Torsors under Reductive Groups and Dualities for Hitchin Systems

扭矩(也称为主丛)的研究始于20世纪初，由物理学家作为一种形式主义来描述电磁学。后来，这一理论被扩展到包括强相互作用和弱相互作用，因此扭矩成为所谓的标准模型的基础，标准模型是一种描述除引力之外的所有基本力的物理理论。标准模型预测了各种粒子的存在，最后一种粒子被称为希格斯玻色子，是在2012年的一次大型强子对撞机实验中发现的。1950年，S的菲尔兹奖牌获得者让-皮埃尔·塞雷认识到了扭量在代数几何中的重要性。在他1958年的开创性论文中，他给出了扭力的第一个现代定义，并提出了某种深刻的猜想。这个项目的第一部分旨在证明这一猜想，这是数学中最古老的悬而未决的基础问题之一。该项目的第二部分与所谓的希格斯束有关，这可以被认为是希格斯玻色子的数学化身。更准确地说，PI建议证明参数化希格斯丛的空间具有一定的对偶性。这种对偶性是描述电磁场的麦克斯韦方程相对于交换的电场和磁场是对称的这一事实的广泛推广。对偶性是著名的朗兰兹计划的一部分，该计划统一了数论、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。更详细地说，Grothendieck和Serre的一个猜想预测，如果Zariski拓扑中的一个扭矩在Zariski拓扑中是合理平凡的，那么它在Zariski拓扑中是局部平凡的。Ivan Panin和PI在等特征情形下证明了这一猜想。这一猜想在混合特征情形下仍远未得到解决，尽管在这方面已经有了重要的结果。PI建议解决非分支情况下的猜想，即对于正则局部环，其在整数环上的纤维是正则的。一个更雄心勃勃的目标是证明扭矩的纯度猜想，从某种意义上说，这是继Grothendieck-Serre猜想之后的下一步。第二个项目致力于研究Hitchin系统的朗兰兹对偶，预言朗兰兹对偶群的Higgs丛的模堆是等价的。这一猜想可视为几何朗兰兹对偶的经典极限。通过与通常的全局范畴朗兰兹对偶进行类比，PI提出了猜想的局部版本以及局部猜想和全局猜想之间的基本相容。PI将尝试基于PI构建的Hodge模块的几何Satake等价来证明本地猜想。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 torsors (also known as principal bundles) began in the early 20th century by physicists as a formalism to describe electromagnetism. Later, this was extended to encompass strong and weak interactions, so that torsors became a basis for the so-called Standard Model - a physical theory describing all fundamental forces except for gravitation. The standard model predicted the existence of various particles, the last of which, called the Higgs boson, was found in a Large Hadron Collider experiment in 2012. In 1950's Fields medalist Jean-Pierre Serre recognized the importance of torsors in algebraic geometry. In his 1958 seminal paper he gave the first modern definition of a torsor and formulated a certain deep conjecture. The first part of this project is aimed at proving this conjecture, which is among the oldest unsolved foundational questions in mathematics. The second part of the project is related to the so-called Higgs bundles, which can be thought of as mathematical incarnations of the Higgs bosons. More precisely, the PI proposes to prove a certain duality for the spaces parameterizing Higgs bundles. This duality is a vast generalization of the fact that the Maxwell equations describing electromagnetic fields are symmetric with respect to interchanging electrical and magnetic fields. The duality is a part of 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 will also be part of the proposed activity.In more detail, a conjecture of Grothendieck and Serre predicts that a torsor under a reductive group scheme over a regular scheme is trivial locally in the Zariski topology if it is rationally trivial. This conjecture was settled by Ivan Panin and the PI in the equal characteristic case. The conjecture is still far from resolution in the mixed characteristic case, though there are important results in this direction. The PI proposes to resolve the conjecture in the unramified case; that is, for regular local rings whose fibers over the ring of integers are regular. A more ambitious goal is to prove the purity conjecture for torsors, which is, in a sense, the next step after the Grothendieck–Serre conjecture. The second project is devoted to Langlands duality for Hitchin systems, predicting that moduli stacks of Higgs bundles for Langlands dual groups are derived equivalent. This conjecture may be viewed as the classical limit of the geometric Langlands duality. By analogy with the usual global categorical Langlands duality, the PI formulates a local version of the conjecture and the basic compatibility between the local and the global conjecture. The PI will attempt to give a proof of the local conjecture based on the geometric Satake equivalence for Hodge modules constructed by the PI.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.