New Hodge theoretic invariants in geometry and physics

这是对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、弦理论和量子物理之间提供了新的接口。第一个问题是构造辛流形和复流形的新的Hodge理论不变量。这些不变量的建立和计算需要发展Deligne上同调、Griffiths群和非对易几何中的正规函数的新的基本形式。第二个问题寻求分析Clalabi-Yau型微分分次范畴中物体模的精细几何结构的新方法。第三个问题将这种方法应用于辛拓扑中的一个问题，并引入了Fukaya范畴的一个新的具体几何描述。第四个问题讨论了Del Pezzo曲面的镜像映射的构造，并在此背景下描述了证明同调镜像对称猜想的一种策略。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，以适合在广泛的应用中实用的方式。除了对代数几何和拓扑学的自然应用外，所提出的工作将立即与量子引力和宇宙学中的深层问题相关。该项目概述了弦对偶和三维场理论量子化的具体跨学科应用。该项目还旨在组织一项集中努力，以增强和建立适用于代数循环理论、辛拓扑和高能物理的新的几何库。这将通过在数学和物理方面培训一批年轻的研究人员和研究生来实现，并通过开发关于镜像对称、非交换几何和Fukaya范畴的代数构造的课程来实现。还讨论了研究生和博士后在几何和弦理论的界面上的具体研究机会。

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.