Non UBC
DSpace
Kaloshin, Vadim
2018-09-16T05:01:06Z
2018-03-19T19:33
M. Kac popularized the question {\em Can you hear the shape of a drum?}
Mathematically, consider a bounded planar domain $\Omega$ and the associated
Dirichlet problem $\Delta u + \lambda^2 u = 0$ with $u|_{\partial \Omega}$ = 0. The set of
$\lambda$s such that this equation has a solution, denoted $\mathcal{L}(\Omega)$
is called the Laplace spectrum of $\Omega.$
Does Laplace spectrum determine $\Omega$? In general, the answer is negative.
Consider the billiard problem inside ?. Call the length spectrum the
closure of the set of perimeters of all periodic orbits of the billiard. Due
to deep properties of the wave trace function, generically, the Laplace
spectrum determines the length spectrum. We show that any generic axis
symmetric planar domain with is dynamically spectrally rigid, i.e. can't be
deformed without changing the length spectrum. This partially answers a
question of P. Sarnak. This is joint works with J. De Simoi, A. Figalli, and
J. De Simoi, Q. Wei.
https://circle.library.ubc.ca/rest/handle/2429/67192?expand=metadata
45 minutes
video/mp4
Author affiliation: University of Maryland
Banff (Alta.)
10.14288/1.0372059
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Faculty
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Dynamical systems and ergodic theory
Probability theory and stochastic processes
Dynamical systems
Can you hear the shape of a drum and deformational spectral rigidity of planar domains?
Moving Image
http://hdl.handle.net/2429/67192