Automorphic Representations

自同构表示是承载具体信息的数学抽象对象。这一信息具有重要意义。它被认为掌握着许多数学领域的大统一的关键，可能还包括物理学的基本领域。然而，信息被很好地隐藏了起来。它被嵌入到众所周知的朗兰兹计划的中心。 朗兰兹计划的核心是一个非常深刻但非常明确的猜想，被称为函数性原则，它代表了自同构表示中数据的基本组织原则。近年来，关于这一猜想的研究取得了重大进展。它是通过内窥镜理论对许多自同构表示进行分类的双重产物。然而，功能互补的情况更深，一直被认为是难以捉摸的。他们落入朗兰兹所称的“超越内窥镜”的领域，在2000年提出了一种非同寻常但相当投机的战略来攻击他们。 我建议研究功能性的一般案例，特别是朗兰兹在《超越内窥镜》一书中的观点。作为一个整体，这个问题包括了数学中一些最难的问题。成功绝非答案。然而，任何可以取得的进展都将对数学的许多广泛领域产生巨大影响。 我还将致力于扩展内窥镜的实际理论。这本身就是一个非常大的问题，但它更容易理解。它包含了许多有趣的问题，对于研究生和年轻的数学家来说，这是一个丰富的问题来源。内窥镜检查理论的进一步发展最终将导致对一类非常广泛的群体进行自同构表示的分类。

Automorphic representations are abstract objects from mathematics that carry concrete information. The information is of great significance. It is believed to hold the keys for a grand unification of many areas of mathematics, and possibly also fundamental areas of physics. However the information is well hidden. It is embedded at the centre of what has become known as the Langlands program. The heart of the Langlands program is a very deep yet quite explicit conjecture, known as the principle of functoriality, which represents a fundamental organizing principle for the data in automorphic representations. There has been significant progress on this conjecture in recent years. It comes as a biproduct of a classification of many automorphic representations through what is known as the theory of endoscopy. However, the complementary cases of functoriality are deeper, and have always been regarded as intractible. They fall into a domain Langlands has called "beyond endoscopy", in suggesting a remarkable but rather speculative strategy for attacking them in 2000. I am proposing to work on general cases of functoriality, and in particular, Langlands' ideas in "beyond endoscopy". Taken as a whole, this problem includes some of the most difficult questions in mathematics. Success is by no means answered. However, any progress that can be made would have an enormous impact on many broad areas of mathematics. I will also work on extending the actual theory of endoscopy. This is a very large problem in its own right, but it is more accessible. It contains many interesting questions, which are a rich source of problems for graduate students and young mathematicians. Further progress in the theory of endoscopy should eventually lead to a classification of automorphic representations for a very wide class of groups.