Geometric Langlands conjecture in de Rham settings

计划中的研究引入了一种新的范畴朗兰兹对应的表述。它取代了朗兰兹通信的“天真”版本，专家们知道这种通信是不一致的。首席调查员建议研究新的陈述，并验证它是否通过了自然一致性检查。然后，他计划从一些特殊情况开始验证新的提法。研究人员还建议研究具有平面奇点的曲线的紧化雅可比，并证明它们是自对偶的。在他的早期工作中，主要研究者在约化的不可约曲线的情况下建立了自对偶，他计划将他的结果推广到可约和不可约的曲线上。朗兰兹计划始于1967年，由罗伯特·朗兰兹提出的一系列猜想开始。它起源于数论和表示论，后来发展成为连接和影响许多数学领域的框架。具体地说，几何朗兰兹程序是用代数几何的形式表述的，但也可以用物理术语解释为量子场论中的对偶。这个项目将新的几何思想引入朗兰兹计划，应该可以解决一些长期存在的问题。

The planned research introduces a new formulation of the categorical Langlands correspondence. It replaces the `naive' version of the Langlands correspondence, which is known to specialists to be inconsistent. The principal investigator proposes to study the new statement and verify that it passes natural consistency checks. He then plans to verify the new formulation, starting with some special cases. The investigator also proposes to study the compactified Jacobians of curves with planar singularities and show that they are self-dual. In his earlier work, the principal investigator established the autoduality in the case of reduced irreducible curves; he plans to extend his results to curves that may be reducible and non-reduced.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, but can also be interpreted in physical terms as a duality in quantum field theory. This project introduces new geometric ideas into the Langlands program that should resolve some long-standing issues.