Local positivity 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 their solutions is intimately tied to the intrinsic geometric nature of the shapes those equations describe. Diophantine geometry is one of the most far-reaching realizations of this philosophy. In general terms it connects the global behaviour of rational number solutions to a system of polynomial equations with the global behaviour of the curvature on the corresponding shape defined by those equations. Previous research by the applicant has extended classic approximation results on the line to arbitrary projective varieties. One of the new ideas introduced was to consider the local influence of curvature on questions of local approximation by rational points. This proposal aims to combine this insight with new ideas from birational and asymptotic algebraic geometry to establish new results in Diophantine geometry. The proposal also aims to contribute new results to questions surrounding positivity in algebraic geometry. The proposed research will train four undergraduate students, three master's students, three doctoral students, and two postdoctoral fellows.