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[PAST EVENT] Mathematics Colloquium: Hyunchul (Hoon) Park, William and Mary
September 6, 2013
2pm - 3pm
Abstract:
In 1966, Mark Kac asked if one could hear the shape of a drum. This means if one can figure out the geometry of the domain (drum) when one has perfect pitch so that he or she can hear all the fundamental tones (eigenvalues of the Laplacian
or Brownian motions) of that drum. This turns out to be false. There exist isospectral but not isometric domains but one can still extract important geometric information such as area, perimeter, or even the Euler Characteristic from the eigenvalues of Dirichlet Laplacian of the domain.
A very natural question is what happens when we replace Brownian motions by other Levy processes. Relativisitic stable processes (RSP) are pure jump Levy processes whose jump rate is similar to stable processes for a small scale but have an exponential decay for a large scale. In a recent work with Song, we prove that one can identify area and perimeter of the domain from the information of eigenvalues of RSP.
This talk will assume no knowledge of graduate probability theory and will be accessible to undergraduate students.
In 1966, Mark Kac asked if one could hear the shape of a drum. This means if one can figure out the geometry of the domain (drum) when one has perfect pitch so that he or she can hear all the fundamental tones (eigenvalues of the Laplacian
or Brownian motions) of that drum. This turns out to be false. There exist isospectral but not isometric domains but one can still extract important geometric information such as area, perimeter, or even the Euler Characteristic from the eigenvalues of Dirichlet Laplacian of the domain.
A very natural question is what happens when we replace Brownian motions by other Levy processes. Relativisitic stable processes (RSP) are pure jump Levy processes whose jump rate is similar to stable processes for a small scale but have an exponential decay for a large scale. In a recent work with Song, we prove that one can identify area and perimeter of the domain from the information of eigenvalues of RSP.
This talk will assume no knowledge of graduate probability theory and will be accessible to undergraduate students.
Contact
Chi-Kwong Li