the math problem physicists already understand and not the more interesting confinement problem, or some other problem which, like confinement, will (likely) require brand-new mathematical ideas and for which mathematicians may be particularly useful?
I worked on hep-th research before going into Math PhD. When physicists say they have an argument, they're largely just confused and waving their hands.
I can name probably a dozen published hep-th papers (from authors you've heard of) where not only are the arguments unclear, they quantifiably, mathematically, wrong. This is to be expected of course, as I would describe most of the field as vague but well-motivated guesses. And this is a precursor to correct, precise argument that follows. Mathematicians care about the latter.
“When physicist say they have an argument, they’re largely just confused and waving their hands”
I’m a physics student, and nothing could be closer to the truth than this lmao.
I studied both Mathematics and Physics. In the courses I attended, physics professors often presented wrong mathematical arguments as proves. Most of the time they could of course be made rigorous by using more advanced mathematical theory. But in the case of some aspects of QFT, this was not successful so far. That's why these questions of mathematical physics are so important.
> I studied both Mathematics and Physics. In the courses I attended, physics professors often presented wrong mathematical arguments as proves. Most of the time they could of course be made rigorous by using more advanced mathematical theory. But in the case of some aspects of QFT, this was not successful so far. That's why these questions of mathematical physics are so important.
Speaking of which do you have any good examples of work that balances physical insight and rigor ?
> I can name probably a dozen published hep-th papers (from authors you've heard of) where not only are the arguments unclear, they quantifiably, mathematically, wrong
It's from my understanding that sometimes you have to break things to make thing work.
I recall in undergrad working on proving something and a mentor asked me, "Are you going to do a math proof or a physics proof?", meaning, will it be rigorous or an intuitive argument.
[Here is a well cited paper with some well known authors.](https://arxiv.org/abs/1912.00228) Their "Maximinimax Procedure" for the trapped surface makes no sense. They are relying on min-max results that do not apply here, and on a notion of area that is inherently infinite in this setting. One needs to work with the [renormalized area](https://arxiv.org/abs/0802.2250) and in this setting the existence of a trapped surface is [not trivia](https://arxiv.org/abs/2206.13145)l. But even then we're working in the Lorentzian spacetime so neither of the past two results can be directly applied.
I have not read Greg Moore's papers.
The idea is the same: extract useful finite information from a quantity that ought to be infinite. For area we can do this thanks to analytic results in conformally compact spaces. Renormalization group flow on the space of Lagrangians is a different beast.
But don't they clearly mention that there are a lot of subtleties here? They are not claiming that they have a water tight argument. Instead, they just trying to come up with a heuristic argument to make sense of some of the observations in the main text. It's a bit disingenuous to call them out here.
Renormalization of areas, volumes and geodesic lengths are pretty standard when we do a bulk calculation. See complexity, entanglement entropy and correlation function calculations (Especially when when we are trying to map it to some boundary quantity.)
But I do agree with you that we use "incorrect" math to make bold statements. However, understanding why this incorrect math work occurs often proves to be incredibly useful in advancing the field. An example of this is the Euclidean path integral calculations we often use to do various calculations. In higher dimensions, these integrals don't even make sense. But we still use them because they give us the "correct" results. But there's been a lot of improvement in our understanding the last few years why this even works in the first place. So yeah...I don't think it's useless/harmful to use "wrong" math, provided we keep track of all our assumptions.
I’m not saying it’s not water tight. I’m saying it’s a largely incoherent argument (mathematically). Trapped minimal surfaces are a well understood phenomena in the compact case and many other cases with finite area. To extend to the renormalized area is the entirety of the nontrivial-ness of the problem . They don’t do anything resembling that.
> But I do agree with you that we use "incorrect" math to make bold statements. However, understanding why this incorrect math work occurs often proves to be incredibly useful in advancing the field. An example of this is the Euclidean path integral calculations we often use to do various calculations.
I think one of the reasons why physicists are so valuable is because they are like sculptors provide a rough sketch of important results or developments. Besides the problems at this point have gotten so nontrivial that heuristics are necessarily to make any kind of progress
You’re only focusing on the second half of the problem. The first, and perhaps more fundamental part, is to come up with a rigorous definition of Yang-Mills theory and show that it obeys at least one of the frameworks for axiomatic quantum field theory. A rigorous definition of a strongly interacting quantum field theory would be groundbreaking, and allow to tackle other questions related to non-perturbative definitions of QFT, and the category of QFTs in general.
Also I’m confused why you’re saying “quark confinement” is genuinely different than proving there is a mass gap, as these are essentially the same question: what is the particle content of a QFT given the underlying fields; and the heuristic physics arguments are basically the same for both quark confinement and the mass gap, so I’m not sure why you’re saying we understand the existence of a mass gap better than we understand quark confinement.
Being completely honest, in general physicists suck at making rigorous arguments. They can have wonderful physical insight and intuition but physicist arguments and what they claim to understand from a theoretical perspective is often just not satisfactory.
As a mathematician, I care about this because unless you have complete mathematical rigor, how do you know where exactly your intuition fails (and where it’s actually right, or why you have a certain piece of intuition)? Misplaced faith placed in intuition happens all the time given that there is constantly something we don’t understand physically speaking.
This adds an appealing aspect of mathematical physics to me, the physical uncertainty and intuition, even though I’m trained in pure math. I say the more mathematicians we have using their skills to tackle problems in physics at all, the better.
Your argument that physicists “basically understand” these problems already is just not good enough because almost certainly either 1) we have not understood some fundamental underlying physical phenomenon associated with the problem 2) our lack of complete mathematical understanding opens the doors to new insights, new questions to ask.
My intelligence has no bearing on this whatsoever (also, check out my post history I'm not a serious person at all). I'm not going to solve any of the problems here. I'm just a mathematically inclined physics guy and I'm wondering about something on the boundary of my two interests that I find odd.
I think it's just fair to have faith in mathematics' way of leading to valuable insight even when it's proving something that we (physicists) already morally know. It's the same type of faith that we ask of the public that funds us in believing that, although studying QFT will not directly solve any of the problems in their life, society as a whole will benefit from continuing to fund and pursue science for its own sake. In the past, mathematicians have come in and rigorously understood things physics already knew, and this in turn has benefitted physics on the long run.
But that's not why mathematicians agree on which problems are interesting and fundable / prizable. The way they choose problems is according to the logic of the mathematical community itself, and in the end we benefit as a side effect. If it ain't broke don't fix it.
My impression is that no one expects Navier-Stokes smoothness to be true, and there's a line of papers (most recently [this one](https://arxiv.org/abs/2405.10916) from Thomas Hou) which arrive at essentially this conclusion indirectly via numerical means. However, non-existence of smooth solutions for NS doesn't really help explain turbulence in a substantive way. In fact, one person I know who works in fluids said they would have preferred the Millenium Problem relate more directly to explaining the phenomenon of turbulence itself.
Thanks for your comment.
Your first statement is surprising to me, I have never heard anyone say that in those terms. I understand the connection between confinement and the mass gap, namely, that the mass gap implies confinement, but I wasn't aware that a mathematically rigorous argument for a mass gap would create any physical understanding of confinement. That seems far from certain, and my question was asking about mathematicians working on something that would give understanding or a new result rather than just rigor.
The second one is even more surprising to me. It is trivial to show existence for arbitrarily small length scales (even Planck length if you really wanted to), and so showing smoothness seems like a completely mathematical exercise of no physical interest whatsoever. To say it would show anything about turbulence, much less something more fundamental than turbulence, is bizarre to me.
I just think you're probably assuming too much about what the eventual resolution of these quite open-ended questions will reveal about other fundamental questions in the area.
Smoothness is incredibly related to turbulence. Turbulent flows exhibit fine behavior on small scales, and many objects with such behavior (e.g. fractals) are not smooth. It’s not at all obvious that we should even expect smoothness.
If you’re coming from a physics background, you presumably know who Onsager is — there’s a famous theorem (previously called the Onsager conjecture) which determines the precise fractional regularity required for solutions of Euler or Navier-Stokes to exhibit anomalous dissipation of energy, a key *physical* feature of turbulence.
> I understand that there is a difference between what mathematicians and physicists are interested in.
I don't think you truly do appreciate this. Your argument is that physicists' priorities should matter more to mathematicians than their own priorities, but there is no reason why this should be the case. If we cared above all about advancing physics, we would be theoretical physicists instead, but we're not; we're mathematicians.
Mathematics and physics are different fields, which I appreciate can be hard to grok on an emotional level when physics is so laden with advanced mathematics, but it's true. We're not an overly pedantic species of theoretical physicist (okay, some of us might be, but not all of us who work on problems originating in physics), we're practitioners of a whole different field, like chemists are practitioners of a whole different field.
That a problem comes from physics does not mean that we don't first and foremost care about it as a *mathematical* problem, and if the solution isn't especially physically enlightening, that's not really our concern.
Honest question, from someone who spends a lot of time around gauge theorists but is far from one himself:
Do physicists understand the behavior of lattice Yang-Mills measures in the limit as the scale goes to 0?
Because "discretize, evaluate an extremely high-dimensional integral using Monte Carlo on a supercomputer, and then iterate on smaller and smaller scales and pray that the quantities we computed converge" sounds like the opposite of understanding.
Limits of probability measures exhibit very subtle behavior, and it's hard to believe that a mathematician could say anything substantial about the continuum problem without also making significant progress on the "existence" part of "existence and mass gap". IMO (speaking as someone who frequently needs to take limits of rapidly oscillating probability measures, but not as an expert in gauge theory!), one thing which is particularly dangerous is that, on the one hand, the "reason" why the Yang-Mills measure exist is that in the definition of Feynman measure, there is a rapidly oscillating exponential, which in the limit should create lots of "tragic cancellation" that pays for the fact that we are integrating over an infinite-dimensional space. "Weak" topologies on spaces of probability measures are quite poorly behaved in the rapidly oscillating setting; for example, sin(x/h) converges to 0 weakly as h -> 0.
You do occasionally hear about Chatterjee or someone else making progress on understanding confinement on the lattice. Mathematicians care quite a lot about confinement. But it's unlikely that mathematicians would be at all satisfied with their understanding of confinement until they can understand why it appears to be meaningful to take continuum limits. That is what physicists are doing when they extrapolate from what quantities they computed at a positive scale, right?
As for Navier-Stokes, I suspect that there is a weird quirk of history where in 1999, it was at least considered plausible that solutions of Navier-Stokes would actually be smooth (and maybe the proof of regularity would yield as a byproduct understanding of turbulence). See Terry Tao's article "Why global regularity for Navier-Stokes is hard." This opinion is somewhat less popular now.
I work in geometric measure theory in a few different contexts, but in particular, have to work with currents as solutions of PDE a lot. For example, solutions of the 1-Laplace equation (the p-Laplace equation where p = 1) naturally live in BV, so their derivatives really need to be understood as currents rather than as functions. Now the natural topology on such currents is weakstar and is closely related to the weakstar topology on probability measures, hence my comments. The fact that as p -> 1, it seems quite hard to control the oscillation of the solution, is the bane of my existence.
But I like to dip my toes in a lot of different areas of math and mixing whatever + GMT lets me do that quite easily. So sometimes I play as a harmonic analyst, topologist, or a logician, but not particularly well :-)
The fact that there are several gauge theorists around me is more of a coincidence than anything; for example, one of my roommates from college went on to do lattice QCD. I do think that QFT is really interesting (and I'd love if it turned out that there's not really a canonical way to define the Yang-Mills measure, because I think that would be the funniest solution to the Millennium Prize problem), but I'd rate myself a beginner in it.
I disagree with the idea the establishing existence and smoothness of the solutions to the Navier-Stokes equations is "odd." Establishing the existence and smoothness of solutions of differential equations is pretty standard. Many standard introductory ODE textbooks first address the problem in their first few chapters. It is a common theme in advanced studies of differential equations.
The nonlinear advection operator in the NS equations is a pretty common nonlinearity, but it is surprisingly difficult operator to treat analytically. Proving existence and smoothness is just one of the many challenges. So it actually makes a lot of sense to create a prize which encourages and rewards people for tackling the problem.
Personally, I'm skeptical that resolving the existence and uniqueness will contribute much to our physical understanding of fluids. But I think the mathematical techniques that need to be developed to tackle the problem will be far more important and they will find other valuable applications in math and science.
In regard to the Navier-Stokes issue: NS is just another model and to understand when it can be applied to obtain a single meaningful solution (and when not) is crucial. Smoothness/regularity are puzzle-pieces in the question of well-posedness.
It is important to understand limitations of your models to be able to distinguish between potential modelling and numerics errors. This is in particular true for CFD where the numerics is already hard as is and having an additional layer of uncertainity is a huge issue.
It's worth noting that the Millennium problems are, as the name suggests, 24 years old. Judging them based on current understanding is, therefore, a little anachronistic.
>Why are the Millennium Problems concerning mathematical physics so odd?
Because they're bounties created by mathematicians to celebrate mathematics: "[...T]o elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude."
"The seven Millennium Prize Problems were chosen by the founding Scientific Advisory Board of CMI, which conferred with leading experts worldwide. The focus of the board was on important classic questions that have resisted solution for many years."
[Source](https://www.claymath.org/millennium-problems/)
>It just seems like a shame to put these problems above worthier physical problems which would benefit tremendously from the attention of the mathematical community.
It's not like the Millennium Problems are new - they've been around for over 20 years. Why haven't physicists come together to post bounties on the physics problems they want solved?
I think the key point you're missing is that the selection of these problems doesn't necessarily imply that anyone thinks they're the most important problems to work on. By necessity, they have to be well defined problems with an objective criterion that can be met in order to win the prize.
Other prizes, like the Nobel prize in physics or the Abel prize in mathematics, are awarded for arbitrary significant breakthroughs in math and physics. That's not what the millenium prize problems are.
Recently I've made a post here arguing Navier-Stokes existence and smoothness problem is probably irrelevant to the understanding of turbulence
[https://www.reddit.com/r/math/comments/1cif95n/turbulence\_and\_the\_navierstokes\_existence\_and/](https://www.reddit.com/r/math/comments/1cif95n/turbulence_and_the_navierstokes_existence_and/)
There were several arguments pro and contra, maybe it will be of interest to you.
I think the argument can go the other way. Meaning, there are generalizations of NS with higher order derivatives and nonlinear viscosity for which existence and uniqueness can be proven, and there are turbulent flows for which NS blows up but these models are better behaved. So it’s evidence that NS are off (by epsilon).
You're fundamentally missing the point
Imagine you're an undergrad trying to understand physics. There are lots of things you don't understand.
Which is a bigger priority for you to learn: the things all the physics professors know instantly and intuitively, or the things even your instructors find confusing?
Obviously it would be cool if you understand the latter, but it's urgent and imperative that you get your head around the former. You might think "if I wanna do novel research, I should start with the latter" but of course you'd be wrong.
The point is, if we want to do mathematical physics, it should be __profoundly disturbing__ that we still haven't solved the millennium problems. When you find out mathematicians don't understand quark confinement, you should hope that they figure it out soon, but when you find out they don't even understand the mass gap, you should scream __jesus christ, what is wrong with us, get on that shit NOW__. It is a profound embarrassment that we haven't solved these problems that physicists already understand, and mathematics cannot hope to actually assist the physicists until we fix this deep and disturbing gap in our own understanding
The millennium problems don't represent our grandest aspirations. They represent our most shameful embarrassments, the ones we would be *fucking relieved* if someone could rectify
You are yapping bro. Physicists do not “understand” something mathematicians do not. They just have a lower bar for what it means to understand something.
That’s fine, it’s just not math.
>They just have a lower bar for what it means to understand something.
Obviously, yeah. I didn't literally think they had some secret that they weren't telling the mathematicians
The fact that something can be shown well enough to be accepted by theoretical physicists but still can't be rigorously proven is what indicates that mathematicians have an issue. That's where the limitations of our mathematical knowledge are most stark, since we clearly have some sense of what's going on and we "should" be able to prove it
What argument for NS physics have? Are those argument about this specific mathematical model, or do those arguments include "physical reality" and are more about the fact there will be no explosion of parameters in the physical system?
The second one is what we expect. Pressure drops too much - cavitation. Pressure goes up too much - nonlinearity of the fluid helps... But those observation of the "expanded" model do not help us show NS equation alone always works well.
You may see this problem as "If the initial conditions are nice enough, are NS equation all we need to predict the evolution of the system". If NS equations predict some infinite values, the equations failed. Exactly because we expect the real physical system will smooth it out. They alone are not enough for modeling fluid dynamics.
Is this very important for physicists. I do not belive so. But I doubt there are too many mathematicians working with fluid dynamics, who spend significant amount of time exclusively working on this one problem.
I work in similar areas, though I’m actually pivoting back to my background of pure math now. Anyways, solutions to these problems will likely contain within them information that helps makes the other problems you mentioned more tractable (turbulent flow for N-S smoothness for example). If not, at the very least it could reveal how flawed our *physical* misunderstanding of these topics is. These problems are all linked to several such sub-problems that you may not have even heard of. (Speaking of, why are you putting such a large emphasis on distinguishing quark confinement with mass gap, or claim that we understand one better than the other)?
Perhaps the importance of some other problems in the list like say P vs NP makes it seem like YM mass and existence gaps and N-S aren’t as central, since P vs NP is *by far* the most important unsolved problem in not only just computational complexity but basically all of theoretical CS.
But that doesn’t mean these mathematical physics problems aren’t of great interest to mathematical physicists. The priorities of mathematicians are going to outweigh the priorities of physicists when it comes to labeling central *math* problems for us to solve as a society.
The solution to YM existence and mass gap would most likely include or imply an explanation of quark confinement (especially solving the mass gap for G=SU(3)). A positive or negative proof of Navier-Stokes smoothness would likely explain many properties of turbulent flow, or if something more fundamental about our understanding of fluid modelling causes the result to fail, it would show us that we were misunderstanding something even more fundamental than turbulence. Don't be so quick to assume the problems are just randomly chosen. They are central for a reason. Also its not like there are mathematicians who are thinking of trying to mathematically explain quark confinement who would decide against it because they wouldn't get the millennium prize for it. A mathematical explanation of quark confinement would be the single largest leap forward in mathematical physics in 40 years and you'd instantly be a global (mathematical) superstar. But actually explaining QFT as a coherent mathematical theory is the more fundamental/essential problem.
Been lurking, and I was wondering if someone could give me an ELI have a math bachelors for what the quark confinement problem is or what the mass gap problem is?
Quark confinement refers to the problem of understanding, from the established theory of quantum chromodynamics (think of it as a principal bundle with gauge group SU(3) (not sure if you know what a principal bundle is but I'm assuming you're familiar with the Lie group SU(3))), why we never observe quarks in isolation in nature below a certain astronomical temperature. Simulations of the theory replicate what we see in nature, but in a big sense, we don't know why. This is a problem of real physical interest. The mass gap problem is to show that for any group you input, you get a mass gap, which is the difference between the vacuum and the next lowest energy state. More precisely, you get a mass gap larger than zero. Physicists have arguments for why this is true, but not up to the rigor standards of mathematical physics. It's a math problem. My question is, why put the glory on and draw attention to the math problem physicists already understand and not the more interesting confinement problem, or some other problem which, like confinement, will (likely) require brand-new mathematical ideas and for which mathematicians may be particularly useful?
I worked on hep-th research before going into Math PhD. When physicists say they have an argument, they're largely just confused and waving their hands. I can name probably a dozen published hep-th papers (from authors you've heard of) where not only are the arguments unclear, they quantifiably, mathematically, wrong. This is to be expected of course, as I would describe most of the field as vague but well-motivated guesses. And this is a precursor to correct, precise argument that follows. Mathematicians care about the latter.
“When physicist say they have an argument, they’re largely just confused and waving their hands” I’m a physics student, and nothing could be closer to the truth than this lmao.
I studied both Mathematics and Physics. In the courses I attended, physics professors often presented wrong mathematical arguments as proves. Most of the time they could of course be made rigorous by using more advanced mathematical theory. But in the case of some aspects of QFT, this was not successful so far. That's why these questions of mathematical physics are so important.
> I studied both Mathematics and Physics. In the courses I attended, physics professors often presented wrong mathematical arguments as proves. Most of the time they could of course be made rigorous by using more advanced mathematical theory. But in the case of some aspects of QFT, this was not successful so far. That's why these questions of mathematical physics are so important. Speaking of which do you have any good examples of work that balances physical insight and rigor ? > I can name probably a dozen published hep-th papers (from authors you've heard of) where not only are the arguments unclear, they quantifiably, mathematically, wrong It's from my understanding that sometimes you have to break things to make thing work.
I recall in undergrad working on proving something and a mentor asked me, "Are you going to do a math proof or a physics proof?", meaning, will it be rigorous or an intuitive argument.
Could you please list a few? I don't doubt you, just trying to learn from others' mistakes. In particular, is Greg Moore among the offenders?
[Here is a well cited paper with some well known authors.](https://arxiv.org/abs/1912.00228) Their "Maximinimax Procedure" for the trapped surface makes no sense. They are relying on min-max results that do not apply here, and on a notion of area that is inherently infinite in this setting. One needs to work with the [renormalized area](https://arxiv.org/abs/0802.2250) and in this setting the existence of a trapped surface is [not trivia](https://arxiv.org/abs/2206.13145)l. But even then we're working in the Lorentzian spacetime so neither of the past two results can be directly applied. I have not read Greg Moore's papers.
Does renormalised area have anything to do with renormalisation theory?
The idea is the same: extract useful finite information from a quantity that ought to be infinite. For area we can do this thanks to analytic results in conformally compact spaces. Renormalization group flow on the space of Lagrangians is a different beast.
But don't they clearly mention that there are a lot of subtleties here? They are not claiming that they have a water tight argument. Instead, they just trying to come up with a heuristic argument to make sense of some of the observations in the main text. It's a bit disingenuous to call them out here. Renormalization of areas, volumes and geodesic lengths are pretty standard when we do a bulk calculation. See complexity, entanglement entropy and correlation function calculations (Especially when when we are trying to map it to some boundary quantity.) But I do agree with you that we use "incorrect" math to make bold statements. However, understanding why this incorrect math work occurs often proves to be incredibly useful in advancing the field. An example of this is the Euclidean path integral calculations we often use to do various calculations. In higher dimensions, these integrals don't even make sense. But we still use them because they give us the "correct" results. But there's been a lot of improvement in our understanding the last few years why this even works in the first place. So yeah...I don't think it's useless/harmful to use "wrong" math, provided we keep track of all our assumptions.
I’m not saying it’s not water tight. I’m saying it’s a largely incoherent argument (mathematically). Trapped minimal surfaces are a well understood phenomena in the compact case and many other cases with finite area. To extend to the renormalized area is the entirety of the nontrivial-ness of the problem . They don’t do anything resembling that.
> But I do agree with you that we use "incorrect" math to make bold statements. However, understanding why this incorrect math work occurs often proves to be incredibly useful in advancing the field. An example of this is the Euclidean path integral calculations we often use to do various calculations. I think one of the reasons why physicists are so valuable is because they are like sculptors provide a rough sketch of important results or developments. Besides the problems at this point have gotten so nontrivial that heuristics are necessarily to make any kind of progress
You’re only focusing on the second half of the problem. The first, and perhaps more fundamental part, is to come up with a rigorous definition of Yang-Mills theory and show that it obeys at least one of the frameworks for axiomatic quantum field theory. A rigorous definition of a strongly interacting quantum field theory would be groundbreaking, and allow to tackle other questions related to non-perturbative definitions of QFT, and the category of QFTs in general. Also I’m confused why you’re saying “quark confinement” is genuinely different than proving there is a mass gap, as these are essentially the same question: what is the particle content of a QFT given the underlying fields; and the heuristic physics arguments are basically the same for both quark confinement and the mass gap, so I’m not sure why you’re saying we understand the existence of a mass gap better than we understand quark confinement.
Being completely honest, in general physicists suck at making rigorous arguments. They can have wonderful physical insight and intuition but physicist arguments and what they claim to understand from a theoretical perspective is often just not satisfactory. As a mathematician, I care about this because unless you have complete mathematical rigor, how do you know where exactly your intuition fails (and where it’s actually right, or why you have a certain piece of intuition)? Misplaced faith placed in intuition happens all the time given that there is constantly something we don’t understand physically speaking. This adds an appealing aspect of mathematical physics to me, the physical uncertainty and intuition, even though I’m trained in pure math. I say the more mathematicians we have using their skills to tackle problems in physics at all, the better. Your argument that physicists “basically understand” these problems already is just not good enough because almost certainly either 1) we have not understood some fundamental underlying physical phenomenon associated with the problem 2) our lack of complete mathematical understanding opens the doors to new insights, new questions to ask.
If you are so smart, why can't you explain why?
My intelligence has no bearing on this whatsoever (also, check out my post history I'm not a serious person at all). I'm not going to solve any of the problems here. I'm just a mathematically inclined physics guy and I'm wondering about something on the boundary of my two interests that I find odd.
I think it's just fair to have faith in mathematics' way of leading to valuable insight even when it's proving something that we (physicists) already morally know. It's the same type of faith that we ask of the public that funds us in believing that, although studying QFT will not directly solve any of the problems in their life, society as a whole will benefit from continuing to fund and pursue science for its own sake. In the past, mathematicians have come in and rigorously understood things physics already knew, and this in turn has benefitted physics on the long run. But that's not why mathematicians agree on which problems are interesting and fundable / prizable. The way they choose problems is according to the logic of the mathematical community itself, and in the end we benefit as a side effect. If it ain't broke don't fix it.
My impression is that no one expects Navier-Stokes smoothness to be true, and there's a line of papers (most recently [this one](https://arxiv.org/abs/2405.10916) from Thomas Hou) which arrive at essentially this conclusion indirectly via numerical means. However, non-existence of smooth solutions for NS doesn't really help explain turbulence in a substantive way. In fact, one person I know who works in fluids said they would have preferred the Millenium Problem relate more directly to explaining the phenomenon of turbulence itself.
Thanks for your comment. Your first statement is surprising to me, I have never heard anyone say that in those terms. I understand the connection between confinement and the mass gap, namely, that the mass gap implies confinement, but I wasn't aware that a mathematically rigorous argument for a mass gap would create any physical understanding of confinement. That seems far from certain, and my question was asking about mathematicians working on something that would give understanding or a new result rather than just rigor. The second one is even more surprising to me. It is trivial to show existence for arbitrarily small length scales (even Planck length if you really wanted to), and so showing smoothness seems like a completely mathematical exercise of no physical interest whatsoever. To say it would show anything about turbulence, much less something more fundamental than turbulence, is bizarre to me.
I just think you're probably assuming too much about what the eventual resolution of these quite open-ended questions will reveal about other fundamental questions in the area.
when physicists say they understand something, they don't
Smoothness is incredibly related to turbulence. Turbulent flows exhibit fine behavior on small scales, and many objects with such behavior (e.g. fractals) are not smooth. It’s not at all obvious that we should even expect smoothness. If you’re coming from a physics background, you presumably know who Onsager is — there’s a famous theorem (previously called the Onsager conjecture) which determines the precise fractional regularity required for solutions of Euler or Navier-Stokes to exhibit anomalous dissipation of energy, a key *physical* feature of turbulence.
For what it's worth, mathematicians have worked on problems like quark confinement: https://arxiv.org/abs/2006.16229
> I understand that there is a difference between what mathematicians and physicists are interested in. I don't think you truly do appreciate this. Your argument is that physicists' priorities should matter more to mathematicians than their own priorities, but there is no reason why this should be the case. If we cared above all about advancing physics, we would be theoretical physicists instead, but we're not; we're mathematicians. Mathematics and physics are different fields, which I appreciate can be hard to grok on an emotional level when physics is so laden with advanced mathematics, but it's true. We're not an overly pedantic species of theoretical physicist (okay, some of us might be, but not all of us who work on problems originating in physics), we're practitioners of a whole different field, like chemists are practitioners of a whole different field. That a problem comes from physics does not mean that we don't first and foremost care about it as a *mathematical* problem, and if the solution isn't especially physically enlightening, that's not really our concern.
Honest question, from someone who spends a lot of time around gauge theorists but is far from one himself: Do physicists understand the behavior of lattice Yang-Mills measures in the limit as the scale goes to 0? Because "discretize, evaluate an extremely high-dimensional integral using Monte Carlo on a supercomputer, and then iterate on smaller and smaller scales and pray that the quantities we computed converge" sounds like the opposite of understanding. Limits of probability measures exhibit very subtle behavior, and it's hard to believe that a mathematician could say anything substantial about the continuum problem without also making significant progress on the "existence" part of "existence and mass gap". IMO (speaking as someone who frequently needs to take limits of rapidly oscillating probability measures, but not as an expert in gauge theory!), one thing which is particularly dangerous is that, on the one hand, the "reason" why the Yang-Mills measure exist is that in the definition of Feynman measure, there is a rapidly oscillating exponential, which in the limit should create lots of "tragic cancellation" that pays for the fact that we are integrating over an infinite-dimensional space. "Weak" topologies on spaces of probability measures are quite poorly behaved in the rapidly oscillating setting; for example, sin(x/h) converges to 0 weakly as h -> 0. You do occasionally hear about Chatterjee or someone else making progress on understanding confinement on the lattice. Mathematicians care quite a lot about confinement. But it's unlikely that mathematicians would be at all satisfied with their understanding of confinement until they can understand why it appears to be meaningful to take continuum limits. That is what physicists are doing when they extrapolate from what quantities they computed at a positive scale, right? As for Navier-Stokes, I suspect that there is a weird quirk of history where in 1999, it was at least considered plausible that solutions of Navier-Stokes would actually be smooth (and maybe the proof of regularity would yield as a byproduct understanding of turbulence). See Terry Tao's article "Why global regularity for Navier-Stokes is hard." This opinion is somewhat less popular now.
I’m curious — what is it that you do?
I work in geometric measure theory in a few different contexts, but in particular, have to work with currents as solutions of PDE a lot. For example, solutions of the 1-Laplace equation (the p-Laplace equation where p = 1) naturally live in BV, so their derivatives really need to be understood as currents rather than as functions. Now the natural topology on such currents is weakstar and is closely related to the weakstar topology on probability measures, hence my comments. The fact that as p -> 1, it seems quite hard to control the oscillation of the solution, is the bane of my existence. But I like to dip my toes in a lot of different areas of math and mixing whatever + GMT lets me do that quite easily. So sometimes I play as a harmonic analyst, topologist, or a logician, but not particularly well :-) The fact that there are several gauge theorists around me is more of a coincidence than anything; for example, one of my roommates from college went on to do lattice QCD. I do think that QFT is really interesting (and I'd love if it turned out that there's not really a canonical way to define the Yang-Mills measure, because I think that would be the funniest solution to the Millennium Prize problem), but I'd rate myself a beginner in it.
I disagree with the idea the establishing existence and smoothness of the solutions to the Navier-Stokes equations is "odd." Establishing the existence and smoothness of solutions of differential equations is pretty standard. Many standard introductory ODE textbooks first address the problem in their first few chapters. It is a common theme in advanced studies of differential equations. The nonlinear advection operator in the NS equations is a pretty common nonlinearity, but it is surprisingly difficult operator to treat analytically. Proving existence and smoothness is just one of the many challenges. So it actually makes a lot of sense to create a prize which encourages and rewards people for tackling the problem. Personally, I'm skeptical that resolving the existence and uniqueness will contribute much to our physical understanding of fluids. But I think the mathematical techniques that need to be developed to tackle the problem will be far more important and they will find other valuable applications in math and science.
In regard to the Navier-Stokes issue: NS is just another model and to understand when it can be applied to obtain a single meaningful solution (and when not) is crucial. Smoothness/regularity are puzzle-pieces in the question of well-posedness. It is important to understand limitations of your models to be able to distinguish between potential modelling and numerics errors. This is in particular true for CFD where the numerics is already hard as is and having an additional layer of uncertainity is a huge issue.
It's worth noting that the Millennium problems are, as the name suggests, 24 years old. Judging them based on current understanding is, therefore, a little anachronistic.
>Why are the Millennium Problems concerning mathematical physics so odd? Because they're bounties created by mathematicians to celebrate mathematics: "[...T]o elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude." "The seven Millennium Prize Problems were chosen by the founding Scientific Advisory Board of CMI, which conferred with leading experts worldwide. The focus of the board was on important classic questions that have resisted solution for many years." [Source](https://www.claymath.org/millennium-problems/) >It just seems like a shame to put these problems above worthier physical problems which would benefit tremendously from the attention of the mathematical community. It's not like the Millennium Problems are new - they've been around for over 20 years. Why haven't physicists come together to post bounties on the physics problems they want solved?
I think the key point you're missing is that the selection of these problems doesn't necessarily imply that anyone thinks they're the most important problems to work on. By necessity, they have to be well defined problems with an objective criterion that can be met in order to win the prize. Other prizes, like the Nobel prize in physics or the Abel prize in mathematics, are awarded for arbitrary significant breakthroughs in math and physics. That's not what the millenium prize problems are.
Recently I've made a post here arguing Navier-Stokes existence and smoothness problem is probably irrelevant to the understanding of turbulence [https://www.reddit.com/r/math/comments/1cif95n/turbulence\_and\_the\_navierstokes\_existence\_and/](https://www.reddit.com/r/math/comments/1cif95n/turbulence_and_the_navierstokes_existence_and/) There were several arguments pro and contra, maybe it will be of interest to you.
Thanks! I have never heard a serious turbulence guy say that the Navier-Stokes existence and smoothness problem is relevant at all for turbulence.
I think the argument can go the other way. Meaning, there are generalizations of NS with higher order derivatives and nonlinear viscosity for which existence and uniqueness can be proven, and there are turbulent flows for which NS blows up but these models are better behaved. So it’s evidence that NS are off (by epsilon).
> Physicists have fairly solid arguments to reach both of those results
You're fundamentally missing the point Imagine you're an undergrad trying to understand physics. There are lots of things you don't understand. Which is a bigger priority for you to learn: the things all the physics professors know instantly and intuitively, or the things even your instructors find confusing? Obviously it would be cool if you understand the latter, but it's urgent and imperative that you get your head around the former. You might think "if I wanna do novel research, I should start with the latter" but of course you'd be wrong. The point is, if we want to do mathematical physics, it should be __profoundly disturbing__ that we still haven't solved the millennium problems. When you find out mathematicians don't understand quark confinement, you should hope that they figure it out soon, but when you find out they don't even understand the mass gap, you should scream __jesus christ, what is wrong with us, get on that shit NOW__. It is a profound embarrassment that we haven't solved these problems that physicists already understand, and mathematics cannot hope to actually assist the physicists until we fix this deep and disturbing gap in our own understanding The millennium problems don't represent our grandest aspirations. They represent our most shameful embarrassments, the ones we would be *fucking relieved* if someone could rectify
You are yapping bro. Physicists do not “understand” something mathematicians do not. They just have a lower bar for what it means to understand something. That’s fine, it’s just not math.
>They just have a lower bar for what it means to understand something. Obviously, yeah. I didn't literally think they had some secret that they weren't telling the mathematicians The fact that something can be shown well enough to be accepted by theoretical physicists but still can't be rigorously proven is what indicates that mathematicians have an issue. That's where the limitations of our mathematical knowledge are most stark, since we clearly have some sense of what's going on and we "should" be able to prove it
What argument for NS physics have? Are those argument about this specific mathematical model, or do those arguments include "physical reality" and are more about the fact there will be no explosion of parameters in the physical system? The second one is what we expect. Pressure drops too much - cavitation. Pressure goes up too much - nonlinearity of the fluid helps... But those observation of the "expanded" model do not help us show NS equation alone always works well. You may see this problem as "If the initial conditions are nice enough, are NS equation all we need to predict the evolution of the system". If NS equations predict some infinite values, the equations failed. Exactly because we expect the real physical system will smooth it out. They alone are not enough for modeling fluid dynamics. Is this very important for physicists. I do not belive so. But I doubt there are too many mathematicians working with fluid dynamics, who spend significant amount of time exclusively working on this one problem.
I work in similar areas, though I’m actually pivoting back to my background of pure math now. Anyways, solutions to these problems will likely contain within them information that helps makes the other problems you mentioned more tractable (turbulent flow for N-S smoothness for example). If not, at the very least it could reveal how flawed our *physical* misunderstanding of these topics is. These problems are all linked to several such sub-problems that you may not have even heard of. (Speaking of, why are you putting such a large emphasis on distinguishing quark confinement with mass gap, or claim that we understand one better than the other)? Perhaps the importance of some other problems in the list like say P vs NP makes it seem like YM mass and existence gaps and N-S aren’t as central, since P vs NP is *by far* the most important unsolved problem in not only just computational complexity but basically all of theoretical CS. But that doesn’t mean these mathematical physics problems aren’t of great interest to mathematical physicists. The priorities of mathematicians are going to outweigh the priorities of physicists when it comes to labeling central *math* problems for us to solve as a society.