Geometric Langlands Program and Beyond

DMS-0401164 DMS-0401164 Vladimir Drinfeld，Alexander Beilinson，Dennis GaitsGordon主要研究以下领域：Kac-Moody代数的表示理论，与曲线上任意奇点的联系，曲面上G丛的模叠叠，二维域上的群表示理论，以及几何Langland对应。他们用自然簇上的D-模研究仿射代数在临界水平上的表示。它们计算与曲线上的向量丛V和V上的连接(可以有不规则奇点)相关联的周期同构的行列式。它们将堆叠的G形束压实在一个表面上。他们研究了准向量空间上二维局部场上的群的表示范畴。研究的主题是现代数学的几个领域的交集--朗兰兹规划、表示论、有限维代数几何和无限维代数几何。这将加深我们对朗兰兹计划的理解，并可能导致其二维推广。

DMS-0401164Vladimir Drinfeld, Alexander Beilinson, Dennis GaitsgoryThe principal investigators conduct research in the following areas: representation theory of Kac-Moody algebras, connections with arbitrary singularities on curves, moduli stacks of G-bundles on surfaces, representation theory of groups over two-dimensional fields, and geometric Langlands correspondence. They study representations of affine algebras at the critical level in terms of D-modules on natural varieties. They compute the determinant of the period isomorphism associated to a vector bundle V on a curve and a connection on V (which can have irregular singularities). They compactify the stack of G-bundles on a surface. They study the category of representations of groups over two-dimensional local fields on pro-vector spaces.The subject of the research lies on the intersection of several domains of modern mathematics - the Langlands program, representation theory, finite-dimensional and infinite-dimensional algebraic geometry. It will deepen our understanding of the Langlands program and possibly lead to its 2-dimensional generalization.