The Hot spot conjecture The conjecture seems quite amazing and simple to formulate,it can be even understood by persons "from the street" seems its prediction can be tested experimentally. It is a subject of "polymath project 7".Let me quote:
The hotspots conjecture can be expressed in simple English as:
Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.
In mathematical terms, we consider a two-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We then conjecture that
For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D
This conjecture has been proven for some domains and proven to be false for others. In particular it has been proven to be true for obtuse and right triangles, but the case of an acute triangle remains open. The proposal is that we prove the Hot Spots conjecture for acute triangles! Note: strictly speaking, the conjecture is only believed to hold for generic solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic eigenvectors of the second eigenvalue of the Laplacian attain their maximum on the boundary. A stronger version of the conjecture asserts that
For all non-equilateral acute triangles, the second Neumann eigenvalue is simple; The second Neumann eigenfunction attains its extrema only at the boundary of the triangle.
(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)