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).