Geometric Langlands Correspondence: Further Directions

几何朗兰兹对应：进一步的方向

• 批准号: 2005475
• 负责人: Mark Kisin
• 金额: $ 21万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005475&HistoricalAwards=false
• 关键词: Geometric; Langlands; Correspondence; Further; Directions

The Langlands Program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and representation theory, in a web of deep and as yet only partially understood connections between number theory, representation theory, geometry, and mathematical physics. The classical Langlands program has been studied for more than fifty years, and has found significant applications to solving classical Diophantine equations, such as the solution of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development as it is also connected with other subjects such as geometry and physics, from where one can draw intuition. In the geometric Langlands program, the number fields in the classical Langlands program are replaced by curves and their function fields, and the web of connections in number theory is replaced by concisely formulated equivalences, or correspondences. This project will extend and develop geometric Langlands correspondences in different settings and provide research training opportunities for graduate students.In more detail, one project will establish the Fundamental Local Equivalence, which is an equivalence between the Whittaker category of the affine Grassmannian for a reductive group and the Kazhdan-Lusztig category for its Langlands dual, as factorization categories. This equivalence could be considered as the starting point for the local geometric Langlands program. The project will establish the required equivalence by equating both sides to a combinatorial object that is directly expressible in terms of the root data and the quantum parameter (the so-called factorization algebra Omega). Another project will directly relate the geometric and classical Langlands theories (in the case of function fields) by realizing the space of automorphic functions as the categorical trace of the Frobenius on the category of automorphic sheaves with nilpotent singular support. In the process of doing so, one naturally re-derives V. Lafforgue's decomposition of the space of automorphic functions over the course moduli space of Langlands parameters via shtukas.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.

朗兰兹纲领是一个数学框架，它将许多不同数学领域的问题统一起来，特别是数论和表示论，将数论、表示论、几何和数学物理之间的深刻且迄今为止仅部分理解的联系联系起来。经典的朗兰兹纲领已经被研究了五十多年，并且在求解经典丢番图方程方面找到了重要的应用，例如费马大定理的求解。几何朗兰兹程序相对较新，正在快速发展，因为它还与其他学科（例如几何和物理）相关，从中可以得出直觉。在几何朗兰兹纲领中，经典朗兰兹纲领中的数域被曲线及其函数域所取代，数论中的连接网络被简洁表述的等价或对应所取代。该项目将在不同的环境中扩展和发展几何朗兰兹对应，并为研究生提供研究培训机会。更详细地说，一个项目将建立基本局部等价，这是仿射格拉斯曼的惠特克类别与还原群之间的等价以及朗兰兹对偶的 Kazhdan-Lusztig 范畴，作为因式分解范畴。这种等价性可以被视为局部几何朗兰兹纲领的起点。该项目将通过将两边等同于一个可以直接用根数据和量子参数（所谓的因式分解代数 Omega）来表达的组合对象来建立所需的等价性。另一个项目将通过将自守函数空间实现为具有幂零奇异支持的自守滑轮范畴上的 Frobenius 范畴迹，将几何和经典朗兰兹理论（在函数域的情况下）直接联系起来。在这个过程中，人们很自然地通过shtukas重新推导了V. Lafforgue在朗兰兹参数的过程模空间上对自守函数空间的分解。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。