Automorphic Representations

自守表示

• 批准号: RGPIN-2015-06082
• 负责人: Arthur, James
• 金额: $ 3.35万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=689222
• 关键词: Automorphic; Representations

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.**

自守表示是数学中携带具体信息的抽象对象。这个信息意义重大。人们认为它掌握着许多数学领域以及物理学基本领域大统一的钥匙。但信息隐藏得很好。它被嵌入到后来被称为朗兰兹纲领的中心。朗兰兹纲领的核心是一个非常深刻而又非常明确的猜想，即函性原理，它代表了自守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。它是通过所谓的内窥镜理论对许多自守表示进行分类的双产品。然而，功能性的互补情况更深，一直被认为是棘手的。它们属于朗兰兹所谓的“超越内窥镜”的领域，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，特别是朗兰兹在“超越内窥镜”中的思想。总的来说，这个问题包括了数学中最难的一些问题。成功是没有答案的。然而，任何可能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，但它更容易解决。它包含了许多有趣的问题，这是一个丰富的来源问题的研究生和年轻的数学家。内窥镜检查理论的进一步发展最终会导致一个非常广泛的类别的自守表示的分类。