Geometric Analytic Number Theory

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My project concerns Diophantine equations and related arithmetic and geometric objects. Diophantine equations are polynomial equations where one seeks solutions in the whole numbers. They have been studied since antiquity, but still hold many secrets and are an important area of research: Andrew Wiles' 1995 solution of Fermat's Last Theorem was one of the crowning achievements of 20th Century Mathematics, resolving a 350 year old conjecture of Fermat. Moreover, in the modern era they have surprisingly found numerous applications to information security and underlie many cryptosystems (e.g. Elliptic Curve Cryptography).Given a Diophantine equation, the first challenge is whether a solution actually exists. My project would take this to the next level: I will develop a unifying framework for studying *families* of Diophantine equations. Namely, when one runs through a family of equations given by varying the coefficients, how many have a solution? I will propose new conjectures for such problems as well as new techniques for tackling them, which vastly generalise the case of conics originally investigated by Jean-Pierre Serre (one of the leading mathematicians of the 20th Century). This will open new research directions in mathematics.