| Brief Introduction to Speaker | 
	     In recent years, spectral methods have become a major tool forcomputational scientists and engineers because of their superior
 accuracy and efficiency when properly implemented.  In this talk, I
 shall present essential ingredients to construct fast spectral
 algorithms and to carry out their error analysis. I shall also
 present, as examples of applications, fast spectral methods for
 acoustic/electromagnetic scattering, incompressible flows and
 mutli-phase complex fluid flows,
 and high-dimensional electronic Schrodinger equation.
 
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