Local Geometric Langlands Conjecture in Quasi-Classical Setting

PI将在几何朗兰兹计划领域进行两个方向的研究。朗兰兹计划始于1967年由罗伯特·朗兰兹提出的一系列猜想。从它的起源在数论和表示理论，它发展成为一个框架连接和影响数学的许多领域。特别地，几何朗兰兹程序是用代数几何表述的，并且可以用物理术语解释为量子场论中的对偶。该项目为该地区引入了新的几何理念。在这个项目中开发的技术很可能会在数学的其他领域得到应用。该奖项通过研究为研究生提供培训。更详细地说，PI将用于局部几何朗兰兹猜想的经典极限和形式曲线的紧化雅可比矩阵的对偶性。在早期的工作中，PI研究了全局几何猜想的经典极限；虽然在GL(n)组的特殊情况下取得了进展，但很明显，一般情况下将需要新的工具。这个项目的一个目标是使用当地的方法开发这样的工具。与此同时，研究的重点是几何朗兰兹计划中一个本质上尚未探索的方向：虽然猜想的一般形状是已知的（并在提案中说明），但目前细节尚不清楚。该项目旨在通过使用GL(n)的简单案例来澄清情况，在试图证明它们之前测试各种猜测。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The PI will conduct research in two directions in the area of the geometric Langlands program. The Langlands program started as a series of conjectures proposed by Robert Langlands beginning in 1967. From its origins in number theory and representation theory, it developed into a framework connecting and influencing many areas of mathematics. In particular, the geometric Langlands program is formulated in terms of algebraic geometry and can be interpreted in physical terms as a duality in quantum field theory. This project introduces new geometric ideas into the area. It is likely that the techniques developed in this project will find applications in other areas of mathematics. This award provides training of graduate students through research.In more detail, the PI will work on the classical limit of the local geometric Langlands conjecture and on the duality for compactified Jacobians of formal curves. In earlier work, the PI has studied the classical limit of the global geometric conjecture; while progress in the special case of the group GL(n) was made, it became apparent that the general case will require new tools. One goal of this project is to develop such tools using local methods. At the same time, the research focuses on an essentially unexplored direction in the geometric Langlands program: while the general shape of conjectures is known (and stated in the proposal), the details are unclear at this point. The project aims to clarify the situation by using the easier case of GL(n) to test various conjectures before attempting to prove them in general.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.