Joint Colloquium

Can one hear the shape of a drum?
Thursday, 13 April 2017 - 4:00 pm to 4:30 pm
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SPEAKER: Vladimir Chernousov (University of Alberta)             ABSTRACT: The title of my talk is due to Mark Kac who published the famous paper with the same title in the American Mathematical Monthly in 1966. In differential geometry, given a Riemannian manifold M one considers the following sets of data: spectrum of the Laplace operator (eigenvalues with multiplicities) and length spectrum (lengths of closed geodesics). Then it is naturally to ask if these data determines our manifold M uniquely up to isometry or at least up to a finite-sheeted cover. ``Identifying a Riemannian manifold with a drum'' one can think of spectrum of its Laplace operator and length spectrum as discrete characteristics (frequency, amplitude and etc.) of the sound wave produced by the drum when you hit it with a hammer. Hence we come in a naural way to the question stated in the title of the talk: can one hear the shape of a drum? This question received considerable attention in geometry. In the talk we first explain how it leads us to some natural problems and conjectures in algebra related to a question whether a noncommutative object (like a quaternion algebra) can be determined by its abelian subobjects. Then we will survey recent developments. <p> Short bio: Prof. Chernousov has received his PhD from the University of Minsk in 1983. During 1990's he held numerous visiting position at EPFL, Lausanne, ETH Zurich, Alexander von Humboldt Fellow at the University of Bielefeld, Germany; University of Munster. Since 2003 he is a professor at the University of Alberta and a Canada Research Chair Tier I in Algebra.