Geometric applications of dualities

这是代数几何领域的一项研究-一个研究多项式方程组解的经典课题。该项目解决了四个问题，为复杂几何与弦理论和量子物理之间提供了新的接口。 第一个概述了一种新的方法来提取霍奇理论不变量直接从层理论的交换或非交换空间。 这些线性实体的形式结构将被研究，并通过量子镜像对称的物理概念来产生辛流形的新不变量。在第二个项目中，提出了一种新的方法证明双有理几何中的K-等价猜想。第三个项目涉及的问题的变形量化的几何对偶性和对称性在复杂的分析背景。第四个项目分析了具有非平凡通量的D-膜的超对称约束，这些通量以Calabi-Yau三重形式包裹代数圈。 本文在代数完全可积系统的背景下探讨了膜的大N量子化问题，对这些问题的理解对于统一代数几何、辛拓扑、理论物理和数学物理中的各种线性化过程是必不可少的。该项目为理解代数簇的基本结构奠定了基础，以便在广泛的应用中实用。该项目概述了矩阵量子力学、弦对偶性和拓扑黑洞的具体跨学科应用。该项目还旨在组织一个集中的努力，以增强高能物理和凝聚态理论中使用的几何武器库技术。这将通过培训一批年轻的研究人员，研究生和本科生在数学和物理学，并通过课程的霍奇理论，非交换几何和镜像对称，在thegradiate和本科水平的课程开发。

This is a research in the field of algebraic geometry - a classical subject studying the solutions to systems of polynomial equations. The project addresses four problems providing novel interfaces between complex geometry and string theory and quantum physics. The first one outlines a new way to extract Hodge theoretic invariants directly from the sheaf theory of commutative or noncommutative spaces. The formal structure of these linear entities will be studied and through the physical notion of quantum mirror symmetry used to produce new invariants of symplectic manifolds. In the second project a new method is proposed for proving the K-equivalence conjecture in birational geometry. The third project concerns the problem of deformation quantization of geometric dualities and symmetries in the complex analytic context. The fourth project analyzes the supersymmetry constraints for D-branes with non-trivial fluxes wrapping algebraic cycles in Calabi-Yau threefolds. The large N quantization of the branes is probed in the context of algebraically completely integrable systems.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology,theoretical and mathematical physics. The project sets the stage forunderstanding the basic structure of algebraic varieties in a waysuitable for pragmatic use in a broad spectrum of applications. Theproject outlines concrete interdisciplinary applications to matrixquantum mechanics, string dualities and topological black holes.This project also aims to organize a concentrated effort on enhancingthe geometric arsenal of techniques used in high energy physics andcondensed matter theory. This will be achieved by training a group ofyoung researchers, and graduate and undergraduate students inmathematics and physics, and by a curriculum development of courses onHodge theory, non-commutative geometry and mirror symmetry, on thegraduate and undergraduate level.