Geometric Analytic Number Theory

我的项目涉及丢番图方程和相关的算术和几何对象。丢番图方程是寻求整数解的多项式方程。它们自古以来就被研究，但仍然有许多秘密，是一个重要的研究领域：安德鲁·怀尔斯1995年对费马大定理的解答是20世纪数学的最高成就之一，解决了一个350年前的费马猜想。此外，在现代，它们令人惊讶地发现了许多信息安全应用，并成为许多密码系统（例如椭圆曲线密码）的基础。对于丢番图方程，第一个挑战是解是否真的存在。我的项目将把它提升到一个新的水平：我将开发一个统一的框架来研究丢番图方程的“族”。也就是说，当一个人通过改变系数得到一系列方程时，有多少方程有解？我将针对这类问题提出新的猜想，并提出解决这些问题的新技术，这些新技术极大地推广了让-皮埃尔·塞尔（Jean-Pierre Serre， 20世纪最重要的数学家之一）最初研究的经济问题。这将为数学开辟新的研究方向。

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.