Geometric applications of dualities

这是在代数几何领域的一项研究--这是一门研究多项式方程组的解的经典学科。该项目解决了四个问题，为复杂几何、弦理论和量子物理之间提供了新的接口。第一个概述了一种直接从交换或非交换空间的鞘理论中提取Hodge理论不变量的新方法。我们将研究这些线性实体的形式结构，并通过量子镜像对称性的物理概念来产生新的辛流形不变量。在第二个方案中，提出了证明二次几何中K-等价猜想的一种新方法。第三个项目涉及复杂分析背景下几何对偶和对称性的变形量子化问题。第四个项目分析了具有非平凡通量的D-膜在Calabi-Yau三重代数循环中的超对称性约束。膜的大N量子化是在代数完全可积系统的背景下讨论的，对这些问题的理解对于统一代数几何、辛拓扑、理论和数学物理中的各种线性化过程是至关重要的。该项目为理解代数变种的基本结构奠定了基础，这种方式适合在广泛的应用中实用。该项目概述了矩阵量子力学、弦对偶和拓扑黑洞的具体跨学科应用。该项目还旨在组织一项集中努力，以加强高能物理和凝聚态理论中使用的几何库技术。这将通过培训一批年轻的研究人员以及数学和物理方面的研究生和本科生，并通过在研究生和本科生水平上开发霍奇理论、非交换几何和镜面对称课程来实现。

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.