Non commutative geometry, microlocal analysis, and symplectic geometry

本课题的目的是扩展形变量化代数的非交换微分几何的研究，并将其应用于辛几何。变形量化是流形上依赖于形式参数的函数代数上的一种新的乘法定律。当参数的值为零时，这个乘积就变成了通常的函数乘积。所有这些变形量化都是由Kontsevich分类的。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非交换微分几何的基本不变量（Hochschild和循环同调等）。我们应用这些结果证明了广义Atiyah-Singer指数定理。我们的主要工具是我们所说的非交换微分学，它是经典的形式和多向量代数结构在非交换环境中的扩展。新项目旨在发展变形量化的非交换几何和代数，特别是变形量化环上的模理论，并将它们应用于辛几何，特别是拉格朗日交点的深谷理论和镜像对称。这个项目的主要目的是发展我们所说的非交换微分学。我们的意思是将经典的多变量微积分扩展到变量不再交换的情况，即当乘积的值不再与因子的顺序无关时。这种情况在数学和物理中很自然地出现；在量子力学中，非交换性在数学上表达了海森堡的测不准原理。我们打算将非交换微积分应用于所谓的变形量子化，这是一个非常受量子力学启发的几何设置。我们之前在这个方向上的工作给出了关于偏微分方程解的经典定理的新证明和推广；我们的新项目旨在应用于数学物理的几何问题，如弦理论、镜像对称和拉格朗日交叉点。

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.