Does diffusion determine the drum?




Date(s) : 09/02/2016   iCal
11 h 00 min - 12 h 00 min

The famous question of Kac is whether one can hear the shape of a drum. Or more precisely, whether all eigen frequencies of a drum determine the drum up to congruency. In general the answer to the latter question is negative if one allows polygon shaped drums. The eigen frequencies are equal if and only if there exists a unitary operator which maps the Laplacian on the first drum onto the Laplacian on the second drum. Equivalently, this unitary operator intertwines between the two heat semigroups on the drums. In this talk we discuss what happens if the unitary operator is replaced by an order isomorphism, i.e., if it maps positive functions to positive functions. Then that order isomorphism maps positive solution of the heat equation on the first drum onto positive solutions on the second drum. Phrased differently: the order isomorphism tells that the diffusion on the two drums are the same. Hence the question: if the diffusion on two drums is the same, are they then congruent?

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