This is an expository lecture on the subject of the title delivered at the Park-IAS mathematical institute in Princeton (July, 2000).
On the Diophantine equations $ \sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3 $
Length-Preserving Directions and Some Diophantine Equations
Rational solutions of certain Diophantine equations involving norms
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