Can you hear an orbifold singularity?
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Posted online: 2024-03-13 07:06:25Z by SciLag Admin5
Cite as: P-240313.1
Problem's Description
The question "Can one hear the shape of a drum?" is a classic problem in mathematics (posed by Mark Kac in 1966) that has inspired various related inquiries in different fields. This provoking question which boils down to whether the Laplace spectrum of a manifold could uniquely identify its isometry class. John Milnor swiftly countered this idea by presenting a pair of 16-dimensional tori that were isospectral but not isometric. Subsequently, many mathematicians have pondered the geometric properties that can be deduced from the Laplacian spectrum.
Similar problems has lead to similar investigation in related areas such as Spectral geometry, Inverse problems in imaging, Quantum chaos, Mathematical resonance, Shape reconstruction.
In Differential Geometry there is yet another type of phenomenon that can give rise to a similar question, namely orbifold singularities.
An orbifold singularity is a type of singularity that can occur in a mathematical object called an orbifold. An orbifold is a generalization of a manifold, which is a space that looks like Euclidean space near each point. In an orbifold, however, the space can look like a quotient space of a manifold under the action of a finite group of symmetries.
An orbifold singularity occurs at a point where the structure of the orbifold is not smooth or well-behaved, typically because the finite group of symmetries acting at that point does not act freely or properly. This can lead to geometric irregularities or folding of the space at that point. Orbifold singularities are important in various areas of mathematics, including geometry, topology, and string theory.
Ian Adelstein proposed [1]:Can one hear the presence of an orbifold singularity, i.e. whether or not there exists a pair of isospectral orbifolds, one of which has singular points whereas the other does not and is therefore a manifold.
See [3] for discussions around the topic. Recent research has revealed that G-invariant spectra cannot detect non-orbifold singularities. In [2], the authors demonstrate the construction of a pair of orbit spaces possessing equivalent G-invariant spectra. However, while one space is isometric to an orbifold, the other accommodates a non-orbifold singularity.
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Created at: 2024-03-13 07:06:25Z
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