Principal Bundles and Higgs Bundles in Algebraic Geometry

代数几何中的主丛和希格斯丛

• 批准号: 2001516
• 负责人: Roman Fedorov
• 金额: $ 18万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard 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=2001516&HistoricalAwards=false
• 关键词: Principal; Bundles; Higgs; Algebraic; Geometry

The study of principal bundles began in the early XXth century by physicists as a formalism to describe electromagnetism. Later, this was extended to encompass strong and weak interactions, so that principal bundles became a basis for the so-called standard model - a physical theory describing three out of four fundamental interactions. In mathematics, principal bundles penetrate many areas: geometry, number theory, mathematical physics, and others. In 1950's Fields Medalist Jean-Pierre Serre recognized the importance of principal bundles in algebraic geometry. In his 1958 seminal paper he gave the first modern definition of a principal bundle and formulated a certain deep conjecture. This conjecture, as well as some remaining questions, are among the oldest unsolved foundational questions in mathematics. The first part of this project is aimed at proving some of these conjectures. The remaining parts of the projects are related to the so-called Higgs principal bundles, which can be thought of as mathematical incarnations of the Higgs boson -- a recently found elementary particle. These parts of the project belong to the famous Langlands program unifying number theory, algebraic geometry, harmonic analysis, and mathematical physics. This award will support continuing research in these areas. Advising students and giving talks at conferences are going to be part of the proposed activity.In more detail, the first project originated from the Grothendieck-Serre conjecture on principal bundles, which was settled recently for rings containing fields by Ivan Panin and the PI. The PI plans to extend the proof to the mixed characteristic case as well as to work on the purity conjecture for principal bundles. The purity conjecture is, in a sense, the next logical step after the Grothendieck-Serre conjecture. In the second project, the PI intends to construct and prove the local Langlands duality for Hitchin systems and to derive some cases of the global Langlands duality for Hitchin systems from the local case. The third project is devoted to counting motivic volumes of moduli stacks of principal Higgs bundles. The goal is to generalize the previous results of the PI and other people from the GL(n) case to the case of arbitrary reductive groups.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.

对主要捆绑包的研究始于物理学家的形式主义，以描述电磁主义的形式主义。后来，将其扩展为包含强大和弱的相互作用，因此主束成为所谓的标准模型的基础 - 一种物理理论，描述了四个基本相互作用中的三个。在数学中，主要捆绑包渗透到许多领域：几何，数理论，数学物理学等。在1950年的田野中，奖牌成员让·皮埃尔（Jean-Pierre）认识到代数几何形状中主束的重要性。在1958年的开创性论文中，他给出了一个现代的主要捆绑包，并提出了一定的猜想。这个猜想以及剩下的一些问题是数学中最古老的未解决的基础问题之一。该项目的第一部分旨在证明其中一些猜想。项目的其余部分与所谓的希格斯主束有关，可以将其视为希格斯玻色子的数学化身 - 最近发现的基本粒子。该项目的这些部分属于著名的Langlands计划统一数字理论，代数几何，谐波分析和数学物理学。该奖项将支持这些领域的持续研究。为学生提供咨询并在会议上进行演讲将成为拟议活动的一部分。更详细地说，第一个项目源自Grothendieck-serre的主要捆绑包，该项目最近针对Ivan Panin和Pi的戒指定居。 PI计划将证据扩展到混合特征案例，并为主束的纯度猜想工作。从某种意义上说，纯度猜想是Grothendieck-Serre猜想之后的下一个逻辑步骤。在第二个项目中，PI打算为Hitchin系统构建和证明本地Langlands二元性，并从当地案例中得出Hitchin Systems的全球Langland二元性案例。第三个项目致力于计算主要希格斯捆绑包的模量堆栈的动机量。目的是将PI和其他人的先前结果概括为从GL（N）案件到任意还原组的案例。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估审查标准来通过评估来获得支持的。

项目成果

On the Grothendieck-Serre conjecture about principal bundles and its generalizations

关于主丛的格洛滕迪克-塞尔猜想及其推广

• DOI: 10.2140/ant.2022.16.447
• 发表时间: 2022
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Fedorov, Roman

On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic

混合特征主丛上的格洛腾迪克塞尔猜想

• DOI: 10.1090/tran/8490
• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Fedorov, Roman