The polymath blog

It’s time for another rollover of the  Polymath7 “Hot Spots” conjecture, as the previous research thread has again become full.

Activity has now focused on a numerical strategy to solve the hot spots conjecture for all acute angle triangles ABC.  In broad terms, the strategy (also outlined in this document) is as follows.   (I’ll focus here on the problem of estimating the eigenfunction; one also needs to simultaneously obtain control on the eigenvalue, but this seems to be to be a somewhat more tractable problem.)

1. First, observe that as the conjecture is scale invariant, the only relevant parameters for the triangle ABC are the angles , which of course lie between 0 and and add up to .  We can also order , giving a parameter space which is a triangle between the values .
2. The triangles that are too close to the degenerate isosceles triangle or the equilateral triangle need to be handled by analytic arguments.  (Preliminary versions of these arguments can be found here and Section 6 of these notes  respectively, but the constants need to be made explicit (and as strong as possible)).
3. For the remaining parameter space, we will use a sufficiently fine discrete mesh of angles ; the optimal spacing of this mesh is yet to be determined.
4. For each triplet of angles in this mesh,  we partition the triangle ABC (possibly after rescaling it to a reference triangle , such as the unit right-angled triangle) into smaller subtriangles, and approximate the second eigenfunction (or the rescaled triangle ) by the eigenfunction (or ) for a finite element restriction of the eigenvalue problem, in which the function is continuous and piecewise polynomial of low degree (probably linear or quadratic) in each subtriangle; see Section 2.2 of these notes.    With respect to a suitable basis, can be represented by a finite vector .
5. Using numerical linear algebra methods (such as Lanczos iteration) with interval arithmetic, obtain an approximation to , with rigorous bounds on the error between the two.  This gives an approximation to or with rigorous error bounds (initially of L^2 type, but presumably upgradable).
6. After (somehow) obtaining a rigorous error bound between and (or and ), conclude that stays far from its extremum when one is sufficiently far away from the vertices A,B,C of the triangle.
7. Using stability theory of eigenfunctions (see Section 5 of these notes), conclude that stays far from its extremum even when is not at a mesh point.  Thus, the hot spots conjecture is not violated away from the vertices.  (This argument should also handle the vertex that is neither the maximum nor minimum value for the eigenfunction, leaving only the neighbourhoods of the two extremising vertices to deal with.)
8. Finally, use an analytic argument (perhaps based on these calculations) to show that the hot spots conjecture is also not violated near an extremising vertex.

This all looks like it should work in principle, but it is a substantial amount of effort; there is probably still some scope to try to simplify the scheme before we really push for implementing it.

It’s once again time to roll over the research thread for the Polymath7 “Hot Spots” conjecture, as the previous research thread has again become full.

As the project moves into a more mature stage, with most of the “low-hanging fruit” already collected, progress is now a bit less hectic, but our understanding of the problem is improving.  For instance, in the previous thread, the relationship between two different types of arguments to obtain monotonicity of eigenfunctions – namely the coupled Brownian motion methods of Banuelos and Burdzy, and an alternate argument based on vector-valued maximum principles – was extensively discussed, and it is now fairly clear that the two methods yield a more or less equivalent set of results (e.g. monotonicity for obtuse triangles with Neumann conditions, or acute triangles with two Neumann and one Dirichlet side).  Unfortunately, for scalene triangles it was observed that the behaviour of eigenfunctions near all three vertices basically preclude any reasonable monotonicity property from taking place (in particular, the conjecture in the preceding thread in this regard was false).  This is something of a setback, but perhaps there is some other monotonicity-like property which could still hold for scalene acute triangles, and which would imply the hot spots conjecture for these triangles.  We do now have a number of accurate numerical representations of eigenfunctions, as well as some theoretical understanding of their behaviour, especially near vertices or near better-understood triangles (such as the equilateral or isosceles right-angled triangle) so perhaps they could be used to explore some of these properties.

Another result claimed in previous threads – namely, a theoretical proof of the simplicity of the second eigenvalue – has now been completed, with fairly good bounds on the spectral gap, which looks like a useful thing to have.

It was realised that a better understanding of the geometry of the nodal line would be quite helpful – in particular its convexity, which would yield the hot spots conjecture on one side of the nodal line at least.   We did establish one partial result in this direction, namely that the nodal line cannot hit the same edge of the triangle twice, but must instead straddle two edges of the triangle (or a vertex and an opposing edge, though presumably this case only occurs for isosceles triangles).  Unfortunately, more control on the nodal line is needed.

In the absence of a definitive theoretical approach to the problem, the other main approach is via rigorous numerics – to obtain, for a sufficiently dense mesh of test triangles, a collection of numerical approximations to second eigenfunctions which are provably close (in a suitable norm) to a true second eigenfunction, and whose extrema (or near-extrema) only occur at vertices (or near-vertices).  In principle, this sort of information would be good enough to rigorous establish the hot spots conjecture for such a test triangle as well as nearby perturbations of that triangle.  The details of this approach, though, are still being worked out.  (And given that they could be a bit messy, it may well be a good idea to not proceed too prematurely with the numerical approach, in case some better approach is discovered in the near future).  One proposal is to focus on a single typical triangle (e.g. the 40-60-80 triangle) as a test case in order to fix parameters.

There was also some further exploration of whether reflection methods could be pushed further.   It was pointed out that even in the very simple case of the unit interval [0,1], it is not obvious (even heuristically) from reflection arguments why the hot spots conjecture should be true.  Reflecting around a vertex whose angle does not go evenly into has created a number of technical difficulties which seem to so far be unsatisfactorily addressed (but perhaps getting an alternate proof of hot spots in the model cases where reflection does work, i.e. the 30-60-90, 45-45-90, and 60-60-60 triangles, would be worthwhile).

The previous research thread for the Polymath7 “Hot Spots Conjecture” project has once again become quite full, so it is again time to roll it over and summarise the progress so far.

Firstly, we can update the map of parameter space from the previous thread to incorporate all the recent progress (including some that has not quite yet been completed):

This map reflects the following new progress:

1. We now have (or will soon have) a rigorous proof of the simplicity of the second Neumann eigenvalue for all non-equilateral acute triangles (this is the dotted region), thus finishing off the first part of the hot spots conjecture in all cases.  The main idea here is to combine upper bounds on the second eigenvalue (obtained by carefully choosing trial functions for the Rayleigh quotient), with lower bounds on the sum of the second and third eigenvalues, obtained by using a variety of lower bounds coming from reference triangles such as the equilateral or isosceles right triangle.  This writeup contains a treatment of those triangles close to the equilateral triangle, and it is expected that the other cases can be handled similarly.
2. For super-equilateral triangles (the yellow edges) it is now known that the extreme points of the second eigenfunction occur at the vertices of the base, by cutting the triangle in half to obtain a mixed Dirichlet-Neumann eigenvalue problem, and then using the synchronous Brownian motion coupling method of Banuelos and Burdzy to show that certain monotonicity properties of solutions to the heat equation are preserved.  This fact can also be established via a vector-valued maximum principle.  Details are on the wiki.
3. Using stability of eigenfunctions and eigenvalues with respect to small perturbations (at least when there is a spectral gap), one can extend the known results for right-angled and non-equilateral triangles to small perturbations of these triangles (the orange region).  For instance, the stability results of Banuelos and Pang already give control of perturbed eigenfunctions in the uniform norm; since for right-angled triangles and non-equilateral triangles, the extrema only occur at vertices, and from Bessel expansion and uniform C^2 bounds we know that for any perturbed eigenfunction, the vertices will still be local extrema at least (with a uniform lower bound on the region in which they are extremisers), we conclude that the global extrema will still only occur at vertices for perturbations.
4. Some variant of this argument should also work for perturbations of the equilateral triangle (the dark blue region).  The idea here is that the second eigenfunction of a perturbed equilateral triangle should still be close (in, say, the uniform norm) to some second eigenfunction of the equilateral triangle.  Understanding the behaviour of eigenfunctions nearly equilateral triangles more precisely seems to be a useful short-term goal to pursue next in this project.

But there is also some progress that is not easily representable on the above map.  It appears that the nodal line of the second eigenfunction may play a key role.  By using reflection arguments and known comparison inequalities between Dirichlet and Neumann eigenvalues, it was shown that the nodal line cannot hit the same edge twice, and thus must straddle two distinct edges (or a vertex and an opposing edge).   (The argument is sketched on the wiki.) If we can show some convexity of the nodal line, this should imply that the vertex straddled by the nodal line is a global extremum by the coupled Brownian motion arguments, and the only extremum on this side of the nodal line, leaving only the other side of the nodal line (with two vertices rather than one) to consider.

We’re now also getting some numerical data on eigenvalues, eigenfunctions, and the spectral gap.  The spectral gap looks reasonably large once one is away from the degenerate triangles and the equilateral triangle, which bodes well for an attempt to resolve the conjecture for acute angled triangles by rigorous numerics and perturbation theory.  The eigenfunctions also look reassuringly monotone in various directions, which suggests perhaps some conjectures to make  in this regard (e.g. are eigenfunctions always monotone along the direction parallel to the longest side?).

This isn’t a complete summary of the discussion thus far – other participants are encouraged to summarise anything else that happened that bears repeating here.