Asymptotic algebraic geometry and Diophantine applications

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Algebraic geometry is a rich branch of mathematics which unifies algebra, topology, number theory, and differential geometry. It allows one to reformulate questions which seem to be about algebra or numbers into geometrical questions about shape, or vice versa. Its origins lie in the realization that finding solutions to algebraic equations or understanding the properties of the solutions are intimately tied to the intrinsic geometric nature of the algebraic varieties those equations describe. The Lang-Vojta conjectures are one of the most spectacular realizations of this philosophy. In general terms they connect the global behaviour of rational solutions of equations to the global behaviour of the curvature on the corresponding algebraic variety.