Automorphic Representations

自同构表示是数学中携带具体信息的抽象对象。这个信息很有意义。它被认为是许多数学领域，甚至可能是物理学基本领域的大统一的关键。然而，这些信息隐藏得很好。它是后来被称为朗兰兹计划的核心。***朗兰兹程序的核心是一个非常深刻但非常明确的猜想，称为功能原理，它代表了自同构表示中数据的基本组织原则。近年来，这一猜想取得了重大进展。通过内窥镜理论，它是许多自同构表征分类的双积。然而，功能的补充情况更深入，并且一直被认为是难以解决的。它们属于朗兰兹称之为“超越内窥镜”的领域，他在2000年提出了一个引人注目但相当投机的攻击策略。***我建议研究功能性的一般情况，特别是Langlands在“超越内窥镜”中的想法。总的说来，这个问题包含了数学中一些最难的问题。成功绝不是答案。然而，任何可以取得的进步都将对数学的许多广泛领域产生巨大的影响。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，这些问题是研究生和年轻数学家的丰富问题来源。内窥镜理论的进一步发展将最终导致对非常广泛的群体类别的自同构表示的分类

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.**