Geometric Langlands Program and Infinite-dimensional Algebraic Geometry

主要研究人员在以下领域进行研究：全球几何Langlands对应，de Rham设置中的局部Langlands对应，与Hecke手征代数相关的共形场论，泰特空间家族和相关的无限维代数簇。他们探索类似物的本地朗兰兹对应在德拉姆塞特有关表示的卡茨-穆迪仿射代数与德拉姆本地系统的朗兰兹对偶群的正式穿孔光盘。他们研究手征Hecke代数的表示理论和相关的全球非理性共形场论，其中相关的D-模块形成Hecke本征层，以了解de Rham设置中的全球几何Langlands对应。他们构造并研究了GL（2）局部系统的Langalnds变换的普适族。他们研究无穷维代数簇的代数几何，类似于从穿孔形式圆盘到光滑代数簇的映射空间。研究的主题是现代数学和数学物理的几个领域的交叉-朗兰兹纲领，几何表示理论，无穷维代数几何和共形场论。互补的思想和方法的融合是非常富有成效的-特别是，它导致建设的几何版本的赫克本征形通过一个适当的量子场论。

The principal investigators conduct research in the following areas: global geometric Langlands correspondence, local Langlands correspondence in the de Rham setting, conformal field theories related to Hecke chiral algebras, families of Tate spaces and related infinite-dimensional algebraic varieties. They explore analogs of the local Langlands correspondence in the de Rhamsetting relating representations of Kac-Moody affine algebras with de Rham local systems for the Langlands dual group on the formal punctured disc. They study the representation theory of chiral Hecke algebras and related global non-rational conformal field theories in which the correlator D-modules form Hecke eigensheaves in order to understand the global geometric Langlands correspondence in the de Rham setting. They construct and study the universal family of Langalnds transforms of GL(2) local systems. They study the algebraic geometry of infinite-dimensional algebraic varieties similar to the space of maps from the punctured formal disk to a smooth algebraic variety.The subject of the research lies on the intersection of several domains of modern mathematics and mathematical physics - the Langlands program, geometric representation theory, infinite-dimensional algebraic geometry, and conformal field theory. The blend of complementary ideas and methods is very fruitful - in particular, it leads to construction of a geometric version of Hecke eigenforms by means of an appropriate quantum field theory.