Explicit Correspondences in Number Theory

数学中最基本的对象是计数数1，2，3，.这就是所谓的自然数。它们是如此基本的东西，以至于似乎令人惊讶的是，关于它们还有什么东西有待发现-然而，这种表面上的简单掩盖了一种令人惊讶的复杂结构，这种结构还远远没有被完全理解。数论本质上是对自然数的研究。数论家问的一个问题是某个方程是否能解--也就是说，是否存在使方程正确的自然数，如果存在，它们是什么。对于某些方程，这可能是直截了当的，而对于其他方程，例如著名的费马大定理中的方程，它可能是极其困难的--一般来说，没有办法知道。在处理纯数学中的难题时，一种取得很大成功的技术是尝试用其他术语重新表述问题--也就是说，将问题转换为数学的另一部分。例如，费马大定理最终由怀尔斯解决，他首先把这个问题转化为椭圆曲线问题，然后（这是怀尔斯工作的精髓）转化为模形式问题，后者的对应关系位于一个巨大的预测对应关系网络的中间，这个网络以加拿大数学家罗伯特·朗兰兹的名字命名。朗兰兹纲领则试图在数论和纯数学的一个领域--表示论之间建立这些对应关系。这个项目就是在这一广阔的领域中展开的。粗略地说，表示论试图用对称性来描述数学对象。例如，群{1，-1}可以被认为是直线的对称：1固定直线，而-1反转直线（交换两端）。随着数学对象变得越来越复杂，对称性也变得越来越复杂--在朗兰兹纲领中出现的对象的情况下，我们得到的对称性不是在1维、2维或3维空间中，而是在无限维空间中！最近的工作给出了一个非常明确的描述的一些代表性牵连的方案和该项目的目的是用它来描述部分朗兰兹对应在一个非常精确的方式。人们希望能够使用这些信息来回答数论中的问题。

The most fundamental objects in Mathematics are the counting numbers 1,2,3,..., which are know as the Natural Numbers. They are such basic things that it may seem surprising that there is anything left to discover about them - and yet this apparent simplicity masks an amazingly complex structure which is far from being fully understood. Number Theory is, in essence, the study of the natural numbers.One type of question which Number Theorists ask is whether some equation can be solved - that is, whether there are natural numbers which make the equation correct and, if so, what they are. For some equations this can be straightforward, while for others, for example the equation in Fermat's famous Last Theorem , it can be exceedingly difficult -- in general, there is no way of knowing.A technique which has found much success when approaching difficult problems in Pure Mathematics is to try to rephrase the question in other terms - that is, to translate the question to another part of Mathematics. For example, Fermat's Last Theorem was finally solved by Wiles, first by translating the question to one about Elliptic Curves and then (which was the essence of Wiles' work) into one about Modular Forms .This latter correspondence sits in the middle of a vast web of predicted correspondences, named collectively after a Canadian Mathematician Robert Langlands. The Langlands Programme then seeks to establish these correspondences between Number Theory and an area of Pure Mathematics called Representation Theory. It is in this broad area that the project lies.Representation Theory, roughly speaking, seeks to describe mathematical objects in terms of symmetries. For example, the group {1,-1} can be thought of as the symmetries of a straight line: 1 fixes the line, while -1 reverses it (swaps the two ends). As the mathematical objects get more complicated, so too do the symmetries -- in the case of the objects which arise in the Langlands Programme, we get symmetries not in 1-, 2-, or 3-dimensional space, but in infinite-dimensional space! Recent work gives a very explicit description of some of the representations implicated in the programme and the aim of the project is to use this to describe parts of the Langlands correspondence in a very precise way. One would expect to be able to use this information to answer questions from Number Theory.