Principal Bundles and Higgs Bundles in Algebraic Geometry

主丛的研究开始于世纪初，由物理学家作为描述电磁学的一种形式。后来，这被扩展到包括强相互作用和弱相互作用，因此主丛成为所谓的标准模型的基础-一种描述四种基本相互作用中的三种的物理理论。在数学中，主丛渗透到许多领域：几何、数论、数学物理等。1950年菲尔兹奖获得者让-皮埃尔·塞尔认识到主丛在代数几何中的重要性。在他1958年的开创性论文中，他给出了主丛的第一个现代定义，并提出了一个深刻的猜想。这个猜想，以及一些遗留的问题，是数学中最古老的未解决的基础问题之一。本项目的第一部分旨在证明其中的一些特性。该项目的其余部分与所谓的希格斯主束有关，希格斯主束可以被认为是希格斯玻色子的数学化身-最近发现的基本粒子。这个项目的这些部分属于著名的朗兰兹纲领，它统一了数论、代数几何、调和分析和数学物理。该奖项将支持这些领域的持续研究。建议学生和在会议上发表演讲将是拟议活动的一部分。更详细地说，第一个项目起源于主丛上的Grothendieck-Serre猜想，该猜想最近由Ivan Panin和PI解决了包含域的环。PI计划将证明扩展到混合特征的情况，并致力于主丛的纯度猜想。从某种意义上说，纯度猜想是格罗滕迪克-塞尔猜想之后的下一个逻辑步骤。在第二个项目中，PI打算构造和证明Hitchin系统的局部Langlands对偶，并从局部情况导出Hitchin系统的全局Langlands对偶的一些情况。第三个项目是专门计算主希格斯包的模栈的motivic卷。该奖项反映了NSF的法定使命，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 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.