Quantization, Noncommutative Geometry, and Applications

该提案探索了经典几何对象的新结构，如流形，矢量束等，这些结构可以最自然地从非交换几何的角度来解释，而不是交换几何。这涉及到发展一个新的广义理论的变形量子化的矢量束在代数变化。该理论为最近发现的与拉格朗日子流形交点奇异性相关的不变量提供了一种解释。其他的应用包括一个重要的二维代数曲面的变形量化的显式构造，称为del Pezzo曲面。量子化的主题有着悠久的历史，它起源于狄拉克、海森堡、泡利等人关于量子力学的经典著作。在数学中，量化的思想涉及到用适当的非交换类似物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一些变形。由此产生的理论通常被称为非交换几何。本课题关注的是一个更具体的方向，即非交换代数几何。这是一个相对较新的领域，10-15岁，在代数，几何和理论物理的十字路口。弦理论是描述高能基本粒子基本定律的理论物理的一部分，非交换代数几何的发展受到弦理论的强烈影响，并在弦理论中有重要的应用。

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.