Rational points on algebric varieties

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One of the fundamental goals of number theory is to find solutions in whole numbers to certain kinds of equations called Diophantine equations.  These equations can often be interpreted in a geometric manner, and therefore geometric techniques can be brought to bear on the solution of the problem. Paul Vojta has made some wide-ranging conjectures on what kinds of solutions Diophantine equations should have, based on what kind of geometrical  properties the equations have.  In my future research, I propose to study these conjectures, try to improve on my previous proofs of various special cases of them, and try to use existing results to gain further insight into the solutions of Diophantine equations.