What are good, still unsolved problems in mathematical physics that are in vogue? I always get the same answers: reference to Millennium Problems by the Clay Institute, or "there's still a lot to do in this or that..." or the three body problem. Could you be more specific?

You could start with Michael Aizenman's list of a dozen specific problems from a variety of areas of mathematical physics. The list is two decades old, but most of these problems are still wide open. Fifteen problems in the field of Schrödinger operators are in Barry Simon's list, (1984), with a more recent update (2000).

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    Some of Simon's (2000) problems have been solved. By far the most prominent problem on the list is (1), "extended states" for the Anderson model. – Christian Remling Jul 9 '17 at 16:19

In low-energy nuclear structure, we have certain theories that work for certain types of nuclei, but there are some nuclei, called transitional nuclei, for which there is essentially no tractable theoretical description.

We can classify nuclei along a continuous scale from spherical through transitional to deformed nuclei. A nucleus's location on this scale can be estimated based on the product of the number of valence neutrons and protons (counting a hole as a positive number). When the product is small, the nucleus is spherical, when large, deformed.

The nuclear shell model, developed by Maria Goeppert Mayer in the 1950s, works very well for spherical nuclei. For deformed nuclei we also have good models, such as the cranked shell model with pairing. But for the intermediate case, transitional nuclei, there is basically nothing.

The closest thing to a working theory for transitional nuclei is the interacting boson model (IBM, also known as the interacting boson approximation, IBA). But the IBM has many adjustable parameters, especially for odd and odd-odd nuclei, and nobody knows how to predict these parameters a priori for a particular nucleus, so the model's predictive value is extremely limited.

More generally, this is an example of the quantum many-body system, where we need an effective approximation scheme. It's an odd situation, because we actually have an effective approximation scheme when the number of bodies (valence particles) is larger, but none for the case when it's intermediate. The state of ignorance is extreme, in the sense that we can't reliably predict even the simplest properties of transitional nuclei, such as the excitation energy of the first 2+ state in an even-even nucleus. We expect to have difficulties with many-body systems, but not to be as completely powerless as this. In a classical analogy, it would be as though we couldn't predict the future motion of the planets in our solar system, even on time-scales of hours. Another good comparison is with atomic physics, where there are tractable approximation schemes allowing extremely precise predictions, even for very large atoms.

I suspect that a lot of talented people would be deterred from working on this problem by an incorrect perception that it would require learning a lot of grotty details about nuclear forces. In fact, all the physical phenomena that are salient to this problem are generic phenomena for systems of fermions interacting through an attractive short-range force. Experimentally, we see clusters of atoms that demonstrate the same "magic numbers" (shell closures) as clusters of neutrons and protons. If you could make progress on this problem for a toy model of identical fermions interacting through a delta-function potential, your work would almost certainly be immediately generalizable to nuclei.

In a Notices article last month, Charles Radin presented some open problems understanding the statistical mechanics of crystallization in a system of particles with simple pairwise interactions. In addition to some problems where physicists know what answer is expected, but a formal mathematical proof is missing, Radin also gives the following problem as being of real interest in advancing the theoretical understanding of crystallization:

Problem: Prove, in two dimensions, whether or not there is some region of $(𝑃, 𝑇)$ values throughout which the optimal Wulff shape $𝑆_𝑊(𝑃, 𝑇)$ is not the circle.

$P$ and $T$ are the pressure and temperature respectively, and the Wulff shape is the limit of the equilibrium shape of a droplet of $N$ particles, scaled by a factor of $N^{-1/d}$ in the limit $N\to\infty$.

Most of the Millennium Problems ore not mathematical physics. But one or two are. You want specific? Here you go:

Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

source

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    This question is largely irrelevant for physics, since the NS equation itself is an approximation and becomes invalid precisely when its solution structure is difficult. Thus arguably, this is a question in physical mathematics, not in physics. – Walter Jul 9 '17 at 18:20
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    @Walter The OP was asking about mathematical physics, not physics. – Robert Israel Jul 10 '17 at 0:43
  • @RobertIsrael Either mathematical physics is part of physics, or it is inappropriately named. – Walter Jul 10 '17 at 12:34
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    Well, in that case I guess I'll just have to send back my PhD. – Robert Israel Jul 10 '17 at 15:45

The following contains several open problems (as of 2001, but most are still open I believe) in topological fluid dynamics by Moffatt :

Some Remarks on Topological Fluid Mechanics

Exact formulas, or better approximations in minimum times in quantum control, a.k.a., the quantum speed limit, is largely unsolved.

Expressed mathematically:

Given $a,b \in \mathfrak{su}(n)$ which are bracket generating and some $G \in SU(n)$, consider a system obeying the equation $\dot{U}_t = (a + f(t)b)U_t$. What is the minimum time $T=T^*$, over all controls $f$ (where all functions, or even delta functions are permitted), to achieve $U_T = G$.

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