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Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

TBA

Series: Analysis Seminar

TBA

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider
the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions
of this type arise in physics as modes of periodic vibration
of drums and membranes. They also represent stationary states of a free
quantum particle on a Riemannian manifold. In the first part of the
lecture, I will give a survey of results which demonstrate how the
geometry of $M$ affects the behaviour of these special
functions, particularly their “size” which can be quantified by
estimating $L^p$ norms.
In joint work with Malabika Pramanik (U. British Columbia),
I will present in the second part of my lecture a result on the $L^p$ restriction of
these eigenfunctions to random Cantor-type
subsets of $M$. This, in some sense, is complementary to the smooth submanifold $L^p$ restriction results
of Burq-Gérard-Tzetkov ’06 (and later work of other authors). Our
method includes concentration inequalities from probability theory in
addition to the analysis of singular Fourier integral operators on
fractals.