Hyperbolicity and classification theory in complex algebraic geometry

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My work involves the structure and classification of algebraic varieties, which are objects defined by homogeneous polynomials, up to a natural equivalence, called birational equivalence. Two such objects are birationally equivalent if their spaces of rational functions, are naturally isomorphic. The behavior of holomorphic (i.e., complex differentiable) or algebraic functions from the complex number plane with values in them lies at the center of my investigation. Besides the classical motivation of generalizing Picard's theorems (Lang's conjecture on the pseudo-hyperbolicity of varieties of general type), there is also strong motivation from number theory, such as the Mordell conjecture on the finiteness of solutions to a set of polynomials in terms of rational numbers if the variety defined is a curve of general type (solved by G. Faltings, for which he obtained the highest honor in Mathematics, the Fields Medal). Faltings gave two solutions to the Mordell conjecture, the second using an idea of Vojta-Mazur on the parallel between the value distribution theory of holomorphic curves, Pioneered by Nevanlinna, and the distribution of rational or algebraic points (i.e. solutions by integers or by radicals of integers of the homogeneous equations defining the variety). It is therefore hoped that a deeper understanding of the structure of curves in an algebraic variety would eventually lead to unlocking the mysteries of the similarities seen between natural questions in number theory and those concerning the behavior of holomorphic and algebraic curves.