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The Covering Spectrum and Isospectrality

Ruth Gornet
University of Texas at Arlington

Thursday, April 2, 2009
007 Kemeny Hall, 4 pm
Tea 3:30 pm, 300 Kemeny Hall

Abstract: "Can one hear the shape of a drum?" is a question popularized by M. Kac. The question asks whether the geometry of a shape can by "heard"; i.e., determined by the frequencies it makes. Mathematically translated, can geometric properties be determined by the eigenvalues of the Laplace-Beltrami operator. We will focuse on the length spectrum (the set of all lengths of closed geodesics), and a weaker/related notion, the Covering Spectrum, which was recently defined by C. Sormani and G. Wei. In particular, we compare and contrast the group theoretic conditions used by Sunada to construct isospectral manifolds and the group theoretic conditions used by DeSmit-Gornet-Sutton to construct manifolds with the same covering spectrum.

This talk is aimed at graduate students. Definitions will be carefully presented and discussed on elementary examples.

This talk will be accessible to graduate students.