Mathematical Sciences: Algebraic Topology of Algebraic Varieties, Conformal Field Theory

小行星9625768 该项目涉及朗兰兹对应的几何版本，其中自守形式的作用由复曲线上主丛模空间上的D-模发挥，而伽罗瓦表示的作用由曲线上的局部系统发挥。 看来手征代数的理论提供了一个强有力的方法来构造“自守形式”对应的“伽罗瓦表示”，这可能最终导致完全理解朗兰兹代数的几何情况。 朗兰兹定理似乎是现代数论中最深刻的观点之一;目前似乎没有任何线索来全面理解它们。 然而，在几何情形下的平行陈述（从技术上讲，复数曲线上的有理数函数域代替了数域）似乎更容易管理，因为手征代数的机制（起源于数学物理）。 这些并行的语句将在本项目中处理。 ***

9625768 Beilinson The project deals with a geometric version of Langlands' correspondence in which the role of automorphic forms is played by D-modules on the moduli space of principal bundles on a complex curve, and that of Galois representations, by the local systems on the curve. It appears that the theory of chiral algebras provides a powerful method for constructing the 'automorphic forms' corresponding to the 'Galois representations,' which may eventually lead to complete understanding of the Langlands conjectures in the geometric situation. The Langlands conjectures appear to be among the deepest visions of modern number theory; at the moment there seems to be no clue to understanding them in full generality. However, the parallel statements in the geometric situation (technically, with the number field replaced by the field of rational functions on a complex curve) seem to be much more manageable, because of the machinery of chiral algebras (originated in mathematical physics). These parallel statements are to be addressed in this project. ***