Non commutative geometry, microlocal analysis, and symplectic geometry

非交换几何、微局域分析和辛几何

• 批准号: 0306624
• 负责人: Boris Tsygan
• 金额: $ 13.95万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306624&HistoricalAwards=false
• 关键词: Non; commutative; geometry; microlocal; analysis

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.

该项目的目的是扩展变形量化代数的非交换微分几何的研究，并将其应用于辛几何。形变量化是流形上函数代数的一种新的乘法定律，它取决于形式参数。当参数的值为零时，该乘积就成为函数的通常乘积。所有此类变形量化均由 Kontsevich 进行分类。其中最简单的例子来自流形上微分算子的代数。在我们之前的工作中，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）。我们应用这些结果来证明广义 Atiyah-Singer 指数定理。我们的主要工具是我们所说的非交换微分微积分，它是具有形式和多向量的经典代数构造到非交换设置的扩展。新项目旨在发展非交换几何和变形量化代数，特别是变形量化环上的模理论，并将它们应用于辛几何，特别是拉格朗日交集的 Fukaya 理论和镜像对称。该项目的主要目的是发展我们所谓的非交换微分微积分。我们的意思是，将经典多变量微积分扩展到变量不再交换的情况，即乘积的值不再独立于因子的顺序。这种情况在数学和物理学中很自然地出现。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化，这是一种很大程度上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产生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，例如弦理论、镜像对称和拉格朗日交集。