数学中最深刻的洞见往往来自意想不到的类比。
The Langlands program is an invitation to explore the unknown territories of mathematics.
朗兰兹纲领是对探索数学未知领域的邀请。
The search for reciprocity laws is a fundamental theme in the history of mathematics.
寻找互反律是数学史上的一个基本主题。
The arithmetic of elliptic curves is a microcosm of the deeper structures in number theory.
椭圆曲线的算术是数论中更深层结构的一个缩影。
The Langlands correspondence is not just a theorem but a guiding principle for future discoveries.
朗兰兹对应不仅仅是一个定理，而是未来发现的指导原则。
The study of Shimura varieties bridges the gap between arithmetic geometry and automorphic forms.
志村簇的研究架起了算术几何和自守形式之间的桥梁。
The true test of a mathematical idea is its ability to inspire further questions.
数学思想的真正考验是它能否激发更多的问题。
Mathematics advances not by the efforts of individuals alone but through the collective work of generations.
数学的进步不仅仅依靠个人的努力，而是通过几代人的集体工作。
The connection between Galois groups and automorphic forms is one of the most profound mysteries in mathematics.
伽罗瓦群和自守形式之间的联系是数学中最深刻的奥秘之一。
The conjectures of the Langlands program are not merely problems to be solved but a vision of how mathematics should be.
朗兰兹纲领的猜想不仅仅是待解决的问题，而是关于数学应该如何的愿景。
Representation theory provides the language in which many number-theoretic phenomena can be expressed.