Polya's Conjecture

Polya's Conjecture is a mathematical hypothesis related to the vibration modes of a drum-shaped object. It suggests that for a given shape, the frequencies (or eigenvalues) of its vibrations are always at least as high as those predicted by a simple, idealized model—specifically, the frequencies of a circular drum with the same area. In essence, it proposes a lower bound for these natural vibration frequencies, implying that the shape's geometry influences the vibrational behavior in a predictable way. While proved for some cases, it remains unproven in full generality.