In 1877, Lord Rayleigh asserted (with good evidence!) that

"If the area of a membrane be given, there must evidently be

some form of boundary for which the pitch (of the principal

tone) is the gravest possible, and this form can be no other

than the circle."

In modern terminology, he was claiming that among all domains of a given

area, the one that minimizes the first eigenvalue of the Laplacian

(assuming Dirichlet boundary conditions) is a disk.

Raleigh's conjecture has motivated an impressive succession of work: Faber

and Krahn found proofs in the 1920s, by inventing the symmetric decreasing

rearrangement of functions; Polya and Szego's 1951 monograph on

"Isoperimetric Inequalities in Mathematical Physics" created a field of

the same name; Luttinger (1973) insightfully unified Rayleigh's conjecture

with the classical isoperimetric theorem via the trace of the heat kernel;

Bossel (1986) gave a new proof of Rayleigh's conjecture by "reverse

rearrangement"; and Hamel, Nadirashvili and Russ (2007) added a lower

order term to the Laplacian, thus handling "diffusion with drift".

This survey talk will sketch the main ideas of these historic developments,

and describe some interesting problems that remain open to this day.