Representations of algebraic groups over 2-dimensional fields, G-bundles on surfaces and mathematical physics

二维场上的代数群、曲面上的 G 丛和数学物理的表示

• 批准号: 0600851
• 负责人: Alexander Braverman
• 金额: $ 14.38万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600851&HistoricalAwards=false
• 关键词: Representations; algebraic; groups; over; dimensional

The PI plans to study several problems related to algebraic groups over 2-dimensional local and global fields and their representations. More specifically the PI would like to introduce and study the notion of Hecke eigen-forms for 2-dimensional semi-global fields. In the second part of his project the PI suggests to attack several (mathematically well-posed) problems related to 4-dimensional gauge theory. Both subjects can be reformulated in terms of various questions about G-bundles on algebraic surfaces. The PI believes that above questions may also be connected to the (not yet formulated) 2-dimensional geometric Langlands duality.The research lies on the border of such fields as number theory, representation theory and mathematical physics; sucesful implementation of the project might shed some new light on the connection between these fields. For example, number theory is perhaps one of the oldest mathematical subjects and one of its most important parts is called the Langlands program. Recently it has been realized that geometric aspects of the Langlands program have many connections with modern mathamtical physics (such as 4-dimensional quantum field theory). The research project should confirm the existence of such links as well broaden and generalize them.

PI计划研究与2维局部和全局域上的代数群及其表示相关的几个问题。更具体地说，PI想介绍和研究二维半全局域的Hecke本征形式的概念。在他的项目的第二部分，PI建议攻击几个与四维规范理论有关的（数学适定）问题。这两个主题可以重新制定的各种问题的G-丛代数曲面。PI认为，上述问题也可能与二维几何朗兰兹对偶（尚未公式化）有关。该研究处于数论、表示论和数学物理等领域的边缘;该项目的成功实施可能会为这些领域之间的联系提供一些新的线索。例如，数论可能是最古老的数学学科之一，它最重要的部分之一被称为朗兰兹纲领。最近人们意识到朗兰兹纲领的几何方面与现代数学物理学（如四维量子场论）有许多联系。该研究项目应确认这种联系的存在，并扩大和概括它们。