Research Interests
Eigenvalues are physical attributes of various systems, of interest in
their own right. They appear as physical characteristics of vibrating
structures and mechanical systems; as rates of heat dissipation through a
body; and as energy levels for quantum systems. They encode, in a rather
complicated fashion, various geometric features of manifolds. They are at
the heart of the old colorful inverse question of M. Kac on "hearing the
shape of a drum": Rebuild the underlying structure of a manifold
(geometry, topology, etc.) from knowing its spectrum.
My scientific research is focused on proving, extending, or improving
various universal and domain dependent inequalities for eigenvalues of
elliptic operators arising in some of the physical problems described
above.
Extending existing inequalities means to enlarge the class of operators
for which inequalities of the same kind exist. Improving means to
strengthen existing bounds. While universal bounds are inequalities which
hold true with no reference to the underlying domain, domain-dependent
inequalities involve either its volume or any of its moments.
Of interest to me are the following aspects of a very rich area of
mathematics:
(1) Bounds for Riesz means and their consequences
(2) Weyl-type bounds for power means of eigenvalues
(3) Universal eigenvalue inequalities
(4) Applications of Eigenvalues to Computer Vision
