Quantization, Noncommutative Geometry, and Applications

该提案探索了经典几何对象上的新结构，如流形、矢量丛等，最自然地可以从非对易几何的角度来解释，而不是从对易几何的角度来解释。这涉及到发展代数簇上向量丛的形变量子化的新的一般理论。该理论为最近发现的与拉格朗日子流形的交点的奇异性有关的不变量提供了解释。其他应用包括显式构造一类重要的二维代数曲面的形变量子化，称为del Pezzo曲面。量子化的主题有很长的历史，其起源于Dirac，Heisenberg，Pauli等人关于量子力学的经典著作。在数学中，量子化的概念包括用适当的非交换类比代替熟悉的几何对象，这些类似物被认为是相应几何对象的某些变形。由此产生的理论通常被称为非对易几何。本项目涉及一个更具体的方向，即非对易代数几何。这是一个相对较新的领域，有10-15年的历史，处于代数、几何和理论物理的十字路口。非对易代数几何的发展受到弦理论的强烈影响，并对弦理论有重要的应用。弦理论是理论物理的一部分，描述了基本粒子在非常高能量下的基本定律。

The Proposal explores new structures on classical geometric objects, like manifolds, vector bundles, etc., that can be interpreted, most naturally, from the point of view of noncommutative, rather than commutative, geometry. This involves developing a new general theory of deformation quantization of vector bundles on an algebraic variety. The theory provides an explanation for recently discovered invariants associated with singularities of intersections of lagrangian submanifolds. Other applications include explicit constructions of deformation quantizations of an important class of 2-dimentional algebraic surfaces known as del Pezzo sufaces.The subject of quantization has a long history and takes its origins in classic works on quantum mechanics by Dirac, Heisenberg, Pauli, and others. In mathematics, the idea of quantization involves replacing familiar geometric objects by appropriate noncommutative analogues thought of as some deformations of the corresponding geometric objects. The resulting theory is often referred to as noncommutative geometry. The present Project is concerned with a more specific direction known as noncommutative algebraic geometry. This is a relatively recent area, 10-15 years old, at the crossroads between algebra, geometry and theoretical physics. The developments in noncommutative algebraic geometry were strongly influenced by, and have important applications to, string theory, a part of theoretical physics describing fundamental laws of elementary particles at very high energy.