New Hodge theoretic invariants in geometry and physics

这是代数几何领域的一项研究。 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。第一个问题的目的是构造新的辛流形和复流形的霍奇理论不变量。这些不变量的建立和计算需要发展新的基础形式主义的德利涅上同调，格里菲斯群，和正常的功能在非交换几何。第二个问题是寻求一种新的方法来分析Clalabi-Yau型微分分次范畴中对象的模的精细几何结构。第三个问题将该方法应用于辛拓扑中的一个问题，并引入了福谷范畴的一个新的具体几何描述。第四个问题解决的镜像映射的德尔佩佐表面的建设，并描述了一种策略，证明同调镜像对称猜想在这种情况下。这些问题的理解是必不可少的统一各种线性化程序在代数几何，辛拓扑，理论和数学物理。该项目为理解代数簇的基本结构奠定了基础，适合在广泛的应用中实用。除了代数几何和拓扑学的自然应用外，所提出的工作将与量子引力和宇宙学中的深层问题直接相关。该项目概述了弦对偶性和三维场论量子化的具体跨学科应用。该项目还旨在组织一个集中的努力，以增强和建立一个新的几何武器库的技术适用于理论的代数循环，辛拓扑和高能物理。这将通过培训一批年轻的研究人员和数学和物理学研究生，以及通过镜像对称、非交换几何和福谷范畴代数结构课程的课程开发来实现。 还讨论了研究生和博士后的几何和弦理论的接口的具体研究机会。

This is a research in the field of algebraic geometry. The project addresses four problems providing novel interfaces between complex geometry and string theory and quantum physics. The first problem aims to construct new Hodge theoretic invariants of symplectic and complex manifolds. The building and computation of these invariants requires the development of new foundational formalism of Deligne cohomology, Griffiths groups, and normal functions in non-commutative geometry. The second problem seeks a new method for analysing the fine geometric structure of the moduli of objects in differential graded categories of Clalabi-Yau type. The third problem applies the method to a problem in symplectic topology and introduces a new concrete geometric description of the Fukaya category. The fourth problem addresses the construction of the mirror map for del Pezzo surfaces, and describes a strategy for proving the homological mirror symmetry conjecture in this context. The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry, symplectic topology, theoretical and mathematical physics. The project sets the stage for understanding the basic structure of algebraic varieties in a way suitable for pragmatic use in a broad spectrum of applications. Aside from the natural applications to algebraic geometry and topology, the work proposed will be immediately relevant to deep questions in quantum gravity and cosmology. The project outlines concrete interdisciplinary applications to string dualities and the quantization of three dimensional field theories.The project also aims to organize a concentrated effort on enhancing and building a new geometric arsenal of techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by training a group of young researchers, and graduate students in mathematics and physics, and by a curriculum development of a course on Mirror Symmetry, non-commutative geometry, and algebraic constructions of Fukaya categories. Specific research opportunities on the interface of geometry and string theory for graduate students and postdocs are also discussed.