Serre-type conjectures and mod p Langlands correspondences

数论是研究整数和方程在整数中的可解性的学科。它是数学中最古老的分支之一。一个特别著名的问题是费马最后定理，它涉及某些简单方程的整数的不可解性。大约15年前，通过在椭圆曲线(几何对象)和模形式(分析对象和对称理论)之间建立深层联系，才解决了这个问题。这种深刻的联系是朗兰兹计划的一个特例，朗兰兹计划由一些非常普遍和相互关联的猜想组成，特别是有助于解释数论中的许多现象。

Number theory is the study of whole numbers and the solvability of equations in whole numbers. It is one of the oldest branches of mathematics. One particularly famous problem is Fermat's Last Theorem, which concerns the insolvability in the integers of certain simple equations. It was only solved some 15 years ago, by establishing a deep connection between elliptic curves (objects of geometry) and modular forms (objects of analysis and the theory of symmetries). This deep connection is a special instance of the Langlands Program, which consists of a number of very general and interlinked conjectures that, in particular, help to explain many phenomena in number theory.