Mathematicians model the sound an object makes by its spectrum, a collection of numbers. “Hearing shapes” means reconstructing geometric properties from the spectrum. Famously posed as “Can you hear the shape of a drum?” spectral geometry asks which properties of shape can be “heard,” i.e. determined by its spectrum.
Roughly speaking, manifolds are “smooth” objects, while orbifolds are “almost smooth.” A major open question is “Can a smooth manifold and a not-entirely smooth orbifold have the same spectrum? We will give a basic introduction to spectral geometry, manifolds, and orbifolds, and describe some recent results related to this question.
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