Geometric Langlands Program and Infinite-dimensional Algebraic Geometry

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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.