Non commutative geometry, microlocal analysis, and symplectic geometry

本项目的目的是扩展变形量子化代数的非交换微分几何的研究，并将其应用于辛几何。形变量子化是流形上函数代数上依赖于形式参数的一个新的乘法律。当参数的值为零时，这个乘积就成为函数的通常乘积。Kontsevich对所有这些形变量子化进行了分类。它们最简单的例子来自流形上的微分算子代数。在我们以前的工作中，我们计算了所有变形代数的基本不变量的非交换微分几何（Hochschild和循环同调等）。我们应用这些结果证明了广义Atiyah-Singer指标定理。我们的主要工具是我们所谓的非交换微分学，这是一个扩展的经典代数结构的形式和多向量的非交换设置。这个新项目的目的是发展非交换几何和变形量子化代数，特别是变形量子化环上的模理论，并将它们应用于辛几何，特别是拉格朗日交点的福谷理论和镜像对称。这个项目的主要目的是发展我们所说的非交换微分学。这意味着经典的多变量微积分的扩展，当变量不再交换时，即当产品的价值不再独立于因素的顺序时。这种情况在数学和物理学中非常自然地出现;在量子力学中，非对易性在数学上表达了海森堡的不确定性原理。我们打算将非对易演算应用于所谓的变形量子化，这是一种非常受量子力学启发的几何设置。我们以前的工作在这个方向上产生了新的证明和推广的经典定理的解决方案偏微分方程;我们的新项目的目的是应用到几何问题的数学物理，如弦理论，镜像对称，拉格朗日相交。

The aim of this project is to extend the study of non-commutative differential geometry of deformation quantization algebras, and to apply it to symplectic geometry. A deformation quantization is a new multiplication law on an algebra of function on a manifold which depends on a formal parameter. When the value of the parameter is zero, then this product becomes the usual product of functions. All such deformation quantizations were classified by Kontsevich. The simplest examples of them arise from algebras of differential operators on manifolds. In our previous work, we computed for all deformed algebras basic invariants of non-commutative differential geometry (Hochschild and cyclic homology, etc.). We applied these results to prove generalized Atiyah-Singer index theorems. Our main tool was what we call non-commutative differential calculus, which is an extension of classical algebraic constructions with forms and multi-vectors to non-commutative setting. The new project is aimed at developing both non-commutative geometry and algebra of deformation quantization, in particular a theory of modules over deformation quantization rings, and at applying them to symplectic geometry, in particular to the Fukaya theory of Lagrangian intersections and to mirror symmetry.The main aim of this project is to develop what we call non-commutative differential calculus. By this we mean an extension of the clasical multi-variable calculus to the case when the variables no longer commute, i.e. when the value of the product is no longer independent of the order of factors. Such situations arise very naturally in mathematics and physics; in quantum mechanics, the non-commutativity expresses mathematically the uncertainty principle of Heisenberg. We intend to apply the non-commutative calculus to so called deformation quantization, a geometric setup very much motivated by quantum mechanics. Our previous work in this direction yielded new proofs and generalizations of classical theorems about solutions of partial differential equations; our new project aims at applications to geometric questions of mathematical physics, such as string theory, mirror symmetry, and Lagrangian intersections.