Principal bundles on local schemes and a duality for Hitchin systems

该奖项支持与物理领域有关的两个数学项目的工作。它的数学范围涉及数论、表示论和代数几何，并与现在所知的几何朗兰兹计划有关。罗伯特·朗兰兹在20世纪60年代提出了一个纲领，S提出了一个纲领，它是数论中一系列定理的顶峰，这些定理始于高斯在1800年左右证明的二次互易定理。Alexander Beilinson和Vladimir Drinfeld提出了一个几何版本的朗兰兹程序，在这个程序中，他们将朗兰兹的想法与数学和物理的其他领域的工具和想法结合在一起：代数几何、层理论、模空间、可积系统等等。后来，Anton Kapustin和Edward Witten在一篇突破性的论文中将几何朗兰兹计划与超对称性和电磁二元性联系起来。这两个项目的中心对象是约化代数群的主丛；PI计划在几何朗兰兹计划的背景下研究有关主丛的某些问题。在第一个项目中，PI将部分与Ivan Panin合作，将几何朗兰兹计划的技术(即仿射Grassmannians)应用到主丛的局部研究中。约化群方案的主丛(即扭量)在Zariski拓扑中不一定是平凡的。然而，Grothendieck和Serre的一个猜想预测，如果它是理性平凡的，并且基是光滑的，那么它是平凡的。这一猜想已经在某些情况下得到了证明，但PI的目的是在更广泛的一般性上证明它。PI还计划描述将分式域上的主丛扩展到闭点邻域的充要条件。第二个项目部分是与德米特里·阿林金合作的，目的是证明朗兰兹范畴对应的准经典版本。这是约化群上的Hitchin可积系的派生范畴与朗兰兹对偶群上的Hitchin系的派生范畴的等价性。这应该被看作是阿贝尔变种的傅里叶-穆凯变换到阿贝尔变种的奇异退化的推广。有强有力的迹象表明，证明Hitchin系统的对偶性将有助于证明最初的几何朗兰兹猜想。

This award supports work on two projects in mathematics that have bearing on areas of physics. Its mathematical scope involves number theory, representation theory, and algebraic geometry and relates to what is now known as the Geometric Langlands Program. Robert Langlands in the 1960's proposed a program, which is a culmination of a sequence of theorems in number theory that began with the quadratic reciprocity theorem, proved by Gauss around 1800. Alexander Beilinson and Vladimir Drinfeld suggested a geometric version of the Langlands program, in which they brought the ideas of Langlands together with tools and ideas from other areas of mathematics and physics: algebraic geometry, sheaf theory, moduli spaces, integrable systems, and many others. Later, Anton Kapustin and Edward Witten in a breakthrough paper related the Geometric Langlands Program to supersymmetry and electro-magnetic duality. The central objects in the two projects are principal bundles for reductive algebraic groups; the PI plans to study certain questions about them in the context of the Geometric Langlands Program.In the first project, partly in collaboration with Ivan Panin, the PI will apply techniques of the Geometric Langlands Program (namely, affine Grassmannians) to the local study of principal bundles. A principal bundle (that is, a torsor) for a reductive group scheme is not necessarily trivial in the Zariski topology. However, a conjecture of Grothendieck and Serre predicts that it is trivial if it is rationally trivial and the base is smooth. This conjecture is already proved in certain cases, but the PI aims to prove it in wider generality. The PI also plans to describe the necessary and sufficient conditions for extending a principal bundle over the fraction field to a neighborhood of a closed point. The second project, partly in collaboration with Dmitry Arinkin, is aimed at proving the quasi-classical version of the categorical Langlands correspondence. This is the equivalence between the derived category of the Hitchin integrable system for a reductive group and the derived category of the Hitchin system for the Langlands dual group. This should be viewed as an extension of the Fourier-Mukai transform for abelian varieties to singular degenerations of abelian varieties. There are strong indications that a proof of the duality for Hitchin systems will help with the proof of the original Geometric Langlands conjectures.