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Technion
 
                        SPECIAL MATHEMATICS COLLOQUIUM
 
                 Please note the unusual day and unusual time. 
 
Speaker: Isaac Pesenson, Temple University,
 
Title:   Approximation of eigenvalues and eigenfunctions
         on Riemannian manifolds and combinatorial graphs
 
Date:    Thursday June 2 at 2:30 p.m.
 
Place:   Amado 232, Technion.
 
Abstract:

The abstract is available (temporarily) at
 <http://www.math.technion.ac.il/~techm/temp/20110602PESE.pdf>

Or you can read it here:

Key words and phrases. Riemannian manifold, Laplace-Beltrami operator, 
Rayleigh-Rit method, Poincare inequality, polyharmonic spline.
  
We will show that eigenvalues of Laplace-Beltrami operators on
compact Riemannian manifolds can be determined as limits of eigenvalues of
certain finite-dimensional operators in finite-dimensional spaces of
polyharmonic functions with singularities. In particular, a bounded set of
eigenvalues can be determined using a space of such polyharmonic functions
with a fixed set of singularities. We will also show that corresponding
eigenfunctions can be reconstructed as uniform limits of the same
polyharmonic functions with appropriate fixed set of singularities.
 
These results for Laplace-Beltrami operators on compact manifolds have
natural analogs for combinatorial Laplacians on finite combinatorial
graphs.
 
We prove that eigenfunctions of a combinatorial Laplace operator that
correspond to small eigenvalues can be reconstructed as limits of the
so-called variational splines. We also prove that small eigenvalues of the
combinatorial Laplacian can be approximated by eigenvalues of certain
matrices acting in spline spaces.
 
These results have potential applications to various problems such as
high- dimensional data dimension reduction, image processing, computer
graphics, visualization and learning theory. It is joint work with Meyer
Z. Pesenson from Computing and Mathematical Sciences Deptartment, Caltech,
 <mzp@cms.caltech.edu.>
 
Our work was partially supported by the National Geospatial-Intelligence
Agency, NURI, Grant HM1582-08-1-0019 and by AFOSR, MURI, Award FA9550-
09-1-0643.


Department of Mathematics, Temple University, Philadelphia, PA 19122
E-mail address:  <pesenson@temple.edu>


 
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