Explicit Correspondences in Number Theory

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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.